Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 631795 7 pageshttpdxdoiorg1011552013631795

Research ArticleFractional Black-Scholes Model and TechnicalAnalysis of Stock Price

Song Xu1 and Yujiao Yang2

1 Department of Mathematics and Computer Science Huainan Normal University Huainan 232038 China2 School of Finance and Statistics East China Normal University Shanghai 200241 China

Correspondence should be addressed to Song Xu songxuxsgmailcom

Received 20 July 2013 Accepted 20 November 2013

Academic Editor Roberto Barrio

Copyright copy 2013 S Xu and Y Yang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the stock market some popular technical analysis indicators (eg Bollinger bands RSI ROC etc) are widely used to forecastthe direction of prices The validity is shown by observed relative frequency of certain statistics using the daily (hourly weeklyetc) stock prices as samples However those samples are not independent In earlier research the stationary property and the lawof large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have beendiscussed Since the fitness of both Black-Scholes model and short-range dependent process has been questioned we extend theabove results to fractional Black-Scholes model with Hurst parameter 119867 gt 12 under which the stock returns follow a kind oflong-range dependent process We also obtain the rate of convergence

1 Introduction

Liu et al discussed in [1] the Bollinger bands for the Black-Scholes model They introduced the corresponding statistics119880

(119899)119905 calculated according to the formulation of the Bollinger

bands which is a stationary process and then they gavethe law of large numbers since 119880(119899)

119905+119896119899119896=12 are mutually

independent for each fixed 119905 ge 0 Thus the Bollinger bandsproperty which seems unthinkable at first glance was provedThe related results have been extended to stochastic volatilitymodel in [2] and AR-ARCHmodel in [3]

It has been noted in [4] that ldquotechnical analysis is asecurity analysis discipline for forecasting the direction ofprices through the study of past market data primarilyprice and volumerdquo Technical analysis has been widely usedamong traders and financial professionals in stock marketsand foreign exchange markets in recent decades Howevertechnical analysis has not received the same level of academicscrutiny and acceptance as more traditional approaches suchas fundamental analysis since ldquoa simulated sample is only onerealization of geometric Brownian motionrdquo and ldquoit is difficultto draw general conclusions about the relative frequenciesrdquo

(see [5]) However given the stock price models we showhere that we can do statistics based on relative frequency ofoccurrence for some technical analysis indicators

The fitness of both Black-Scholes model and short-rangedependent process has been questioned Since Willinger etal [6] gave the empirical evidence of long-range dependencein stock price returns there have been several empiricalstudies that lent further support to such property of long-range dependence (see eg [7ndash10]) We consider the processof alternatives to short-range dependence a model driven bythe fractional Brownian motion (fBm) which is long-rangedependent In the following discussion we assume that thestock price satisfies the fractional Black-Scholes model (seeeg [11])

119878119905 = 1198780 exp (120583 + ]) 119905 + 120590119861119867

119905 119905 isin [0 119879] (1)

where120583 ] isin R are constants120590 is a positive real number119861119867119905 isa fBmwith Hurst parameter119867 and119867 isin (0 1)The fractionalBrownian motion is a continuous-time Gaussian process 119861119867119905on [0 119879] which starts at zero with expectation zero for all

2 Journal of Applied Mathematics

119905 isin [0 119879] and has the following covariance function (seeeg [12 13])

119864 [119861119867(119905) 119861

119867(119904)] =

1

2(|119905|

2119867+ |119904|

2119867minus |119905 minus 119904|

2119867) 119904 119905 ge 0

(2)

where 119867 is a real number in (01) called the Hurst indexor Hurst parameter associated with the fractional Brownianmotion In contrast to Brownian motion the increments ofthe fBm are not independent if 119867 = 12 The fBm is self-similar that is 119861119867(119886119905) (=) |119886|119867119861119867(119905) and the incrementsare stationary that is 119861119867(119905) minus 119861119867(119904) (=) 119861119867(119905 minus 119904) and theincrements exhibit long-range dependence if 119867 gt 12 thatis suminfin

119899=1 119864[119861119867(1)(119861

119867(119899 + 1) minus 119861

119867(119899))] = infin119867 gt 12 where

119883(=)119884 denotes that119883 and119884 have the same distribution Notethat the fBm is in fact a regular Brownian motion if119867 = 12

In this paper we discuss the statistical properties ofsome popular technical indicators such as Bollinger bandsRelative Strength Index (RSI) and Rate of Change (ROC)Under fractional Black-Scholes model (1) we show that thecorresponding statistics are stationary and the law of largenumbers holds for frequencies of stock prices falling out ofnormal scope of the technical indicators

This paper is organized as follows Section 2 introducessome technical indicators Section 3 gives the ratios ofBollinger bands RSI and ROC falling in the correspondingsets In Section 4 by constructing a statistic 119880(119899)

119905 we investi-gate the stationary properties of corresponding statistics InSection 5 we obtain the law of large numbers for frequenciesof the statistics And we give some comments of the results inSection 6

2 Definitions of Technical Indicators

Let 119878119905 be current stock price and 119878119905minus119894120575 the stock price 119894periods ago where 120575 is the length of the period between twoobservation spots (the period can be day minute etc) Werecall the definitions of technical indicators in the following

(1) Bollinger BandsDeveloped by John Bollinger in the 1980sBollinger Bands are volatility bands placed above and belowa moving average denoted by

119878(119899)

119905 =1

119899

119899minus1

sum

119894=0

119878119905minus119894120575 119861119899med119905 =

1

sum119899

119894=1 119894

119899minus1

sum

119894=0

(119899 minus 119894) 119878119905minus119894120575

119904(119899)

119905 = radic1

119899 minus 1

119899minus1

sum

119894=0

(119878119905minus119894120575 minus 119878(119899)

119905 )

2

119905 ge (119899 minus 1) 120575

(3)

The curve 119861119899med119905 is called the middle Bollinger band the

curve119861119899upp119905 = 119861119899med119905 +2119904

(119899)119905 is called the upper Bollinger band

and 119861119899low119905 = 119861119899med119905 minus 2119904

(119899)119905 is called the lower Bollinger band

where 119899 is the number of selected periods The Bollingerbands of SampP500 are shown in Figure 1 Usually we take119899 = 12 or 20 120575 = one day According to Bollinger [14]and Liu et al [1] the bands contain more than 88-89of price action which makes a move outside the bands

0 50 100 150 200 2501050

1100

1150

1200

1250

1300

1350

1400

1450

Bollinger

Figure 1 SampP500 annual Bollinger bands until March 27 2012

significant Technically prices are relatively high when abovethe upper band and relatively low when below the lowerband However relatively high should not be regarded asbearish or as a sell signal Likewise relatively low shouldnot be considered bullish or as a buy signal As with otherindicators Bollinger bands are seldom used alone Tradersshould combine Bollinger bandswith basic trend analysis andother indicators for confirmation

(2) Relative Strength Index (RSI) The RSI was developed byWilder [15] and it is classified as a momentum oscillatormeasuring the velocity and magnitude of directional pricemovements If we denote

Δ119878+

119905 = (119878119905+120575 minus 119878119905) or 0 Δ119878minus

119905 = (119878119905 minus 119878119905+120575) or 0 (4)

the 119899-period RSI is defined as

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

119905 ge 119899120575 (5)

The RSI of SampP500 is shown in Figure 2 Usually we take119899 = 14 120575 = one day RSI oscillates between zero and 100with high and low levels marked at 70 and 30 More extremehigh and low levels (80 and 20 or 90 and 10) occur lessfrequently but indicate stronger momentum TraditionallyRSI readings greater than the 70 level are considered to bein overbought territory and RSI readings lower than the 30level are considered to be in oversold territory If the RSI isbelow 50 it generally means that the market is in a weektrend When the RSI is above 50 it generally means that themarket is in a strong trend Zhu [16] discussed the statisticalproperty and the forecasting ability of RSI

(3) Rate of Change (ROC) The ROC is a pure momentumoscillator that measures the percent of change in price fromone period to the next The 119899-period ROC is defined as

ROC(119899)

119905 = 100 times119878119905 minus 119878119905minus119899120575

119878119905minus119899120575

119905 ge 119899120575 (6)

Journal of Applied Mathematics 3

Table 1 Ratio of SPY in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9571 9791 7344 9004 98762009 9828 9160 6875 9000 1002010 9760 9159 8241 100 1002011 9569 9706 8083 9750 100

Table 2 Ratio of QQQ in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9700 9582 5892 8797 98762009 9612 9160 6625 9125 1002010 9663 8178 7407 9954 1002011 9440 9664 8000 9667 100

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250020406080100

RSI

(b)

Figure 2 SampP500 annual RSI until March 27 2012

TheROCof SampP500 is shown in Figure 3 Usually we take 119899 =12 120575 = one day Prices are constantly increasing as long asthe ROC remains positive Conversely prices are fallingwhenthe ROC is negative The ROC has its antennas and groundswhich are indefinite and can give identifiable extremes thatsignal overbought and oversold conditions In general it istime to sell out when the ROC rises to the first ultra-buy line(5) and then the rising trend mostly ends when it reachesthe second ultra-buy line (10) It is time to buy in when ROCdrops to the first ultra-sell line (minus5) and then the droppingtrend mostly ends when it reaches the second ultra-sell line(minus10) Li [17] discussed the empirical evidence of ROC Likeall technical indicators the ROC oscillator should be used inconjunction with other aspects of technical analysis

