INTEGRATION WITH RESPECT TOFRACTAL

FUNCTIONS AND STOCHASTIC CALCULUSII

M. Zahle

Mathematical InstituteUniversity of Jena

D–07737 Jena

e-mail: zaehle@minet.uni-jena.de

Abstract

The link between fractional and stochastic calculus established in part Iof this paper is investigated in more detail. We study a fractional integraloperator extending the Lebesgue–Stieltjes integral and introduce a relatedconcept of stochastic integral which is similar to the so–called forwardintegral in stochastic integration theory. The results are applied to ODEdriven by fractal functions and to anticipative SDE whose noise processespossess absolutely continuous generalized covariation processes. A surveyon this approach may be found in [21].

Mathematics Subject Classification: Primary 60H, Secondary26A42, 34A

0 Introduction

In part I we have introduced an extension of Lebesgue–Stieltjes integrals forintegrands and integrators of unbounded variation:

b∫a

f dg = (−1)αb∫

a

Dαa+ fa+(x)D1−α

b− gb−(x) dx+ f(a+)(g(b−)− g(a+)

)(1)

where Dαa+ and D1−α

b− are the left– and right–sided Weyl–Marchaud derivatives oforders α and 1− α, respectively, on the interval (a, b) ⊂ R and

fa+(x) := 1(a,b)(x)(f(x)− f(a+)

)gb−(x) := 1(a,b)(x)

(g(x)− g(b−)

)are the corrected functions continuously vanishing at a+ and b−, respectively.(The one-sided limits f(a+) and g(b−) are supposed to exist.)In the present paper we will continue to study this integral and a stochastic exten-sion. Then we will apply our notions to associated deterministic and stochasticdifferential equations.Section 1 is based on the definitions from part I. We recall the notion of the inte-gral and the corresponding function spaces. Theorem 1.2 provides an importantcontinuity property of the integral specified to Besov (or Slobodeckij) spaces oftype W α

2 .Section 2 deals with the associated integral operator

f →(·)∫

0

a(f, ϕ) dg

for fractal parameter functions ϕ from the same spaces and smooth transfor-mation functions a. Using Theorem 1.2 we show its continuity and under astronger smoothness condition on a the local contraction property. The higher–dimensional version is also formulated.Section 3 presents the classical change–of–variable formula for the integral in thecase where the fractional degree of differentiability of the integrator is greaterthan 1/2. Here the results of section 2 are used as an approximation tool.Section 4 contains the following extension of our integral:

b∫a

f dg := limε0

ε

1∫0

uε−1

b∫a

f(x)gb−(x+ u)− gb−(x)

udx du .

In particular, if the corresponding Riemann–Stieltjes sums converge uniformlythen the above limit exists and agrees with the Riemann–Stieltjes integral. (In a

2

classical paper of L. C. Young [23] uniform convergence of the sums was provedfor functions of finite p– and q–variations with 1/p+ 1/q > 1.) We further applythis approach to random functions, variable upper bounds of the integrals anduniform convergence in probability. Thus we obtain a modification of a stochasticintegral introduced by Russo and Vallois [16] which is more adapted to fractionalcalculus. For the special case of the Wiener process as integrator we formulaterather general conditions for existence of the integral in terms of Malliavin cal-culus. Then the integral may be interpreted as the trace corrected Skorohodintegral which has been considered by several authors under more restrictive as-sumptions.Section 5 provides a related concept of generalized quadratic variations and co-variations of stochastic processes. It is an extension of a notion of Russo andVallois [17]. We establish relationships to the above fractional calculus and provethe simple version of the Ito formula for such processes. Our approach admits tocalculate the bracket of a stochastic integral by means of that of the integratorunder rather general conditions (Theorem 5.4).Section 6 is concerned with differential equations driven by functions with frac-tal degree of differentiability greater than 1/2. Working within the Besov–typespaces mentioned above we are able to use the contraction theorem from section2 in order to prove existence and uniqueness of a local solution. It can be de-termined by means of Picard’s iteration method. In the one–dimensional case(or under certain algebraic conditions on the vector fields) the solutions may berepresented as smooth functions of the driving processes (or their iterated inte-grals). This turns out from the results of section 7. In particular, this provides apathwise solution procedure for a large class of stochastic processes like fractionalBrownian motion with time dependent Hurst exponents.Section 7 treats stochastic differential equations driven by processes with abso-lutely continuous generalized covariation processes. Applying the above stochas-tic integrals and the simple Ito formula we construct a local pathwise solution forthe case of commuting anticipative random vector fields. (The method of reduc-ing to classical (partial) differential equations has already been used by Doss [4]and Sussman [19] in terms of adapted Ito or Stratonovitch integration in a specialcase.) We also prove uniqueness in the class of all processes satisfying the generalIto transformation formula. Then we extend this approach to the case wherethe Lie algebra generated by the vector fields is nilpotent of rank p > 1. Sincewe do not require any kind of adaptedness we have to assume that the iteratedintegrals of order ≤ p of the driving processes satisfy the Ito formula. For thealgebraic part which is the same as in the adapted Stratonovitch approach werefer to Ikeda and Watanabe [7] and Yamato [22].

3

1 Generalized Stieltjes integrals in W α2

In part I we defined the integralb∫a

f dg according to (1) provided that fa+ ∈

Iαa+ (Lp) , g(a+) exists, gb− ∈ I1−αb− (Lq) for some 1/p+1/q ≤ 1 , 0 ≤ α ≤ 1, where

Lp = Lp(a, b) and Iαa+ (Lp) denotes the space of functions which are representableas Iαa+– (resp. Iαb−–) integral of some Lp–function on the interval (a, b). Forαp < 1 this integral agrees with

b∫a

f dg = (−1)αb∫

a

Dαa+ f(x)D1−α

b− gb−(x) dx (1′)

which is also determined for general f ∈ Iαa+ (Lp) with lim supxa

f(x) < ∞. (This

means that in this case we need no correction of f at the left endpoint.The sets Iαa+

(b−)

(Lp) become Banach spaces by the norms

‖f‖Iαa+

(b−)

(Lp) := ‖f‖Lp + ‖Dαa+

(b−)f‖Lp ∼ ‖Dα

a+

(b−)f‖Lp .

For αp < 1 the spaces Iαa+ (Lp) and Iαb− (Lp) agree up to norm equivalence. Simi-larly, for any −∞ ≤ a < x < y < b ≤ +∞ the restriction of f ∈ Iαa+ (Lp) to theinterval (x, y) belongs to Iαx+

(Lp(x, y)

)and the continuation of f ∈ Iαx+

(Lp(x, y)

)by zero beyond (x, y) is an element of Iαa+ (Lp). (This results from the Hardy–Littlewood inequality, cf. Samko, Kilbas and Marichev [18], chapter 13.) Thefunctions need not be continuous or bounded.

For αp > 1 we have embedding in a Holder space, i.e.,

Iαa+(b−)

(Lp) → Hα−1/p

and the functions vanish at a+ of order o((x − a)α−1/p

)(at b− of order o

((b −

x)α−1/p)). In this section we will specify to p = q = 2 and consider the following

Besov– (or Slobodeckij–) type spaces Wα2 (with modifications) given by the (semi)

4

norms

‖f‖Wα2

:=

( b∫a

b∫a

(f(x)− f(y)

)2|x− y|2α+1

dx dy

)1/2

‖f‖Wα2

:= ‖f‖L2 + ‖f‖Wα2

‖f‖Wα2,∞

:= ‖f‖L∞ + ‖f‖Wα2

‖f‖Wα2 (a+) :=

( b∫a

f(x)2

(x− a)2αdx

)1/2

+ ‖f‖Wα2

‖f‖Wα2 (b−) :=

( b∫a

f(x)2

(b− x)2αdx

)1/2

+ ‖f‖Wα2.

When restricting to some subinterval (x, y) ⊂ (a, b) we will use the notations

Wα2 (x, y) , W α

2 (x, y) , W α2,∞(x, y) , W α

2 (a+)(b−)

(x, y).

For (a, b) = R we will write Iαa+(b−)

= Iα± , Wα2 (R), etc. The above Besov spaces

are closely related to Iαa+(b−)

(L2). Below we will need the following relationships.

Suppose 0 < α < 1.

1.1 Theorem.

(i) ‖f‖Iαa+

(b−)

(L2)+‖f‖L∞ ∼ ‖f‖Wα2,∞

and ‖f‖Iαa+

(b−)

(L2) ≤ const(α) ‖1(a,b) f‖Wα2 (R)

if 0 < α < 1/2 .

(ii) Wα+δ2 (a+)

(b−)→ Iαa+

(b−)

(L2) , δ > 0 .

(iii) Iα+δa+

(b−)

(L2) → Wα2 , 0 < δ < 1− α (see Feyel, de LaPradelle [5], Theorem

27).

(iv) g ∈ Wα2 implies gy− ∈ Wα

2 (y−) (x, y) for any x ∈ [a, b] and Lebesgue almostall y ∈ (x, b) (similarly for x+).

Remark. The constants in the norm estimates in (i) and (ii) tend to infinity asα 1/2 and δ 0, respectively.

5

Proof. (i) follows from the norm equivalences

‖ · ‖Iα± (L2(R)) ∼ ‖ · ‖Wα2 (R)

(see Mazja and Nagel [13]),

‖1(a,b) f‖Iα+(L2(R)) ∼ ‖f‖Iαa+(L2 (a,b))

(cf. [18], 13.3),‖1(a,b) f‖Wα

2,∞(R) ∼ ‖f‖Wα2,∞(a,b)

and the inequality‖1(a,b) f‖L2 ≤ const ‖1(a,b) f‖Wα

2 (R).

(The last two facts follow from the definitions.)

(ii) ‖f‖Iαa+(L2) up to the summand( b∫a

f(x)2

(x−a)2α dx)1/2

(which is included into the

norm of the Besov-type space) and up to some constants does not exceed thelimit as ε 0 of( b∫

a

( x−ε∫a

f(x)− f(y)

(y − x)2α+1dy

)2

dx

)1/2

=

( b∫a

( x−ε∫a

f(x)− f(y)

(y − x)α+2δ

1

(y − x)1−2δdy

)2

dx

)1/2

≤ const(δ)

( b∫a

x−ε∫a

(f(x)− f(y)

)2(y − x)2(α+2δ)

1

(y − x)1−2δdy dx

)1/2

= const(δ)

( b∫a

x−ε∫a

(f(x)− f(y)

)2(y − x)2(α+δ)+1

dy dx

)1/2

≤ const(δ) ‖f‖Wα+δ2

.

(In the first estimation we have used the Cauchy–Schwarz inequality.) The prooffor b− is similar.(iv) Since

b∫a

b∫a

(g(u)− g(v)

)2|u− v|2α

du dv <∞

we infery∫x

y∫x

(g(u)− g(v)

)2|u− v|2α

du dv <∞ ,

6

i.e. the first part of the assertion, and

b∫x

y∫x

(g(u)− g(y)

)2(y − u)2α+1

du dy <∞

for any a ≤ x < y ≤ b. The last inequality yields

y∫x

(g(u)− g(y)

)2(y − u)2α+1

du <∞

for almost all y ∈ (x, b). Consequently, at these x we get

y∫x

(g(u)− g(y)

)2(y − u)2α

du ≤ (b− a)

y∫x

(g(u)− g(y)

)2(y − u)2α+1

du <∞

which proves the remaining part of the assertion.

We now turn to continuity properties of the integral (1) or (1′) as function of theupper and lower boundaries. They are the key for the main results of this paper.