3 Some Facts from the Stock Market

Liu et al [1] traced 15 years of the DOW SampP500 andNASDAQdaily closing prices and drew the conclusion that inevery year more than 94 of daily closing prices are between

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250

0102030

ROC

minus20

minus10

(b)

Figure 3 SampP500 annual ROC until March 27 2012

the Bollinger bands We give the ratios of Bollinger bandsRSI and ROC falling in the corresponding sets from January2008 to December 2011 in Tables 1 2 and 3 where B-Bdenote the Bollinger bands We can see that more than 95of daily closing prices are between the Bollinger bands morethan 81 of RSI are in the interval [2080] and more than87 of ROC are in the interval [minus10 10] So we show thatthe stationary of the indexes is still maintained even sincethe world economic crisis in 2008 In the following we givea mathematical proof to this fact under the fractional Black-Scholes model

4 Stationary Property

Let 119878119905 denote the observed stock price under the model (1)And let

ℎ (119905 119894 119899) = exp 119861119867119905minus119894120575 minus 119861119867

119905minus119899120575 119894 = 0 1 2 119899 minus 1

119880(119899)

119905 = 119891 (ℎ (119905 0 119899) ℎ (119905 119899 minus 1 119899)) 119905 ge 119899120575

(7)

4 Journal of Applied Mathematics

Table 3 Ratio of DIA in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9657 9623 7635 9212 98762009 9742 9163 6971 9087 1002010 9856 9256 8710 100 1002011 9483 9496 8417 9833 100

where 119891 is a measurable function R119899rarr R Then we have

the following resultsThe process 119880(119899)

119905 119905ge119899120575 is stationary

Remark 1 Let 119878119905 be the stock price generated by themodel (1)119871(119899)119905 = (119878119905 minus119861

119899med119905 )119904

(119899)119905 (119905 ge 119899120575) Then the process 119871(119899)119905 119905ge119899120575 is

stationary

Remark 2 Let 119878119905 be the stock price generated by the model(1) Then the process

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

(119905 ge 119899120575) (8)

is stationary

Remark 3 Let 119878119905 be the stock price generated by the model(1)Then the process ROC(119899)

119905 = 100times(119878119905minus119878119905minus119899120575)119878119905minus119899120575(119905 ge 119899120575)

is stationary

5 Law of Large Numbers

Let 119870(119899)

Γ119894= 119868

119880(119899)

119894120575isinΓ

119894 ge 119899 where Γ is a subset of R And let

119881(119899)

119873Γ =1

119873 + 1

119873

sum

119894=0

119870(119899)

Γ119899+119894 (9)

which is the observed frequency of the events [119880(119899)

(119899+119894)120575isin

Γ] (119894 = 0 1 119873)It is natural to assume 120575 lt 1 that is the length between

two observation spots is less than one year From the abovediscussion we can let119901 = 119875(119880(119899)

119905 isin Γ) 119905 ge 119899120575We denote119880(119899)

119894120575

by 119880(119899)

119894 119894 ge 119899 in the following discussion for convenience

Denote by R119898times119899 the set of119898 times 119899 real matrices and set

119885119896119895 = 119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ) (10)

for each fixed 119895 and 119896 we give the following lemma

Lemma 4 For all Γ sub R there exist 120572 isin (0 1) and a constant119862 gt 0 when119873 is large enough and |1198961 minus1198962| minus 119899 gt 119873120572 one has

119864 (1198851198961 1198951198851198962 119895

) le 119862 |2119867 minus 1|119873120572(119867minus1)2

(11)

Proof Let 119882119896119894 = 119861119867[(119896+1)119899+119895minus119894]120575 minus 119861

119867[(119896+1)119899+119895minus(119894+1)]120575 119896 isin Z+

119894 = 0 1 2 119899 minus 1 And set 119883 = (1198821198961 0 1198821198961 119899minus1

)119879≜

(1198831 119883119899)119879 119884 = (1198821198962 0

1198821198962 119899minus1)119879≜ (1198841 119884119899)

119879

119885 = (119883119879 119884

119879)119879 where 119883119879 is the transpose of 119883 then 119864119883 =

119864119884 = 0 119864119885 = 0 We denote by 119860 = (119886119894119895) the covariancematrix of 119883 denote by 119863 = (119889119894119895) the covariance matrix of119884 and denote by 119861 = (119887119894119895) the covariance matrix of 119883 and119884 Let Σ be the covariance matrix of 119885 then Σ = ( 119860 119861

119861119879119863)

|119860| gt 0 |119863| gt 0 Since the fBm has stationary incrementswe can get 119860 = 119863 and

119886119894119895 = 119889119894119895 =1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 minus 1) 1205751003816100381610038161003816

2119867

+1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 + 1) 1205751003816100381610038161003816

2119867minus1003816100381610038161003816(119894 minus 119895)120575

1003816100381610038161003816

2119867 119867 isin (0 1)

(12)

Similar to the conclusion given by Deng and Barkai [18] wecan get from simple calculation

119887119894119895 = 119864119883119894119884119895 sim 1205752119867(2119867 minus 1)

10038161003816100381610038161198961 minus 1198962 + 119895 minus 1198941003816100381610038161003816

2119867minus2

1198962 minus 1198961 997888rarr +infin

(13)

where 119901119896 sim 119902119896 means lim119896rarrinfin(119901119896119902119896) = 1 So we have 119861 =(119887119894119895) rarr 0

When 119867 = 12 we can easily derive that the conclusionof Lemma 4 holdsWe assume119867 = 12 in the following proofLet 119901(119911) be the probability density function of119885 and 119865(119911) thedistribution function of 119885 and let the marginal distributionsfor 119885 be 119865119894(119911119894) 119894 = 1 2 2119899 where 119911 isin R2119899times1 119911 =

(1199111 1199112 1199112119899)119879 Take the notation Θ = [minus119872119872]119872 gt 0

and Γ1015840 = Θ times Θ times sdot sdot sdot times Θ ≜ Θ119899 Furthermore we put

119892 (119885) = 1198851198961 1198951198851198962 119895

(14)

First we will consider the integral of 119892(119885) on Γ1015840 times Γ1015840Referring to Bernstein [19] we have Σminus1 = Σ1 + Σ2 whereΣ = 119863 minus 119861

119879119860

minus1119861

Σ1 = (119860

minus1 00 119863

minus1)

Σ2 = (119860

minus1119861Σ

minus1119861119879119860

minus1minus119860

minus1119861Σ

minus1

minusΣminus1119861119879119860

minus1119863

minus1119861119879(119860 minus 119861119863

minus1119861119879)minus1119861119863

minus1)

(15)

Journal of Applied Mathematics 5

We take the notation 119889 = 1198891199111 1198891199112 1198891199112119899Then we obtain

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

=

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) (2120587)minus119899(|119860| |119863|)

minus12

times exp minus12119911119879Σ1119911

times (

10038161003816100381610038161003816119863 minus 119861

119879119860

minus111986110038161003816100381610038161003816

minus12

|119863|minus12

exp minus12119911119879Σ2119911 minus 1)119889

(16)

where lim|1198961minus1198962|rarrinfin(|119863 minus 119861119879119860

minus1119861||119863|) = (1|119863|)(|119863 minus

lim|1198961minus1198962|rarrinfin(119861119879119860

minus1119861)|) = 1 Assume lim1198962minus1198961rarrinfin(119872|1198961 minus

1198962|1minus119867) = 0 then 119911119879Σ2119911 rarr 0 so we get forall120598 gt 0 exist1198731

forall|1198961 minus 1198962| gt 1198731

1003816100381610038161003816100381610038161003816exp minus1

2119911119879Σ2119911 minus 1

1003816100381610038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161 minus

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911) minus 1

1003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816

(17)

Choose 120572 isin (0 1) satisfying119873120572gt 1198731 where119873 lt (119873 + 1)119899

then if |1198961 minus 1198962| minus 119899 gt 119873120572 and119873 is large enough there exist

1198620 gt 0 and 1198620 has relation with 119899 such that

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816le 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1)

(18)

Therefore by (16) (17) and (18) we can derive that there exist1198620 gt 0 satisfying

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

le 1199012(120598 + 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1))

le 1198620 |2119867 minus 1|119873120572(119867minus1)

(19)

Then we will consider the integral of 119892(119885) on thecomplementary set of Γ1015840 times Γ1015840 in the following Let Ξ be arandom variable satisfying 119875(Ξ = (minusinfin minus119872)) = 12 119875(Ξ =(119872infin)) = 12 Let Γ119894 be the set that contains all elements ofthe following form

Ξ times sdot sdot sdot times Θ times sdot sdot sdot times Ξ times sdot sdot sdot times Θ ≜ Ξ(119894) (20)

where Θ occurs 119894 times and Ξ occurs 2119899 minus 119894 times in Ξ(119894) 119894 =0 1 2119899minus1 So we can see that Γ119894 is composed of ( 2119899119894 )sdot2