1.2 Theorem. Suppose 0 < α < 1/2 and 0 < β < 1.

(i) If f ∈ Iαa+(L2) and gb− ∈ I1−αb− (L2) then the integral

b∫a

1(x,y)f dg in the sence

of (1′) exists for any a ≤ x < y ≤ b. Moreover,

b∫a

1(x,y)f dg =

y∫x

f dg

whenever the right–hand side is determined in the sense of (1′). (In general,we will use the left–hand side as definition for the right–hand side.)

(ii) If f and g fulfill the conditions of (i) and f is bounded, i.e. f ∈ Wα2,∞, then

x∫a

f dg and

b∫x

f dg

are continuous functions in x ∈ (a, b).

(iii)

max

(∥∥ (·)∫a

f dg∥∥Wβ

2,∞,∥∥ b∫

(·)

f dg∥∥Wβ

2,∞

)≤ const(α, β)

(‖f‖

Wmax(α,α+β−1/2)2,∞

‖gb−‖I1−αb− (L2) + ‖f‖L∞ ‖g‖Wβ2

)7

(iv) ∥∥ (·)∫a

f dg∥∥Wβ

2,∞≤ const (β)

∥∥f∥∥Wβ

2,∞

∥∥gb−∥∥Wβ2 (b−)

provided that β > 1/2.

Remark. The constants tend to infinity as α 1/2 in (iii) and as β 1/2 in(iv).

Proof. (i) Since 2α < 1 the restriction and continuation properties of thespace Iα(·)(L2) mentioned at the beginning of this section imply that 1(x,y)f ∈Iαa+(L2(a, b)

)and f ∈ Iαx+

(L2(x, y)

). Therefore both the integrals are deter-

mined. The equality follows from Theorem 2.5 (i) in part I.(ii) Theorem 2.5 (ii) from part I, i.e. additivity of the integral as function of theboundary for continuous g, yields

x∫a

f dg−x−δ∫a

f dg =

x∫x−δ

f dg =

b∫a

1(x−δ,x)f dg = (−1)αb∫

a

Dαa+(1(x−δ,x)f)(y)D1−α

b− gb−(y) dy .

Below we will show that

L2 − limδ0

Dαa+ (1(x−δ,x)f) ≡ 0.

Then we obtain from the Cauchy–Schwarz inequality that the last integral tends

to zero as δ 0, i.e. continuity of(·)∫a

f dg. (The proof for the lower boundary is

similar.)By definition,

Γ(1− α)∣∣Dα

a+ (1(x−δ,x)f)(y)∣∣ =

∣∣∣1(x−δ,x)(y)f(y)

(y − a)α

+ α

y∫a

1(x−δ,x)(y) f(y)− 1(x−δ,x)(z) f(z)

(y − z)α+1dz∣∣∣.

The first summand of the last sum tends to zero in L2 as δ 0, since f isbounded and α < 1/2. The second summand, say αSδ(y), does so by the followingarguments: First note that Sδ(y) = 0 if y < x− δ. For y > x− δ we get

Sδ(y) = 1(x−δ,x)(y)

( x−δ∫a

f(y)

(y − z)α+1dz +

y∫x−δ

f(y)− f(z)

(y − z)α+1dz

)

− 1[x,b)(y)

x∫x−δ

f(z)

(y − z)α+1dz .

8

The L2–limit as δ 0 of both the last summands is zero, because f is boundedand α < 1/2.(iii) In order to estimate the supremum norm we apply the Cauchy–Schwarzinequality and Theorem 1.1 (i) and obtain for any x ∈ (a, b]

∣∣∣ x∫a

f dg∣∣∣ =

∣∣∣ b∫a

1(a,x)f dg∣∣∣ =

∣∣∣ b∫a

Dαa+ (1(a,x)f)(x)D1−α

b− gb−(x) dx∣∣∣

≤ ‖1(a,x)f‖Iαa+(L2) ‖gb−‖I1−αb− (L2)

≤ const(α) ‖1(a,x)f‖Wα2,∞

‖gb−‖I1−αb− (L2)

≤ const(α) ‖f‖Wα2,∞

‖gb−‖I1−αb− (L2)

(for different constants here and in the sequel). For the last estimate we haveused once more that α < 1/2.We now turn to the W β

2 -seminorm: By the additivity property of the integral wefirst get

y∫a

f dg−x∫a

f dg =

b∫x

1(x,y)f dg =

b∫a

Dαx+ (1(x,y)fx+)(z)D1−α

b− gb−(z) dz+f(x)(g(y)−g(x)

)at all points x of continuity of f . Hence,

( b∫a

y∫a

( y∫a

f dg −x∫a

f dg)2

(y − x)2β+1dx dy

) 12

=

( b∫a

y∫a

(y − x)−2β−1

[ b∫x

Dαx+(1(x,y)fx+)(z)D1−α

b− gb−(z) dz + f(x)(g(y)− g(x)

)]2

dx dy

) 12

≤( b∫

a

y∫a

(y − x)−2β−1

( b∫x

Dαx+ (1(x,y)fx+)(z)D1−α

b− gb−(z) dz

)2

dx dy

) 12

+

( b∫a

y∫a

f(x)2 (g(y)− g(x))2

(y − x)2β+1dx dy

) 12

=: S1 + S2

The summand S2 does not exceed

‖f‖L∞ ‖g‖Wβ2

9

which corresponds to the second summand in the asserted estimate. For S1 weobtain

S1 ≤( b∫

a

y∫a

(y − x)−2β−1 ‖1(x,y)fx+‖2Iαx+(L2(x,b)) ‖gb−‖2

I1−αb− (L2(x,b))dx dy

) 12

.

Regarding‖1(x,y)fx+‖Iαx+(L2(x,b)) ≤ const(α) ‖1(x,y)f‖Wα

2 (R)

(cf. Theorem 1.1 (i)) and

‖gb−‖I1−αb− (L2(x,b)) ≤ ‖gb−‖I1−αb− (L2)

(which follows from the definitions) we infer

S1 ≤ const(α)

( b∫a

y∫a

(y − x)−2β−1 ‖1(x,y)fx+‖2Wα

2 (R)dx dy

) 12

‖gb−‖I1−αb− (L2) .

It remains to estimate the integral factor, say F , in the last product. Recall that

‖1(x,y)fx+‖2Wα

2 (R)=

∫ ∫ (1(x,y)fx+(u)− 1(x,y)fx+(v)

)2|u− v|2α+1

du dv

=

y∫x

y∫x

(f(u)− f(v)

)2|u− v|2α+1

du dv +

x∫−∞

y∫x

(f(u)− f(x)

)2(u− v)2α+1

du dv

+

y∫x

x∫−∞

(f(v)− f(x)

)2(v − u)2α+1

du dv +

∞∫y

y∫x

(f(v)− f(x)

)2(v − u)2α+1

du dv

=

y∫x

y∫x

(f(u)− f(v)

)2|u− v|2α+1

du dv + 2

x∫−∞

y∫x

(f(u)− f(x)

)2(u− v)2α+1

du dv

+

∞∫y

y∫x

(f(v)− f(x))2

(v − u)2α+1du dv

=: S3 + S4 + S5 .

We next can estimate the summands S4 and S5 by changing the order of integra-tion:

S4 ≤ const(α)

y∫x

(f(u)− f(x)

)2(u− x)2α

du,

S5 ≤ const(α)

y∫x

(f(u)− f(x)

)2(y − u)2α

du.

10

From this we infer

const(α)F ≤( b∫

a

y∫a

1

(y − x)2β+1

y∫x

y∫x

(f(u)− f(v)

)2|u− v|2α+1

du dv dx dy

) 12

+

( b∫a

y∫a

1

(y − x)2β+1

y∫x

(f(u)− f(x)

)2(u− x)2α

du dx dy

) 12

+

( b∫a

y∫a

1

(y − x)2β+1

y∫x

(f(u)− f(x))2

(y − u)2αdu dx dy

) 12

=: S6 + S7 + S8 .

Changing again the orders of integration we get

S6 =

( b∫a

b∫a

(f(u)− f(v)

)2|u− v|2α+1

b∫u∨v

u∧v∫a

1

(y − x)2β+1dx dy du dv

) 12

≤ const(β)

( b∫a

b∫a

(f(u)− f(v)

)2|u− v|2α+2β

dx dy

) 12

= const(β) ‖f‖Wα+β−1/22

.

Similarly,

S7 =

( b∫a

u∫a

(f(u)− f(x)

)2(u− x)2α

b∫u

1

(y − x)2β+1dy dx du

) 12

≤ const(β)

( b∫a

u∫a

(f(u)− f(x)

)2(u− x)2α+2β

dx du

) 12

= const(β) ‖f‖Wα+β−1/22

.

Finally,

S8 =

( b∫a

u∫a

(f(u)− f(x)

)2 b∫u

1

(y − x)2β+1

1

(y − u)2αdy dx du

) 12

.

The inner integral equals

b−u∫0

1

(y + u− x)2β+1

1

y2αdy =

1

(u− x)2α+2β

b−uu−x∫0

1

(z + 1)2β+1

1

z2αdz

≤ const(α, β)1

(u− x)2α+2β.

11

Hence,S8 ≤ const(α, β) ‖f‖

Wα+β−1/22

.

This completes the proof for(·)∫a

f dg. The arguments forb∫

(·)f dg are similar.

(iv) is a consequence of (iii): Choose there α such that 1− β < α < 1/2 and useTheorem 1.1 (ii) in order to estimate

‖gb−‖I1−αb− (L2) ≤ const(α, β) ‖gb−‖Wβ2 (b−) .

Remark. An analysis of the estimations in [18] and [13] leading to Theorem 1.1and the proof of the preceding theorem show that const(α, β) tends to infinity asα 1/2, and consequently const(β) does so as β 1/2.

2 An integral operator, continuity and contrac-

tion properties

From now on we will frequently use the ”time“ interval (0, T ) instead of (a, b)having in mind applications to related fractal–type (stochastic) differential equa-tions. Let us fix an integrator g with gT− ∈ W β

2 (T−) for some 1/2 < β < 1, a”parameter“ function ϕ ∈ W β

2,∞ and a transformation mapping a ∈ C1(R×R,R).

By the local Lipschitz property of a for any f ∈ W β2,∞ the function a

(f(·), ϕ(·)

)lies again in this space. Therefore Theorem 1.2 (iv) implies that the non–linearintegral operator

f → x0 +

(·)∫0

a(f, ϕ) dg

for fixed x0 ∈ R acts from W β2,∞ into itself.

Below we will prove continuity of this operator controling the norms (Theorem2.1). Moreover, in the case when both the partial derivatives of a are locallyLipschitz in the first argument we will infer a certain local contraction property(Theorem 2.2). This provides the key for solving related (anticipative stochastic)differential equations in section 6. As a second result we will derive in section 3an integral transformation formula using the norm estimates from Theorem 2.1together with the Lipschitz property of the linear operator

g → x0 +

(·)∫0

a(f, ϕ) dg

12

from W β2 (T−) into L∞.

We first consider the behaviour of the mapping

f → a(f, ϕ) .

For arbitrary functions f, h, ϕ ∈ Wα2,∞, where 0 < α < 1, denote the closed convex

hull of the set(f([0, T ]) ∪ h([0, T ])

)× ϕ([0, T ]) in R× R by K(f, h, ϕ). For any

compact K ⊂ R×R denote L0(a,K) := ‖ ∂a∂x1‖L∞(K) and let wi(a,K; ε), i = 1, 2,

be the moduli of continuity of ∂a∂xi

(x1, x2) on K with respect to the first argument.Li(a,K), i = 1, 2 denote the corresponding Lipschitz constants of the partialderivatives if they exist, i.e.,

wi(a,K; ε) ≤ Li(a,K) ε .