2119899minus119894

mutually disjoint elementsTherefore the complementary set

of Γ1015840 times Γ1015840 should be (Γ1015840 times Γ1015840)119888 = ⋃2119899minus1

119894=0 Γ119894 and Γ119894⋂Γ119895 = Φ119894 = 119895 that is the complementary set of Γ1015840 times Γ1015840 is the union of(sum

2119899minus1

119894=0 (2119899119894 ) sdot 2

2119899minus119894= 3

2119899minus 1)mutually disjoint sets

Since10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119892 (119911) 119901 (119911) 119889

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119901 (119911) 119889 le 119865 (1199111 1199112 1199112119899)

le min1le119894le2119899

119865119894 (119911119894) le 1198651 (minus119872) le120575119867119890minus119872221205752119867

119872

(21)

where119872 occurs 119894 times and minus119872 occurs 2119899 minus 119894 times within1199111 1199112 1199112119899 the second inequality holds because |119892(119911)| le1 and the last inequality holds because int+infin

119872119890minus11990922119889119909 =

intminus119872

minusinfin119890minus11990922119889119909 le 119890

minus11987222119872

Take119872 = 119873120572(1minus119867)2 We conclude from (19) and (21) that

119864 (1198851198961 1198951198851198962 119895

) =

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

+

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

(Γ1015840timesΓ1015840)

119888

119892 (119911) 119901 (119911) 119889

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+

2119899minus1

sum

119894=0

(2119899

119894) sdot 2

2119899minus119894 120575119867119890minus(119872)

221205752119867

119872

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

(22)

Take 119862 = 1198620 + (32119899minus 1)120575

119867|2119867 minus 1| then we have

1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

le 119862 |2119867 minus 1|119873120572(119867minus1)2

(23)

from which the proof immediately follows

Then we obtain the law of large numbers

Theorem 5 There exist 120573 isin (0 1) and a constant 119862 gt 0 suchthat

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2le119862

119873120573 (24)

Proof To simplify notation we put Λ = 119896 0 le 119896119899 + 119895 le 119873and set for each fixed 119895 and 119896

119884119895 = sum

119896isinΛ

[119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ)] (25)

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

119905 isin [0 119879] and has the following covariance function (seeeg [12 13])

119864 [119861119867(119905) 119861

119867(119904)] =

1

2(|119905|

2119867+ |119904|

2119867minus |119905 minus 119904|

2119867) 119904 119905 ge 0

(2)

where 119867 is a real number in (01) called the Hurst indexor Hurst parameter associated with the fractional Brownianmotion In contrast to Brownian motion the increments ofthe fBm are not independent if 119867 = 12 The fBm is self-similar that is 119861119867(119886119905) (=) |119886|119867119861119867(119905) and the incrementsare stationary that is 119861119867(119905) minus 119861119867(119904) (=) 119861119867(119905 minus 119904) and theincrements exhibit long-range dependence if 119867 gt 12 thatis suminfin

119899=1 119864[119861119867(1)(119861

119867(119899 + 1) minus 119861

119867(119899))] = infin119867 gt 12 where

119883(=)119884 denotes that119883 and119884 have the same distribution Notethat the fBm is in fact a regular Brownian motion if119867 = 12

In this paper we discuss the statistical properties ofsome popular technical indicators such as Bollinger bandsRelative Strength Index (RSI) and Rate of Change (ROC)Under fractional Black-Scholes model (1) we show that thecorresponding statistics are stationary and the law of largenumbers holds for frequencies of stock prices falling out ofnormal scope of the technical indicators

This paper is organized as follows Section 2 introducessome technical indicators Section 3 gives the ratios ofBollinger bands RSI and ROC falling in the correspondingsets In Section 4 by constructing a statistic 119880(119899)

119905 we investi-gate the stationary properties of corresponding statistics InSection 5 we obtain the law of large numbers for frequenciesof the statistics And we give some comments of the results inSection 6

2 Definitions of Technical Indicators

Let 119878119905 be current stock price and 119878119905minus119894120575 the stock price 119894periods ago where 120575 is the length of the period between twoobservation spots (the period can be day minute etc) Werecall the definitions of technical indicators in the following

(1) Bollinger BandsDeveloped by John Bollinger in the 1980sBollinger Bands are volatility bands placed above and belowa moving average denoted by

119878(119899)

119905 =1

119899

119899minus1

sum

119894=0

119878119905minus119894120575 119861119899med119905 =

1

sum119899

119894=1 119894

119899minus1

sum

119894=0

(119899 minus 119894) 119878119905minus119894120575

119904(119899)

119905 = radic1

119899 minus 1

119899minus1

sum

119894=0

(119878119905minus119894120575 minus 119878(119899)

119905 )

2

119905 ge (119899 minus 1) 120575

(3)

The curve 119861119899med119905 is called the middle Bollinger band the

curve119861119899upp119905 = 119861119899med119905 +2119904

(119899)119905 is called the upper Bollinger band

and 119861119899low119905 = 119861119899med119905 minus 2119904

(119899)119905 is called the lower Bollinger band

where 119899 is the number of selected periods The Bollingerbands of SampP500 are shown in Figure 1 Usually we take119899 = 12 or 20 120575 = one day According to Bollinger [14]and Liu et al [1] the bands contain more than 88-89of price action which makes a move outside the bands

0 50 100 150 200 2501050

1100

1150

1200

1250

1300

1350

1400

1450

Bollinger

Figure 1 SampP500 annual Bollinger bands until March 27 2012

significant Technically prices are relatively high when abovethe upper band and relatively low when below the lowerband However relatively high should not be regarded asbearish or as a sell signal Likewise relatively low shouldnot be considered bullish or as a buy signal As with otherindicators Bollinger bands are seldom used alone Tradersshould combine Bollinger bandswith basic trend analysis andother indicators for confirmation

(2) Relative Strength Index (RSI) The RSI was developed byWilder [15] and it is classified as a momentum oscillatormeasuring the velocity and magnitude of directional pricemovements If we denote

Δ119878+

119905 = (119878119905+120575 minus 119878119905) or 0 Δ119878minus

119905 = (119878119905 minus 119878119905+120575) or 0 (4)

the 119899-period RSI is defined as

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

119905 ge 119899120575 (5)

The RSI of SampP500 is shown in Figure 2 Usually we take119899 = 14 120575 = one day RSI oscillates between zero and 100with high and low levels marked at 70 and 30 More extremehigh and low levels (80 and 20 or 90 and 10) occur lessfrequently but indicate stronger momentum TraditionallyRSI readings greater than the 70 level are considered to bein overbought territory and RSI readings lower than the 30level are considered to be in oversold territory If the RSI isbelow 50 it generally means that the market is in a weektrend When the RSI is above 50 it generally means that themarket is in a strong trend Zhu [16] discussed the statisticalproperty and the forecasting ability of RSI

(3) Rate of Change (ROC) The ROC is a pure momentumoscillator that measures the percent of change in price fromone period to the next The 119899-period ROC is defined as

ROC(119899)

119905 = 100 times119878119905 minus 119878119905minus119899120575

119878119905minus119899120575

119905 ge 119899120575 (6)

Journal of Applied Mathematics 3

Table 1 Ratio of SPY in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9571 9791 7344 9004 98762009 9828 9160 6875 9000 1002010 9760 9159 8241 100 1002011 9569 9706 8083 9750 100

Table 2 Ratio of QQQ in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9700 9582 5892 8797 98762009 9612 9160 6625 9125 1002010 9663 8178 7407 9954 1002011 9440 9664 8000 9667 100

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250020406080100

RSI

(b)

Figure 2 SampP500 annual RSI until March 27 2012

TheROCof SampP500 is shown in Figure 3 Usually we take 119899 =12 120575 = one day Prices are constantly increasing as long asthe ROC remains positive Conversely prices are fallingwhenthe ROC is negative The ROC has its antennas and groundswhich are indefinite and can give identifiable extremes thatsignal overbought and oversold conditions In general it istime to sell out when the ROC rises to the first ultra-buy line(5) and then the rising trend mostly ends when it reachesthe second ultra-buy line (10) It is time to buy in when ROCdrops to the first ultra-sell line (minus5) and then the droppingtrend mostly ends when it reaches the second ultra-sell line(minus10) Li [17] discussed the empirical evidence of ROC Likeall technical indicators the ROC oscillator should be used inconjunction with other aspects of technical analysis

3 Some Facts from the Stock Market

Liu et al [1] traced 15 years of the DOW SampP500 andNASDAQdaily closing prices and drew the conclusion that inevery year more than 94 of daily closing prices are between

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250

0102030

ROC

minus20

minus10

(b)

Figure 3 SampP500 annual ROC until March 27 2012

the Bollinger bands We give the ratios of Bollinger bandsRSI and ROC falling in the corresponding sets from January2008 to December 2011 in Tables 1 2 and 3 where B-Bdenote the Bollinger bands We can see that more than 95of daily closing prices are between the Bollinger bands morethan 81 of RSI are in the interval [2080] and more than87 of ROC are in the interval [minus10 10] So we show thatthe stationary of the indexes is still maintained even sincethe world economic crisis in 2008 In the following we givea mathematical proof to this fact under the fractional Black-Scholes model