2.1 Proposition.

(i) For f, h, ϕ ∈ W α2,∞ and a ∈ C1 we have

‖a(f, ϕ)− a(h, ϕ)‖L∞ ≤ L0

(a,K(f, h, ϕ)

)‖f − h‖L∞

and

‖a(f, ϕ)− a(h, ϕ)‖Wα2≤ L0

(a,K(f, h, ϕ)

)‖f − h‖Wα

2

+ w1

(a,K(f, h, ϕ); ‖f − h‖L∞

)min

(‖f‖Wα

2, ‖h‖Wα

2

)+ w2

(a,K(f, h, ϕ); ‖f − h‖L∞

)‖ϕ‖Wα

2

(ii) If ∂a∂x1 and ∂a

∂x2 are locally Lipschitz with respect to the first argument thenin (ii) wi may be replaced by Li

(a,K(f, h, ϕ)

)‖f − h‖L∞ , i = 1, 2.

Proof. (i) The mean value theorem implies

‖a(f, ϕ)− a(h, ϕ)‖L∞ ≤ L0

(a,K(f, h, ϕ)

)‖f − h‖L∞ .

Futhermore,∣∣∣a(f(s), ϕ(s))− a(h(s), ϕ(s)

)− a(f(t), ϕ(t)

)+ a(h(t), ϕ(t)

)∣∣∣≤∣∣∣a(f(s), ϕ(s)

)− a(f(t), ϕ(s)

)− a(h(s), ϕ(s)

)+ a(h(t), ϕ(s)

)∣∣∣+∣∣∣a(f(t), ϕ(s)

)− a(f(t), ϕ(t)

)− a(h(t), ϕ(s)

)+ a(h(t), ϕ(t)

)∣∣∣=: S1 + S2 .

13

By the Leibniz rule we obtain for the first summand:

S1 =

∣∣∣∣1∫

0

∂a

∂x1

(λf(s) + (1− λ) f(t), ϕ(s)

)dλ(f(s)− f(t)

)

−1∫

0

∂a

∂x1

(λh(s) + (1− λ)h(t), ϕ(s)

)dλ(h(s)− h(t)

)∣∣∣∣≤∣∣∣∣

1∫0

∂a

∂x1

(λf(s) + (1− λ) f(t), ϕ(s)

)dλ∣∣f(s)− f(t)− h(s) + h(t)

∣∣+

∣∣∣∣1∫

0

(∂a

∂x1

(λh(s) + (1− λ)h(t), ϕ(s)

)− ∂a

∂x1

(λf(s) + (1− λ) f(t), ϕ(s)

))dλ

∣∣∣∣|h(s)− h(t)|

≤ L0

(a,K(f, h, ϕ)

)|f(s)− f(t)− h(s) + h(t)|

+ w1

(a,K(f, h, ϕ); ‖f − h‖L∞

)|h(s)− h(t)| .

Similarly, the second summand may be estimated by

S2 ≤ w2

(a,K(f, h, ϕ); ‖f − h‖L∞

)|ϕ(s)− ϕ(t)| .

14

Hence,∥∥a(f, ϕ)− a(h, ϕ)∥∥Wα

2

=

( T∫0

T∫0

(a(f(s), ϕ(s))− a(h(s), ϕ(s))− a(f(t), ϕ(t)) + a(h(t), ϕ(t))

)2|s− t|−(2α+1) ds dt

)1/2

≤ L0

(a,K(f, h, ϕ)

)‖f − h‖Wα

2+ w1

(a,K(f, h, ϕ); ‖f − h‖L∞

)( T∫

0

T∫0

(h(s)− h(t)

)2|s− t|2α+1

ds dt

)1/2

+ w2

(a,K(f, h, ϕ); ‖f − h‖L∞

)( T∫

0

T∫0

(ϕ(s)− ϕ(t)

)2|s− t|2α+1

ds dt

)1/2

= L0

(a,K(f, h, ϕ)

)‖f − h‖Wα

2

+ w1

(a,K(f, h, ϕ); ‖f − h‖L∞

)‖h‖Wα

2

+ w2

(a,K(f, h, ϕ); ‖f − h‖L∞

)‖ϕ‖Wα

2.

Since the roles of f and h may be exchanged the proof of (i) is completed.(ii) is an immediate consequence of (i).

Remark. An analysis of the proof shows that the function ϕ may also be chosenvector–valued with coordinate functions ϕ1, . . . , ϕk in W α

2,∞. If w2, . . . , wk+1 are

the moduli of continuity of ∂a∂yi

(x, y1, . . . , yk) , i = 1, . . . , k, as function in x then

w2(·) ‖ϕ‖(·) in (i) has to be replaced by

k∑i=1

wi+1

(a,K(f, h, ϕ); ‖f − h‖L∞

)‖ϕi‖Wα

2.

The following main estimation is an immediate consequence of Theorem 1.2,Proposition 2.1 and Theorem 1.1 (iii).

2.2 Theorem.

(i) Let 0 < α < 1/2 , 0 < β < 1 , f, h, ϕ ∈ Wmax(α,α+β−1/2)2,∞ , gT− ∈

15

I1−αT− (L2) , g ∈ W β

2 and a ∈ C1(R× R,R). Then we have

∥∥∥∥(·)∫

0

a(f, ϕ) dg −(·)∫

0

a(h, ϕ) dg

∥∥∥∥Wβ

2,∞

≤ const(α, β)

([L0

(a,K(f, h, ϕ)

)‖f − h‖

Wmax(α,α+β−1/2)2,∞

+ w1

(a,K(f, h, ϕ); ‖f − h‖L∞

)min

(‖f‖

Wmax(α,α+β−1/2)2

, ‖h‖W

max(α,α+β−1/2)2

)+ w2

(a,K(f, h, ϕ); ‖f − h‖L∞

)‖ϕ‖

Wmax(α,α+β−1/2)2

]‖gT−‖I1−αT− (L2) + L0

(a,K(f, h, ϕ)

)‖f − h‖L∞ ‖g‖Wβ

2

)(ii) ∥∥∥∥

(·)∫t0

a(f, ϕ) dg −(·)∫t0

a(h, ϕ) dg

∥∥∥∥Wβ

2,∞(t0,t)

≤ const(α, β)[L0

(a,K(f, h, ϕ)

)‖f − h‖Wβ

2,∞(t0,t)

+ L1

(a,K(f, h, ϕ)

)‖f − h‖L∞(t0,t) min

(‖f‖Wβ

2 (t0,t), ‖h‖Wβ

2 (t0,t)

)+ L2

(a,K(f, h, ϕ)

)‖f − h‖L∞(t0,t) ‖ϕ‖Wβ

2 (t0,t)

]‖gt−‖Wβ

2 (t0,t)

for any 0 ≤ t0 < T and almost all t0 < t ≤ T provided that 1/2 < β <

1 , f, h, ϕ ∈ W β2,∞ , g ∈ W

β2 , a ∈ C1 and the partial derivatives ∂a

∂x1 and ∂a∂x2

are locally Lipschitz with respect to the first argument.

As a corollary we now will formulate a local contraction property of the integraloperator. Denote W β

2,∞(t0, t;x0, 1) the set of functions f on (t0, t) with f(t0+) =x0 and ‖ft0+‖Wβ

2,∞(t0,t)≤ 1.

2.3 Theorem. Let x0, y0 ∈ R and β, g, a be as in Theorem 2.2 (ii). Then forany t0 ∈ (0, T ) and c > 0 there exists some t ∈ (t0, T ) such that for any ϕ ∈W β

2,∞(t0, t; y0, 1) the integral operator A with

Af := x0 +

(·)∫t0

a(f, ϕ) dg

maps W β2,∞(t0, t;x0, 1) into itself and we have

‖Af − Ah‖Wβ2,∞(t0,t)

≤ c‖f − h‖Wβ2,∞(t0,t)

for all f, h ∈ W β2,∞(t0, t;x0, 1).

16

Proof. First note that by Theorem 1.2 (ii) we get

limtt0

(x0 +

t∫t0

a(f, ϕ) dg)

= x0

for any f, ϕ ∈ W β2,∞(t0, t). Further, Theorem 2.2 (ii) implies for any f, h, ϕ as in

the assertion and almost all t > t0∥∥∥∥(·)∫t0

a(f, ϕ) dg −(·)∫t0

a(h, ϕ) dg

∥∥∥∥Wβ

2,∞(t0,t)

≤ const‖f − h‖Wβ2,∞(t0,t)

‖gt−‖Wβ2 (t−)(t0,t)

.

Below we will show that

limk→∞

‖gtk−‖Wβ2 (tk−)(t0,tk)

= 0

for some sequence tk t0. Hence, for large k we get

const‖gtk−‖Wβ2 (tk−)(t0,tk)

≤ c

which leads to the asserted estimation. In order to prove that the norm of theimages does not exceed 1 we use Theorem 1.2 (iv) and obtain for almost all t > t0

‖Aft0+‖Wβ2,∞(t0,t)

≤ const‖a(f, ϕ)‖Wβ2,∞(t0,t)

‖gt−‖Wβ2 (t−)(t0,t)

.

For f, ϕ as before the first norm on the right–hand side is uniformly boundedand in the second norm we may replace t by the sequence tk mentioned above inorder to make it arbitrarily small.

Thus, it remains to show existence of such a sequence tk t0. Recall that

‖gt−‖Wβ2 (t−)(t0,t)

=

( t∫t0

(g(s)− g(t)

)2(t− s)2β

ds

)1/2

+

( t∫t0

t∫t0

(g(s)− g(r)

)2|s− r|2β+1

ds dr

)1/2

.

The second summand goes to zero as t t0, because it is finite for (t0, t) = (0, T ).In order to prove

limk→∞

tk∫t0

(g(tk)− g(s)

)2(tk − s)2β

ds = 0

for some sequence tk t0 we consider for 0 < ∆ < 1 the auxiliary randomvariables

X∆(u) :=

t0+∆u∫t0

(g(t0 + ∆u)− g(s)

)2(t0 + ∆u− s)2β

ds

17

with respect to the normalized Lebesgue measure on the interval (0, T − t0).Below we will show that the mean value of X∆ tends to zero as ∆ 0. Thenthere exists a sequence ∆k 0 such that

limk→∞

X∆k(u) = 0

for almost all u and we may choose tk := t0 + ∆ku for any such u.The expectation of X∆ equals

1

T − t0

T−t0∫0

t0+∆u∫t0

(g(t0 + ∆u)− g(s)

)2(t0 + ∆u− s)2β

ds du

=

t0+∆(T−t0)∫t0

r∫t0

(g(r)− g(s)

)2∆(T − t0)(r − s)2β

ds dr

≤t0+∆(T−t0)∫

t0

r∫t0

(g(r)− g(s)

)2(r − s)2β+1

ds dr

which tends to zero as ∆ 0 by the above arguments for the second summand.

The following higher–dimensional contraction theorem is a straightforward ex-tension.

2.4 Theorem. The statement of Theorem 2.3 remains valid if x0 ∈ Rn, y0 ∈Rk, (gj) ∈ W β

2 , aj ∈ C1(Rn × Rk, Rn) with partial derivatives being locally Lips-chitz in the first n arguments, j = 1, . . . , l, ϕ takes values in Rk and f and h inRn with coordinate functions as before and the operator is given by

Af := x0 +l∑

j=1

(·)∫t0

aj(f, ϕ) dgj

with coordinatewise definition of the integrals.

3 Integral transformation formulae

The well–known change–of–variable formula for integration of smooth functionsremains valid for our fractal–type integral provided the integrator has fractionalderivatives of order greater than 1/2. (This condition makes the generalizedquadratic variation zero, cf. section 5.)

18

3.1 Theorem. Let 0 < α < 1/2 , f ∈ Iα0+(L2) be bounded (i.e., f ∈ W α2,∞),

gT− ∈ I1−αT− (L2) and

h(t) := h(0) +

t∫0

f dg , t ∈ (0, T ] .