4 Stationary Property

Let 119878119905 denote the observed stock price under the model (1)And let

ℎ (119905 119894 119899) = exp 119861119867119905minus119894120575 minus 119861119867

119905minus119899120575 119894 = 0 1 2 119899 minus 1

119880(119899)

119905 = 119891 (ℎ (119905 0 119899) ℎ (119905 119899 minus 1 119899)) 119905 ge 119899120575

(7)

4 Journal of Applied Mathematics

Table 3 Ratio of DIA in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9657 9623 7635 9212 98762009 9742 9163 6971 9087 1002010 9856 9256 8710 100 1002011 9483 9496 8417 9833 100

where 119891 is a measurable function R119899rarr R Then we have

the following resultsThe process 119880(119899)

119905 119905ge119899120575 is stationary

Remark 1 Let 119878119905 be the stock price generated by themodel (1)119871(119899)119905 = (119878119905 minus119861

119899med119905 )119904

(119899)119905 (119905 ge 119899120575) Then the process 119871(119899)119905 119905ge119899120575 is

stationary

Remark 2 Let 119878119905 be the stock price generated by the model(1) Then the process

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

(119905 ge 119899120575) (8)

is stationary

Remark 3 Let 119878119905 be the stock price generated by the model(1)Then the process ROC(119899)

119905 = 100times(119878119905minus119878119905minus119899120575)119878119905minus119899120575(119905 ge 119899120575)

is stationary

5 Law of Large Numbers

Let 119870(119899)

Γ119894= 119868

119880(119899)

119894120575isinΓ

119894 ge 119899 where Γ is a subset of R And let

119881(119899)

119873Γ =1

119873 + 1

119873

sum

119894=0

119870(119899)

Γ119899+119894 (9)

which is the observed frequency of the events [119880(119899)

(119899+119894)120575isin

Γ] (119894 = 0 1 119873)It is natural to assume 120575 lt 1 that is the length between

two observation spots is less than one year From the abovediscussion we can let119901 = 119875(119880(119899)

119905 isin Γ) 119905 ge 119899120575We denote119880(119899)

119894120575

by 119880(119899)

119894 119894 ge 119899 in the following discussion for convenience

Denote by R119898times119899 the set of119898 times 119899 real matrices and set

119885119896119895 = 119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ) (10)

for each fixed 119895 and 119896 we give the following lemma

Lemma 4 For all Γ sub R there exist 120572 isin (0 1) and a constant119862 gt 0 when119873 is large enough and |1198961 minus1198962| minus 119899 gt 119873120572 one has

119864 (1198851198961 1198951198851198962 119895

) le 119862 |2119867 minus 1|119873120572(119867minus1)2

(11)

Proof Let 119882119896119894 = 119861119867[(119896+1)119899+119895minus119894]120575 minus 119861

119867[(119896+1)119899+119895minus(119894+1)]120575 119896 isin Z+

119894 = 0 1 2 119899 minus 1 And set 119883 = (1198821198961 0 1198821198961 119899minus1

)119879≜

(1198831 119883119899)119879 119884 = (1198821198962 0

1198821198962 119899minus1)119879≜ (1198841 119884119899)

119879

119885 = (119883119879 119884

119879)119879 where 119883119879 is the transpose of 119883 then 119864119883 =

119864119884 = 0 119864119885 = 0 We denote by 119860 = (119886119894119895) the covariancematrix of 119883 denote by 119863 = (119889119894119895) the covariance matrix of119884 and denote by 119861 = (119887119894119895) the covariance matrix of 119883 and119884 Let Σ be the covariance matrix of 119885 then Σ = ( 119860 119861

119861119879119863)

|119860| gt 0 |119863| gt 0 Since the fBm has stationary incrementswe can get 119860 = 119863 and

119886119894119895 = 119889119894119895 =1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 minus 1) 1205751003816100381610038161003816

2119867

+1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 + 1) 1205751003816100381610038161003816

2119867minus1003816100381610038161003816(119894 minus 119895)120575

1003816100381610038161003816

2119867 119867 isin (0 1)

(12)

Similar to the conclusion given by Deng and Barkai [18] wecan get from simple calculation

119887119894119895 = 119864119883119894119884119895 sim 1205752119867(2119867 minus 1)

10038161003816100381610038161198961 minus 1198962 + 119895 minus 1198941003816100381610038161003816

2119867minus2

1198962 minus 1198961 997888rarr +infin

(13)

where 119901119896 sim 119902119896 means lim119896rarrinfin(119901119896119902119896) = 1 So we have 119861 =(119887119894119895) rarr 0

When 119867 = 12 we can easily derive that the conclusionof Lemma 4 holdsWe assume119867 = 12 in the following proofLet 119901(119911) be the probability density function of119885 and 119865(119911) thedistribution function of 119885 and let the marginal distributionsfor 119885 be 119865119894(119911119894) 119894 = 1 2 2119899 where 119911 isin R2119899times1 119911 =

(1199111 1199112 1199112119899)119879 Take the notation Θ = [minus119872119872]119872 gt 0

and Γ1015840 = Θ times Θ times sdot sdot sdot times Θ ≜ Θ119899 Furthermore we put

119892 (119885) = 1198851198961 1198951198851198962 119895

(14)

First we will consider the integral of 119892(119885) on Γ1015840 times Γ1015840Referring to Bernstein [19] we have Σminus1 = Σ1 + Σ2 whereΣ = 119863 minus 119861

119879119860

minus1119861

Σ1 = (119860

minus1 00 119863

minus1)

Σ2 = (119860

minus1119861Σ

minus1119861119879119860

minus1minus119860

minus1119861Σ

minus1

minusΣminus1119861119879119860

minus1119863

minus1119861119879(119860 minus 119861119863

minus1119861119879)minus1119861119863

minus1)

(15)

Journal of Applied Mathematics 5

We take the notation 119889 = 1198891199111 1198891199112 1198891199112119899Then we obtain

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

=

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) (2120587)minus119899(|119860| |119863|)

minus12

times exp minus12119911119879Σ1119911

times (

10038161003816100381610038161003816119863 minus 119861

119879119860

minus111986110038161003816100381610038161003816

minus12

|119863|minus12

exp minus12119911119879Σ2119911 minus 1)119889

(16)

where lim|1198961minus1198962|rarrinfin(|119863 minus 119861119879119860

minus1119861||119863|) = (1|119863|)(|119863 minus

lim|1198961minus1198962|rarrinfin(119861119879119860

minus1119861)|) = 1 Assume lim1198962minus1198961rarrinfin(119872|1198961 minus

1198962|1minus119867) = 0 then 119911119879Σ2119911 rarr 0 so we get forall120598 gt 0 exist1198731

forall|1198961 minus 1198962| gt 1198731

1003816100381610038161003816100381610038161003816exp minus1

2119911119879Σ2119911 minus 1

1003816100381610038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161 minus

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911) minus 1

1003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816

(17)

Choose 120572 isin (0 1) satisfying119873120572gt 1198731 where119873 lt (119873 + 1)119899

then if |1198961 minus 1198962| minus 119899 gt 119873120572 and119873 is large enough there exist

1198620 gt 0 and 1198620 has relation with 119899 such that

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816le 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1)

(18)

Therefore by (16) (17) and (18) we can derive that there exist1198620 gt 0 satisfying

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

le 1199012(120598 + 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1))

le 1198620 |2119867 minus 1|119873120572(119867minus1)

(19)

Then we will consider the integral of 119892(119885) on thecomplementary set of Γ1015840 times Γ1015840 in the following Let Ξ be arandom variable satisfying 119875(Ξ = (minusinfin minus119872)) = 12 119875(Ξ =(119872infin)) = 12 Let Γ119894 be the set that contains all elements ofthe following form

Ξ times sdot sdot sdot times Θ times sdot sdot sdot times Ξ times sdot sdot sdot times Θ ≜ Ξ(119894) (20)

where Θ occurs 119894 times and Ξ occurs 2119899 minus 119894 times in Ξ(119894) 119894 =0 1 2119899minus1 So we can see that Γ119894 is composed of ( 2119899119894 )sdot2

2119899minus119894

mutually disjoint elementsTherefore the complementary set

of Γ1015840 times Γ1015840 should be (Γ1015840 times Γ1015840)119888 = ⋃2119899minus1

119894=0 Γ119894 and Γ119894⋂Γ119895 = Φ119894 = 119895 that is the complementary set of Γ1015840 times Γ1015840 is the union of(sum

2119899minus1

119894=0 (2119899119894 ) sdot 2

2119899minus119894= 3

2119899minus 1)mutually disjoint sets

Since10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119892 (119911) 119901 (119911) 119889

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119901 (119911) 119889 le 119865 (1199111 1199112 1199112119899)

le min1le119894le2119899

119865119894 (119911119894) le 1198651 (minus119872) le120575119867119890minus119872221205752119867