Then we get for any C1–function F (x, t) on R× [0, T ] such that ∂F∂x∈ C1 and for

any 0 ≤ t0 < t ≤ T :

F(h(t), t

)− F

(h(t0), t0

)=

t∫t0

∂F

∂x

(h(s), s

)f(s) dg(s) +

t∫t0

∂F

∂t

(h(s), s

)ds .

Remark. Under the more restrictive assumption that g is Holder continuous oforder µ > 1/2 and f is Holder continuous of order λ > 1−µ we have proved thisformula in part I (Theorems 4.3.1 and 4.4.2) for F ∈ C1 such that ∂F

∂x

(h(·), ·

)is

Holder continuous of order λ, in particular for ∂F∂x

∈ C1. The smoothness of ∂F∂x

seems to be the price that we have to pay in order to treat functions of low orderof Holder continuity.

Proof of Theorem 3.1. By the usual kernel smoothing procedure we may approx-imate gT− in the I1−α

T− (L2)–norm by smooth functions gn as n→∞ (cf. part I).Denote

hn(t) := h(0) +

t∫0

f dgn.

Then Theorem 1.2 (iii) for β = α and Theorem 1.1 (iii) for α+ δ = 1− α imply

limn→∞

‖hn − h‖Wα2,∞

= 0 .

From classical calculus we know that

F(hn(t), t

)− F

(hn(t0), t0

)=

t∫t0

∂F

∂x

(hn(s), s

)f(s) dgn(s) +

t∫t0

∂F

∂t

(hn(s), s

)ds.

Since F is continuous the left–hand side converges to F(h(t), t

)−F

(h(t0), t0

)as

n→∞. Similarly,

limn→∞

t∫t0

∂F

∂t

(hn(s), s

)ds =

t∫t0

∂F

∂t

(h(s), s

)ds .

19

It remains to show that the first integral tends to

t∫t0

∂F

∂x

(h(s), s

)f(s) dg(s) .

The difference may be estimated as follows:

∣∣∣∣t∫

t0

∂F

∂x

(hn(s), s

)f(s) dgn(s)−

t∫t0

∂F

∂x

(h(s), s

)f(s) dg(s)

∣∣∣∣≤∣∣∣∣

t∫t0

∂F

∂x

(hn(s), s

)f(s) dgn(s)−

t∫t0

∂F

∂x

(h(s), s

)f(s) dgn(s)

∣∣∣∣+

∣∣∣∣t∫

t0

∂F

∂x

(h(s), s

)f(s) d

(gn(s)− g(s)

)∣∣∣∣ .The last summand vainishes asymptotically by the same arguments as above for‖hn−h‖Wα

2,∞. In order to prove the same property for the first summand we apply

the first part of the proof of Theorem 1.2 (iii) and the remark to Proposition 2.1(ii) to the functions a(x, y1, y2) := ∂F

∂x(x, y1) · y2, if y1 ∈ [0, T ] , ϕ1 := identity,

ϕ2 := 1(t0,t)f and obtain the upper estimate

const ‖hn − h‖Wα2,∞

‖gn‖I1−αT− (L2) .

Since the last factor is uniformly bounded this tends to zero as n→∞.

The higher–dimensional version is again a straight–forward extension:

3.2 Theorem. For i = 1, . . . ,m let 0 < αi < 1/2 , f i ∈ Iαi0+(L2) be bounded,giT− ∈ I

1−αiT− (L2),

hi(t) := hi(0) +

t∫0

f i dgi

and h := (h1, . . . , hm). Then we have for any C1–mapping F : Rm × R → Rn

such that ∂F∂xi

∈ C1, i = 1, . . . ,m, and any 0 ≤ to < t ≤ T

F(h(t), t

)− F

(h(t0), t0

)=

m∑i=1

t∫t0

∂F

∂xi(h(s), s

)f i(s) dgi(s) +

t∫t0

∂F

∂t

(h(s), s

)ds .

20

4 An extension of the integral and its stochastic

version

In part I we have introduced a new notion of stochastic integrals which includesthe Ito integral and the trace corrected Skorohod integral for a related type ofintegrands. Here we will extend these ideas and connect them with the approachof Russo and Vallois [16], [17] for a general anticipative situation.In order to motivate our notion we start with two relationships for the integral(1) or (1′):

4.1 Lemma. If f and g are as in definitions (1) or (1′) then the integral maybe approximated as follows

b∫a

f dg = limε0

b∫a

Iεa+ f dg .

Proof. By definition (1),

limε0

b∫a

Iεa+ f dg = limε0

(−1)αb∫

a

Dαa+ Iεa+ fa+(x) D1−α

b− gb−(x) dx

+ f(a+)(g(b−)− g(a+)

).

Since Dαa+ Iεa+ fa+ = Iεa+ Dα

a+ fa+ and

(Lp)− limε0

Iεa+ Dεa+ fa+ = Dα

a+ fa+

(cf. part I), the assertion follows from the Holder inequality. (The argumentsunder the conditions of (1′) are similar.)

The integralsb∫a

Iεa+ f dg are also determined for f and g of slightly lower order of

differentiability. In order to treat the limit case we introduce the function spaces

Iβ−a+(b−)

(Lp) :=⋂α<β

Iαa+(b−)

(Lp)

and similarly W β−2,∞, etc. The following representation leads to a relationship

between our approach and that of Russo and Vallois.

4.2 Lemma. Suppose 1p

+ 1q≤ 1 and ε > 0.

21

(i)

b∫a

Iεa+ f dg =1

Γ(ε)limδ0

∞∫δ

uε−1

b∫a

f(s)gb−(s+ u)− gb−(u)

uds du

provided that f ∈ Iα−εa+ (Lp) , gb− ∈ I1−αb− (Lq) with αp 6= 1.

(ii) The equation in (i) holds true if f ∈ Iβ−a+ (Lp) , gb− ∈ I(1−β)−b− (Lq) for some

0 < β < 1.