119872

(21)

where119872 occurs 119894 times and minus119872 occurs 2119899 minus 119894 times within1199111 1199112 1199112119899 the second inequality holds because |119892(119911)| le1 and the last inequality holds because int+infin

119872119890minus11990922119889119909 =

intminus119872

minusinfin119890minus11990922119889119909 le 119890

minus11987222119872

Take119872 = 119873120572(1minus119867)2 We conclude from (19) and (21) that

119864 (1198851198961 1198951198851198962 119895

) =

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

+

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

(Γ1015840timesΓ1015840)

119888

119892 (119911) 119901 (119911) 119889

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+

2119899minus1

sum

119894=0

(2119899

119894) sdot 2

2119899minus119894 120575119867119890minus(119872)

221205752119867

119872

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

(22)

Take 119862 = 1198620 + (32119899minus 1)120575

119867|2119867 minus 1| then we have

1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

le 119862 |2119867 minus 1|119873120572(119867minus1)2

(23)

from which the proof immediately follows

Then we obtain the law of large numbers

Theorem 5 There exist 120573 isin (0 1) and a constant 119862 gt 0 suchthat

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2le119862

119873120573 (24)

Proof To simplify notation we put Λ = 119896 0 le 119896119899 + 119895 le 119873and set for each fixed 119895 and 119896

119884119895 = sum

119896isinΛ

[119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ)] (25)

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

Table 1 Ratio of SPY in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9571 9791 7344 9004 98762009 9828 9160 6875 9000 1002010 9760 9159 8241 100 1002011 9569 9706 8083 9750 100

Table 2 Ratio of QQQ in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9700 9582 5892 8797 98762009 9612 9160 6625 9125 1002010 9663 8178 7407 9954 1002011 9440 9664 8000 9667 100

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250020406080100

RSI

(b)

Figure 2 SampP500 annual RSI until March 27 2012

TheROCof SampP500 is shown in Figure 3 Usually we take 119899 =12 120575 = one day Prices are constantly increasing as long asthe ROC remains positive Conversely prices are fallingwhenthe ROC is negative The ROC has its antennas and groundswhich are indefinite and can give identifiable extremes thatsignal overbought and oversold conditions In general it istime to sell out when the ROC rises to the first ultra-buy line(5) and then the rising trend mostly ends when it reachesthe second ultra-buy line (10) It is time to buy in when ROCdrops to the first ultra-sell line (minus5) and then the droppingtrend mostly ends when it reaches the second ultra-sell line(minus10) Li [17] discussed the empirical evidence of ROC Likeall technical indicators the ROC oscillator should be used inconjunction with other aspects of technical analysis

3 Some Facts from the Stock Market

Liu et al [1] traced 15 years of the DOW SampP500 andNASDAQdaily closing prices and drew the conclusion that inevery year more than 94 of daily closing prices are between

0 50 100 150 200 2501000

1100

1200

1300

1400

SampP5

00

(a)

0 50 100 150 200 250

0102030

ROC

minus20

minus10

(b)

Figure 3 SampP500 annual ROC until March 27 2012

the Bollinger bands We give the ratios of Bollinger bandsRSI and ROC falling in the corresponding sets from January2008 to December 2011 in Tables 1 2 and 3 where B-Bdenote the Bollinger bands We can see that more than 95of daily closing prices are between the Bollinger bands morethan 81 of RSI are in the interval [2080] and more than87 of ROC are in the interval [minus10 10] So we show thatthe stationary of the indexes is still maintained even sincethe world economic crisis in 2008 In the following we givea mathematical proof to this fact under the fractional Black-Scholes model

4 Stationary Property

Let 119878119905 denote the observed stock price under the model (1)And let

ℎ (119905 119894 119899) = exp 119861119867119905minus119894120575 minus 119861119867

119905minus119899120575 119894 = 0 1 2 119899 minus 1

119880(119899)

119905 = 119891 (ℎ (119905 0 119899) ℎ (119905 119899 minus 1 119899)) 119905 ge 119899120575

(7)

4 Journal of Applied Mathematics

Table 3 Ratio of DIA in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9657 9623 7635 9212 98762009 9742 9163 6971 9087 1002010 9856 9256 8710 100 1002011 9483 9496 8417 9833 100

where 119891 is a measurable function R119899rarr R Then we have

the following resultsThe process 119880(119899)

119905 119905ge119899120575 is stationary

Remark 1 Let 119878119905 be the stock price generated by themodel (1)119871(119899)119905 = (119878119905 minus119861

119899med119905 )119904

(119899)119905 (119905 ge 119899120575) Then the process 119871(119899)119905 119905ge119899120575 is

stationary

Remark 2 Let 119878119905 be the stock price generated by the model(1) Then the process

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

(119905 ge 119899120575) (8)

is stationary

Remark 3 Let 119878119905 be the stock price generated by the model(1)Then the process ROC(119899)

119905 = 100times(119878119905minus119878119905minus119899120575)119878119905minus119899120575(119905 ge 119899120575)

is stationary

5 Law of Large Numbers

Let 119870(119899)

Γ119894= 119868

119880(119899)

119894120575isinΓ

119894 ge 119899 where Γ is a subset of R And let

119881(119899)

119873Γ =1

119873 + 1

119873

sum

119894=0

119870(119899)

Γ119899+119894 (9)

which is the observed frequency of the events [119880(119899)

(119899+119894)120575isin

Γ] (119894 = 0 1 119873)It is natural to assume 120575 lt 1 that is the length between

two observation spots is less than one year From the abovediscussion we can let119901 = 119875(119880(119899)

119905 isin Γ) 119905 ge 119899120575We denote119880(119899)

119894120575

by 119880(119899)

119894 119894 ge 119899 in the following discussion for convenience

Denote by R119898times119899 the set of119898 times 119899 real matrices and set

119885119896119895 = 119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ) (10)

for each fixed 119895 and 119896 we give the following lemma

Lemma 4 For all Γ sub R there exist 120572 isin (0 1) and a constant119862 gt 0 when119873 is large enough and |1198961 minus1198962| minus 119899 gt 119873120572 one has

119864 (1198851198961 1198951198851198962 119895

) le 119862 |2119867 minus 1|119873120572(119867minus1)2

(11)

Proof Let 119882119896119894 = 119861119867[(119896+1)119899+119895minus119894]120575 minus 119861

119867[(119896+1)119899+119895minus(119894+1)]120575 119896 isin Z+

119894 = 0 1 2 119899 minus 1 And set 119883 = (1198821198961 0 1198821198961 119899minus1

)119879≜

(1198831 119883119899)119879 119884 = (1198821198962 0

1198821198962 119899minus1)119879≜ (1198841 119884119899)

119879

119885 = (119883119879 119884

119879)119879 where 119883119879 is the transpose of 119883 then 119864119883 =

119864119884 = 0 119864119885 = 0 We denote by 119860 = (119886119894119895) the covariancematrix of 119883 denote by 119863 = (119889119894119895) the covariance matrix of119884 and denote by 119861 = (119887119894119895) the covariance matrix of 119883 and119884 Let Σ be the covariance matrix of 119885 then Σ = ( 119860 119861

119861119879119863)

|119860| gt 0 |119863| gt 0 Since the fBm has stationary incrementswe can get 119860 = 119863 and

119886119894119895 = 119889119894119895 =1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 minus 1) 1205751003816100381610038161003816

2119867

+1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 + 1) 1205751003816100381610038161003816

2119867minus1003816100381610038161003816(119894 minus 119895)120575

1003816100381610038161003816

2119867 119867 isin (0 1)

(12)

Similar to the conclusion given by Deng and Barkai [18] wecan get from simple calculation

119887119894119895 = 119864119883119894119884119895 sim 1205752119867(2119867 minus 1)

10038161003816100381610038161198961 minus 1198962 + 119895 minus 1198941003816100381610038161003816

2119867minus2

1198962 minus 1198961 997888rarr +infin

(13)

where 119901119896 sim 119902119896 means lim119896rarrinfin(119901119896119902119896) = 1 So we have 119861 =(119887119894119895) rarr 0

When 119867 = 12 we can easily derive that the conclusionof Lemma 4 holdsWe assume119867 = 12 in the following proofLet 119901(119911) be the probability density function of119885 and 119865(119911) thedistribution function of 119885 and let the marginal distributionsfor 119885 be 119865119894(119911119894) 119894 = 1 2 2119899 where 119911 isin R2119899times1 119911 =

(1199111 1199112 1199112119899)119879 Take the notation Θ = [minus119872119872]119872 gt 0

and Γ1015840 = Θ times Θ times sdot sdot sdot times Θ ≜ Θ119899 Furthermore we put

119892 (119885) = 1198851198961 1198951198851198962 119895

(14)

First we will consider the integral of 119892(119885) on Γ1015840 times Γ1015840Referring to Bernstein [19] we have Σminus1 = Σ1 + Σ2 whereΣ = 119863 minus 119861

119879119860

minus1119861

Σ1 = (119860

minus1 00 119863

minus1)

Σ2 = (119860

minus1119861Σ

minus1119861119879119860

minus1minus119860

minus1119861Σ

minus1

minusΣminus1119861119879119860

minus1119863

minus1119861119879(119860 minus 119861119863

minus1119861119879)minus1119861119863

minus1)

(15)

Journal of Applied Mathematics 5

We take the notation 119889 = 1198891199111 1198891199112 1198891199112119899Then we obtain

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

=

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) (2120587)minus119899(|119860| |119863|)

minus12

times exp minus12119911119879Σ1119911

times (

10038161003816100381610038161003816119863 minus 119861

119879119860

minus111986110038161003816100381610038161003816

minus12

|119863|minus12

exp minus12119911119879Σ2119911 minus 1)119889

(16)

where lim|1198961minus1198962|rarrinfin(|119863 minus 119861119879119860

minus1119861||119863|) = (1|119863|)(|119863 minus

lim|1198961minus1198962|rarrinfin(119861119879119860

minus1119861)|) = 1 Assume lim1198962minus1198961rarrinfin(119872|1198961 minus

1198962|1minus119867) = 0 then 119911119879Σ2119911 rarr 0 so we get forall120598 gt 0 exist1198731

forall|1198961 minus 1198962| gt 1198731

1003816100381610038161003816100381610038161003816exp minus1

2119911119879Σ2119911 minus 1

1003816100381610038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161 minus

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911) minus 1

1003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816

(17)

Choose 120572 isin (0 1) satisfying119873120572gt 1198731 where119873 lt (119873 + 1)119899

then if |1198961 minus 1198962| minus 119899 gt 119873120572 and119873 is large enough there exist

1198620 gt 0 and 1198620 has relation with 119899 such that

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816le 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1)

(18)

Therefore by (16) (17) and (18) we can derive that there exist1198620 gt 0 satisfying

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

le 1199012(120598 + 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1))

le 1198620 |2119867 minus 1|119873120572(119867minus1)

(19)

Then we will consider the integral of 119892(119885) on thecomplementary set of Γ1015840 times Γ1015840 in the following Let Ξ be arandom variable satisfying 119875(Ξ = (minusinfin minus119872)) = 12 119875(Ξ =(119872infin)) = 12 Let Γ119894 be the set that contains all elements ofthe following form

Ξ times sdot sdot sdot times Θ times sdot sdot sdot times Ξ times sdot sdot sdot times Θ ≜ Ξ(119894) (20)

where Θ occurs 119894 times and Ξ occurs 2119899 minus 119894 times in Ξ(119894) 119894 =0 1 2119899minus1 So we can see that Γ119894 is composed of ( 2119899119894 )sdot2

2119899minus119894

mutually disjoint elementsTherefore the complementary set

of Γ1015840 times Γ1015840 should be (Γ1015840 times Γ1015840)119888 = ⋃2119899minus1

119894=0 Γ119894 and Γ119894⋂Γ119895 = Φ119894 = 119895 that is the complementary set of Γ1015840 times Γ1015840 is the union of(sum

2119899minus1

119894=0 (2119899119894 ) sdot 2

2119899minus119894= 3

2119899minus 1)mutually disjoint sets

Since10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119892 (119911) 119901 (119911) 119889

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119901 (119911) 119889 le 119865 (1199111 1199112 1199112119899)

le min1le119894le2119899

119865119894 (119911119894) le 1198651 (minus119872) le120575119867119890minus119872221205752119867

119872

(21)

where119872 occurs 119894 times and minus119872 occurs 2119899 minus 119894 times within1199111 1199112 1199112119899 the second inequality holds because |119892(119911)| le1 and the last inequality holds because int+infin

119872119890minus11990922119889119909 =

intminus119872

minusinfin119890minus11990922119889119909 le 119890

minus11987222119872

Take119872 = 119873120572(1minus119867)2 We conclude from (19) and (21) that

119864 (1198851198961 1198951198851198962 119895

) =

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

+

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

(Γ1015840timesΓ1015840)

119888

119892 (119911) 119901 (119911) 119889

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+

2119899minus1

sum

119894=0

(2119899

119894) sdot 2

2119899minus119894 120575119867119890minus(119872)

221205752119867

119872

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

(22)

Take 119862 = 1198620 + (32119899minus 1)120575

119867|2119867 minus 1| then we have

1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

le 119862 |2119867 minus 1|119873120572(119867minus1)2

(23)

from which the proof immediately follows

Then we obtain the law of large numbers

Theorem 5 There exist 120573 isin (0 1) and a constant 119862 gt 0 suchthat

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2le119862

119873120573 (24)

Proof To simplify notation we put Λ = 119896 0 le 119896119899 + 119895 le 119873and set for each fixed 119895 and 119896

119884119895 = sum

119896isinΛ

[119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ)] (25)

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

Table 3 Ratio of DIA in 2008ndash2011

Year 119878119905 isin B-B RSI isin [20 80] ROC isin [minus5 5] ROC isin [minus10 10] ROC isin [minus20 20]2008 9657 9623 7635 9212 98762009 9742 9163 6971 9087 1002010 9856 9256 8710 100 1002011 9483 9496 8417 9833 100

where 119891 is a measurable function R119899rarr R Then we have

the following resultsThe process 119880(119899)

119905 119905ge119899120575 is stationary

Remark 1 Let 119878119905 be the stock price generated by themodel (1)119871(119899)119905 = (119878119905 minus119861

119899med119905 )119904

(119899)119905 (119905 ge 119899120575) Then the process 119871(119899)119905 119905ge119899120575 is

stationary

Remark 2 Let 119878119905 be the stock price generated by the model(1) Then the process

RSI(119899)119905 = 100 timessum

119899

119894=1 Δ119878+119905minus119894120575

sum119899

119894=1 Δ119878+119905minus119894120575+ sum

119899

119894=1 Δ119878minus119905minus119894120575

(119905 ge 119899120575) (8)

is stationary

Remark 3 Let 119878119905 be the stock price generated by the model(1)Then the process ROC(119899)

119905 = 100times(119878119905minus119878119905minus119899120575)119878119905minus119899120575(119905 ge 119899120575)

is stationary

5 Law of Large Numbers

Let 119870(119899)

Γ119894= 119868

119880(119899)

119894120575isinΓ

119894 ge 119899 where Γ is a subset of R And let

119881(119899)

119873Γ =1

119873 + 1

119873

sum

119894=0

119870(119899)

Γ119899+119894 (9)

which is the observed frequency of the events [119880(119899)

(119899+119894)120575isin

Γ] (119894 = 0 1 119873)It is natural to assume 120575 lt 1 that is the length between

two observation spots is less than one year From the abovediscussion we can let119901 = 119875(119880(119899)

119905 isin Γ) 119905 ge 119899120575We denote119880(119899)

119894120575

by 119880(119899)

119894 119894 ge 119899 in the following discussion for convenience

Denote by R119898times119899 the set of119898 times 119899 real matrices and set

119885119896119895 = 119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ) (10)

for each fixed 119895 and 119896 we give the following lemma

Lemma 4 For all Γ sub R there exist 120572 isin (0 1) and a constant119862 gt 0 when119873 is large enough and |1198961 minus1198962| minus 119899 gt 119873120572 one has

119864 (1198851198961 1198951198851198962 119895

) le 119862 |2119867 minus 1|119873120572(119867minus1)2

(11)

Proof Let 119882119896119894 = 119861119867[(119896+1)119899+119895minus119894]120575 minus 119861

119867[(119896+1)119899+119895minus(119894+1)]120575 119896 isin Z+

119894 = 0 1 2 119899 minus 1 And set 119883 = (1198821198961 0 1198821198961 119899minus1

)119879≜

(1198831 119883119899)119879 119884 = (1198821198962 0

1198821198962 119899minus1)119879≜ (1198841 119884119899)

119879

119885 = (119883119879 119884

119879)119879 where 119883119879 is the transpose of 119883 then 119864119883 =

119864119884 = 0 119864119885 = 0 We denote by 119860 = (119886119894119895) the covariancematrix of 119883 denote by 119863 = (119889119894119895) the covariance matrix of119884 and denote by 119861 = (119887119894119895) the covariance matrix of 119883 and119884 Let Σ be the covariance matrix of 119885 then Σ = ( 119860 119861

119861119879119863)

|119860| gt 0 |119863| gt 0 Since the fBm has stationary incrementswe can get 119860 = 119863 and

119886119894119895 = 119889119894119895 =1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 minus 1) 1205751003816100381610038161003816

2119867

+1

2

1003816100381610038161003816(1003816100381610038161003816119894 minus 119895

1003816100381610038161003816 + 1) 1205751003816100381610038161003816

2119867minus1003816100381610038161003816(119894 minus 119895)120575

1003816100381610038161003816

2119867 119867 isin (0 1)

(12)

Similar to the conclusion given by Deng and Barkai [18] wecan get from simple calculation

119887119894119895 = 119864119883119894119884119895 sim 1205752119867(2119867 minus 1)

10038161003816100381610038161198961 minus 1198962 + 119895 minus 1198941003816100381610038161003816

2119867minus2

1198962 minus 1198961 997888rarr +infin

(13)

where 119901119896 sim 119902119896 means lim119896rarrinfin(119901119896119902119896) = 1 So we have 119861 =(119887119894119895) rarr 0

When 119867 = 12 we can easily derive that the conclusionof Lemma 4 holdsWe assume119867 = 12 in the following proofLet 119901(119911) be the probability density function of119885 and 119865(119911) thedistribution function of 119885 and let the marginal distributionsfor 119885 be 119865119894(119911119894) 119894 = 1 2 2119899 where 119911 isin R2119899times1 119911 =

(1199111 1199112 1199112119899)119879 Take the notation Θ = [minus119872119872]119872 gt 0

and Γ1015840 = Θ times Θ times sdot sdot sdot times Θ ≜ Θ119899 Furthermore we put

119892 (119885) = 1198851198961 1198951198851198962 119895

(14)

First we will consider the integral of 119892(119885) on Γ1015840 times Γ1015840Referring to Bernstein [19] we have Σminus1 = Σ1 + Σ2 whereΣ = 119863 minus 119861

119879119860

minus1119861

Σ1 = (119860

minus1 00 119863

minus1)

Σ2 = (119860

minus1119861Σ

minus1119861119879119860

minus1minus119860

minus1119861Σ

minus1

minusΣminus1119861119879119860

minus1119863

minus1119861119879(119860 minus 119861119863

minus1119861119879)minus1119861119863

minus1)

(15)

Journal of Applied Mathematics 5

We take the notation 119889 = 1198891199111 1198891199112 1198891199112119899Then we obtain

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

=

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) (2120587)minus119899(|119860| |119863|)

minus12

times exp minus12119911119879Σ1119911

times (

10038161003816100381610038161003816119863 minus 119861

119879119860

minus111986110038161003816100381610038161003816

minus12

|119863|minus12

exp minus12119911119879Σ2119911 minus 1)119889

(16)

where lim|1198961minus1198962|rarrinfin(|119863 minus 119861119879119860

minus1119861||119863|) = (1|119863|)(|119863 minus

lim|1198961minus1198962|rarrinfin(119861119879119860

minus1119861)|) = 1 Assume lim1198962minus1198961rarrinfin(119872|1198961 minus

1198962|1minus119867) = 0 then 119911119879Σ2119911 rarr 0 so we get forall120598 gt 0 exist1198731

forall|1198961 minus 1198962| gt 1198731

1003816100381610038161003816100381610038161003816exp minus1

2119911119879Σ2119911 minus 1

1003816100381610038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161 minus

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911) minus 1

1003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816

(17)

Choose 120572 isin (0 1) satisfying119873120572gt 1198731 where119873 lt (119873 + 1)119899

then if |1198961 minus 1198962| minus 119899 gt 119873120572 and119873 is large enough there exist

1198620 gt 0 and 1198620 has relation with 119899 such that

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816le 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1)

(18)

Therefore by (16) (17) and (18) we can derive that there exist1198620 gt 0 satisfying

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

le 1199012(120598 + 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1))

le 1198620 |2119867 minus 1|119873120572(119867minus1)

(19)

Then we will consider the integral of 119892(119885) on thecomplementary set of Γ1015840 times Γ1015840 in the following Let Ξ be arandom variable satisfying 119875(Ξ = (minusinfin minus119872)) = 12 119875(Ξ =(119872infin)) = 12 Let Γ119894 be the set that contains all elements ofthe following form

Ξ times sdot sdot sdot times Θ times sdot sdot sdot times Ξ times sdot sdot sdot times Θ ≜ Ξ(119894) (20)

where Θ occurs 119894 times and Ξ occurs 2119899 minus 119894 times in Ξ(119894) 119894 =0 1 2119899minus1 So we can see that Γ119894 is composed of ( 2119899119894 )sdot2

2119899minus119894

mutually disjoint elementsTherefore the complementary set

of Γ1015840 times Γ1015840 should be (Γ1015840 times Γ1015840)119888 = ⋃2119899minus1

119894=0 Γ119894 and Γ119894⋂Γ119895 = Φ119894 = 119895 that is the complementary set of Γ1015840 times Γ1015840 is the union of(sum

2119899minus1

119894=0 (2119899119894 ) sdot 2

2119899minus119894= 3

2119899minus 1)mutually disjoint sets

Since10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119892 (119911) 119901 (119911) 119889

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119901 (119911) 119889 le 119865 (1199111 1199112 1199112119899)

le min1le119894le2119899

119865119894 (119911119894) le 1198651 (minus119872) le120575119867119890minus119872221205752119867

119872

(21)

where119872 occurs 119894 times and minus119872 occurs 2119899 minus 119894 times within1199111 1199112 1199112119899 the second inequality holds because |119892(119911)| le1 and the last inequality holds because int+infin

119872119890minus11990922119889119909 =

intminus119872

minusinfin119890minus11990922119889119909 le 119890

minus11987222119872

Take119872 = 119873120572(1minus119867)2 We conclude from (19) and (21) that

119864 (1198851198961 1198951198851198962 119895

) =

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

+

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

(Γ1015840timesΓ1015840)

119888

119892 (119911) 119901 (119911) 119889

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+

2119899minus1

sum

119894=0

(2119899

119894) sdot 2

2119899minus119894 120575119867119890minus(119872)

221205752119867

119872

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

(22)

Take 119862 = 1198620 + (32119899minus 1)120575

119867|2119867 minus 1| then we have

1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

le 119862 |2119867 minus 1|119873120572(119867minus1)2

(23)

from which the proof immediately follows

Then we obtain the law of large numbers

Theorem 5 There exist 120573 isin (0 1) and a constant 119862 gt 0 suchthat

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2le119862

119873120573 (24)

Proof To simplify notation we put Λ = 119896 0 le 119896119899 + 119895 le 119873and set for each fixed 119895 and 119896

119884119895 = sum

119896isinΛ

[119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ)] (25)

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

We take the notation 119889 = 1198891199111 1198891199112 1198891199112119899Then we obtain

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

=

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) (2120587)minus119899(|119860| |119863|)

minus12

times exp minus12119911119879Σ1119911

times (

10038161003816100381610038161003816119863 minus 119861

119879119860

minus111986110038161003816100381610038161003816

minus12

|119863|minus12

exp minus12119911119879Σ2119911 minus 1)119889

(16)

where lim|1198961minus1198962|rarrinfin(|119863 minus 119861119879119860

minus1119861||119863|) = (1|119863|)(|119863 minus

lim|1198961minus1198962|rarrinfin(119861119879119860

minus1119861)|) = 1 Assume lim1198962minus1198961rarrinfin(119872|1198961 minus

1198962|1minus119867) = 0 then 119911119879Σ2119911 rarr 0 so we get forall120598 gt 0 exist1198731

forall|1198961 minus 1198962| gt 1198731

1003816100381610038161003816100381610038161003816exp minus1

2119911119879Σ2119911 minus 1

1003816100381610038161003816100381610038161003816=

10038161003816100381610038161003816100381610038161 minus

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911) minus 1

1003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816

(17)

Choose 120572 isin (0 1) satisfying119873120572gt 1198731 where119873 lt (119873 + 1)119899

then if |1198961 minus 1198962| minus 119899 gt 119873120572 and119873 is large enough there exist

1198620 gt 0 and 1198620 has relation with 119899 such that

1003816100381610038161003816100381610038161003816

1

2119911119879Σ2119911 + 119900 (119911

119879Σ2119911)

1003816100381610038161003816100381610038161003816le 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1)

(18)

Therefore by (16) (17) and (18) we can derive that there exist1198620 gt 0 satisfying

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

le 1199012(120598 + 1198620119872

2|119867 (2119867 minus 1)|119873

2120572(119867minus1))

le 1198620 |2119867 minus 1|119873120572(119867minus1)

(19)

Then we will consider the integral of 119892(119885) on thecomplementary set of Γ1015840 times Γ1015840 in the following Let Ξ be arandom variable satisfying 119875(Ξ = (minusinfin minus119872)) = 12 119875(Ξ =(119872infin)) = 12 Let Γ119894 be the set that contains all elements ofthe following form

Ξ times sdot sdot sdot times Θ times sdot sdot sdot times Ξ times sdot sdot sdot times Θ ≜ Ξ(119894) (20)

where Θ occurs 119894 times and Ξ occurs 2119899 minus 119894 times in Ξ(119894) 119894 =0 1 2119899minus1 So we can see that Γ119894 is composed of ( 2119899119894 )sdot2

2119899minus119894

mutually disjoint elementsTherefore the complementary set

of Γ1015840 times Γ1015840 should be (Γ1015840 times Γ1015840)119888 = ⋃2119899minus1

119894=0 Γ119894 and Γ119894⋂Γ119895 = Φ119894 = 119895 that is the complementary set of Γ1015840 times Γ1015840 is the union of(sum

2119899minus1

119894=0 (2119899119894 ) sdot 2

2119899minus119894= 3

2119899minus 1)mutually disjoint sets

Since10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119892 (119911) 119901 (119911) 119889

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Ξ(119894)

119901 (119911) 119889 le 119865 (1199111 1199112 1199112119899)

le min1le119894le2119899

119865119894 (119911119894) le 1198651 (minus119872) le120575119867119890minus119872221205752119867

119872

(21)

where119872 occurs 119894 times and minus119872 occurs 2119899 minus 119894 times within1199111 1199112 1199112119899 the second inequality holds because |119892(119911)| le1 and the last inequality holds because int+infin

119872119890minus11990922119889119909 =

intminus119872

minusinfin119890minus11990922119889119909 le 119890

minus11987222119872

Take119872 = 119873120572(1minus119867)2 We conclude from (19) and (21) that

119864 (1198851198961 1198951198851198962 119895

) =

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

Γ1015840timesΓ1015840

119892 (119911) 119901 (119911) 119889

+

2119899⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

int sdot sdot sdot int

(Γ1015840timesΓ1015840)

119888

119892 (119911) 119901 (119911) 119889

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+

2119899minus1

sum

119894=0

(2119899

119894) sdot 2

2119899minus119894 120575119867119890minus(119872)

221205752119867

119872

le 1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

(22)

Take 119862 = 1198620 + (32119899minus 1)120575

119867|2119867 minus 1| then we have

1198620 |2119867 minus 1|119873120572(119867minus1)

+ (32119899minus 1) 120575

119867119873

120572(119867minus1)2

le 119862 |2119867 minus 1|119873120572(119867minus1)2

(23)

from which the proof immediately follows

Then we obtain the law of large numbers

Theorem 5 There exist 120573 isin (0 1) and a constant 119862 gt 0 suchthat

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2le119862

119873120573 (24)

Proof To simplify notation we put Λ = 119896 0 le 119896119899 + 119895 le 119873and set for each fixed 119895 and 119896

119884119895 = sum

119896isinΛ

[119870(119899)

Γ(119896+1)119899+119895minus 119875 (119880

(119899)

119899 isin Γ)] (25)

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Since the process 119880(119899)119905 119905ge119899120575 is stationary we have 119875(119880(119899)

119905 isin

Γ) = 119864[119868119880(119899)

(119896+1)119899+119895isinΓ] holds for all 119896 gt 0 In addition by C-r

inequality and Lemma 4 it follows that

11986410038161003816100381610038161003816119881

(119899)

119873Γ minus 119875 (119880(119899)

119899 isin Γ)10038161003816100381610038161003816

2

= 119864

10038161003816100381610038161003816100381610038161003816100381610038161003816

1

119873 + 1

119899minus1

sum

119895=0

119884119895

10038161003816100381610038161003816100381610038161003816100381610038161003816

2

le119899

(119873 + 1)2

119899minus1

sum

119895=0

1198641198842

119895

=119899

(119873 + 1)2

[[[

[

119899minus1

sum

119895=0

sum

119896isinΛ

1198641198852

119896119895 +

119899minus1

sum

119895=0

sum

1198961 = 1198962

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2

[[[

[

119873 + 1 +

119899minus1

sum

119895=0

sum

0lt|1198961minus1198962|le119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

+

119899minus1

sum

119895=0

sum

|1198961minus1198962| gt119873120572

1198961 1198962isinΛ

1198641198851198961 1198951198851198962 119895

]]]

]

le119899

(119873 + 1)2[119873 + 1 + 2119899

119873 + 1

119899119873

120572

+ 2119899119873 + 1

119899(119873 minus 119873

120572) 119862 |2119867 minus 1|119873

120572(119867minus1)2]

(26)

Let 120573 = min1 minus 120572 120572(1 minus 119867)2 and 119862 = 3119899 + 2119899119862|2119867 minus 1|then 119864|119881(119899)

119873Γminus 119875(119880

(119899)119899 isin Γ)|

2le 119862119873

120573

Remark 6 From Theorem 5 it is reasonable to use thestationary distribution of 119880(119899)

119899 to calculate 119881(119899)

119873Γ which is the

relative frequency of the technical indicators falling in thecorresponding set

Corollary 7 Let119867(119899)

119894= 119868

119871(119899)

119894120575ge2

119894 ge 119899 then 119864119867(119899)

119894= 119875(119871

(119899)

119894120575ge

2) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (27)

which is the observed frequency of the events [119871(119899)(119899+119894)120575

ge 2] (119894 =0 1 119873) that is the frequency of stock falling out of theBollinger bands Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875 (119871(119899)

119899120575ge 2)

10038161003816100381610038161003816

2le119862

119873120573 (28)

Corollary 8 Let119867(119899)

119894= 119868

119877119878119868(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [0 20] cup[80 100] Then 119864119867(119899)

119894= 119875(119877119878119868

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (29)

which is the observed frequency of the events [119877119878119868(119899)(119899+119894)120575

isin

Γ] (119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119878119868(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (30)

Corollary 9 Let119867(119899)

119894= 119868

119877119874119862(119899)

119894120575isinΓ

119894 ge 119899 where Γ = [minusinfin 120585]cup[120578infin] and 120585 and 120578 are the indefinite ground and antenna ofROC Then 119864119867(119899)

119894= 119875(119877119874119862

(119899)

119894120575isin Γ) Let

119869(119899)

119873 =1

119873 + 1

119873

sum

119894=0

119867(119899)

119899+119894 (31)

which is the observed frequency of the events [119877119874119862(119899)

(119899+119894)120575isin Γ]

(119894 = 0 1 119873) Then there exist 120573 isin (0 1) and a constant119862 gt 0 such that

11986410038161003816100381610038161003816119869(119899)

119873 minus 119875(119877119874119862(119899)

119899120575isin Γ)

10038161003816100381610038161003816

2le119862

119873120573 (32)

6 Conclusion

In the above discussion we considered a class of long-rangedependent processes of which the rate of decay is slower thanthe exponential one typically a power-like decay We derivedthe rate of convergence of the ergodic theorem for severalstationary processes associated with the technical analysis inthe securitymarket and extended the previous results (see [1ndash3 16 17]) Thus we established the theoretical foundation oftechnical analysis for fractional Black-Scholes model

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the Natural ScienceFoundation of the Anhui High Education Institutions ofChina (KJ2013B261) the Natural Science Foundation of theAnhui High Education Institutions of China (KJ2012A257)

References

[1] W Liu X Huang and W Zheng ldquoBlack-Scholesrsquo model andBollinger bandsrdquo Physica A vol 371 no 2 pp 565ndash571 2006

[2] W Liu and W A Zheng ldquoStochastic volatility model andtechnical analysis of stock pricerdquo Acta Mathematica Sinica vol27 no 7 pp 1283ndash1296 2011

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

[3] X Huang and W Liu ldquoProperties of some statistics for AR-ARCH model with application to technical analysisrdquo Journalof Computational and Applied Mathematics vol 225 no 2 pp522ndash530 2009

[4] C D I I Kirkpatrick and J R Dahlquist Technical AnalysisThe Complete Resource for Financial Market Technicians Pear-son Education Upper Saddle River NJ USA 2nd edition 2011

[5] AW Lo HMamaysky and JWang ldquoFoundations of technicalanalysis computational algorithms statistical inference andempirical implementationrdquo Journal of Finance vol 55 no 4 pp1705ndash1770 2000

[6] W Willinger M S Taqqu and V Teverovsky ldquoStock marketprices and long-range dependencerdquoFinance and Stochastics vol3 no 1 pp 1ndash13 1999

[7] A W Lo ldquoFat tails long memory and the stock market sincethe 1960srdquo Economic Notes vol 26 no 2 pp 219ndash252 1997

[8] P Robinson Ed Time Series with Long Memory AdvancedTexts in Econometrics Oxford University Press New York NYUSA 2003

[9] R Cont ldquoLong range dependence in financial marketsrdquo inFractals in Engineering E Lutton and J Vehel Eds SpringerLondon UK 2005

[10] D O Cajueiro and BM Tabak ldquoTesting for time-varying long-range dependence in real state equity returnsrdquo Chaos Solitonsand Fractals vol 38 no 1 pp 293ndash307 2008

[11] P Cheridito ldquoArbitrage in fractional BrownianmotionmodelsrdquoFinance and Stochastics vol 7 no 4 pp 533ndash553 2003

[12] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000

[13] Y Hu B Oslashksendal and A Sulem ldquoOptimal consumptionand portfolio in a Black-Scholes market driven by fractionalBrownian motionrdquo Infinite Dimensional Analysis QuantumProbability and Related Topics vol 6 no 4 pp 519ndash536 2003

[14] J Bollinger Bollinger on Bollinger Bands McGraw-Hill NewYork NY USA 2002

[15] J W Wilder New Concepts in Technical Trading Systems TrendResearch Greensboro NC USA 1st edition 1978

[16] W Zhu Statistical analysis of stock technical indicators [MSthesis] East China Normal University Shanghai China 2006

[17] W LiAnew standard of effectiveness of moving average technicalindicators in stock markets [MS thesis] East China NormalUniversity Shanghai China 2006

[18] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

[19] D S Bernstein Matrix Mathematics Theory Facts and For-mulas with Application to Linear Systems Theory PrincetonUniversity Press Princeton NJ USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of