Proof. (i) In order to transform the right–hand side into the left–hand side wewill use the special composition formula

Iα−εa+ Dα−εa+ f = f

together with the integration–by–part rule

b∫a

Iα−εa+ ϕ(s) ψ(s) ds = (−1)α−εb∫

a

ϕ(s) Iα−εb− ψ(s) ds

for ϕ := Dα−εa+ f and ψ(s) := gb−(s + u) − gb−(s) , u > 0 (cf. part I). Then we

obtain

b∫a

f(s)(gb−(s+ u)− gb−(s)

)ds =

b∫a

Iα−εa+ Dα−εa+ f(s)

(gb−(s+ u)− gb−(s)

)ds

= (−1)α−εb∫

a

Dα−εa+ f(s)

[Iα−εb− gb−(s+ u)− Iα−εb− gb−(s)

]ds

=: Φ(u)

Note that the right–hand side of the assertion agrees with

limδ0

∞∫δ

1

Γ(ε)uε−2 Φ(u) du .

According to Fubini’s theorem we have

∞∫δ

1

Γ(ε)uε−2 Φ(u) du = (−1)α−ε

b∫a

Dα−εa+ f(s)

∞∫δ

1

Γ(ε)

Iα−εb− gb−(s+ u)− Iα−εb− gb−(s)

u1−ε+1du ds

and the inner integral converges in Lq as δ ↓ 0 to

(−1)εD1−εb− Iα−εb− gb−(s) = (−1)εD1−α

b− gb−(s) .

22

Using Dα−εa+ fa+ = Dα

a+ Iεa+ fa+ and the Holder inequality we infer

limδ↓0

1

Γ(ε)

∞∫δ

uε−2 Φ(u) du = (−1)αb∫

a

Dαa+ I

εa+ f(s)D1−α

b− gb−(s) ds

=

b∫a

Iεa+ f(s) dg(s)

according to definition (1′) if αp < 1 and definition (1) if αp > 1, since in thelatter case Iεa+ f(a+) = 0.

(ii) is a consequence of (i) if we replace there α by β + ∆ε for some 0 < ∆ < 1such that (β + ∆ε) p 6= 1.

Remark. In the stochastic calculus below we will consider β = 1/2 and p = q = 2.An analysis of the last proof and Theorem 1.1 (i) show that in this case the upperboundary b in the integrals may be replaced by any t ∈ (a, b] and convergenceholds uniformly in t.

Lemmas 4.1 and 4.2 suggest the following extension of our integral (1) or (1′)(regarding that Γ(ε) is equivalent to ε−1 as ε 0):

Definition.

b∫a

f dg := limε0

ε limδ0

1∫δ

uε−1

b∫a

f(s)gb−(s+ u)− gb−(s)

uds du (2)

whenever the right–hand side exists.

Note, that the kernel ε uε−1 acts as the δ–function as ε 0. If

lim∆0

b∫a

f(s)gb−(s+ ∆)− gb−(s)

∆ds

exists then the integral (2) is determined and agrees with this limit. The notionof general Riemann–Stieltjes integral in the sense of uniform convergence withrespect to a “random” starting point may be considered as special case:

4.3 Proposition. If the Riemann–Stieltjes sums

S∆(x) :=

[ b−a∆ ]−1∑k=0

f(x+ k∆)(g(x+ (k + 1)∆

)− g(x+ k∆)

)

23

converge uniformly in x ∈ (a, a+ ∆) to a limit (R− S)b∫a

f dg as ∆ 0 then we

have

(R− S)

b∫a

f dg = lim∆0

b∫a

f(s)gb−(s+ ∆)− gb−(s)

∆ds

provided that the Lebesgue integrals on the right–hand side exist.

Remark. In Young [23] such a convergence has been proved for f and g beingof finite p– and q–variations, where 1

p+ 1

q> 1. We conjecture that this may

be extended to a notion of generalized p–variation (see section 5 for p = 2) andconvergence in the sense of (2).

Proof of Proposition 4.3. By the assumption we get

lim∆0

1

∆

∆∫0

S∆(x) dx = (R− S)

b∫a

f dg

and in the definition of S∆ the function g may be replaced by gb− without changingthe limit. Using

∆∫0

S∆(x) dx =∑k

∆∫0

f(x+ k∆)(gb−(x+ (k + 1)∆

)− gb−(x+ k∆)

)dx

=∑k

(k+1)∆∫k∆

f(s)(gb−(s+ ∆)− gb−(s)

)ds

=

b∫a

f(s)(gb−(s+ ∆)− gb−(s)

)ds

we obtain the assertion.

We now will apply this approach to stochastic processes on the interval [0, T ].The integrals as functions of the upper boundary t will again be considered asrandom processes and for limits we will use uniform convergence in probabilityfor t ∈ [0, T ] briefly lim

(ucp), or convergence in the mean square denoted by l.i.m.

Suppose that Y is a caglad (left continuous with right limits) process and Z acadlag (right continuous with left limits) process on [0, T ].

Definition.( t−∫0+

Y dZ =

) t∫0

Y dZ := limε0

ε

1∫0

uε−1

t∫0

Y (s)Zt−(s+ u)− Zt−(s)

uds du (3)

24

whenever the right–hand side is determined, where lim stands for one of the

stochastic limits above and1∫0

for limδ0

1∫δ

with probability 1.

If the right–hand side exists for lim(ucp)

as well as for l.i.m. then the resulting

processes are stochastically equivalent.

We immediately obtain the following sample path property of our integral (3) ifit is determined via (ucp): The process

X(t) :=

t+∫0

Y dZ

is cadlag and X(t) − X(t−) = Y (t)(Z(t) − Z(t−)

). Moreover, continuity of Z

implies that of X.

Remark. Russo and Vallois use (up to replacing Y (s) by Y (s+)) the followingdefinition of stochastic (forward) integral:

t∫0

Y dZ := limu0

(ucp)

t∫0

Y (s)Zt−(s+ u)− Zt−(s)

uds (3′)

i.e., a limit procedure without averaging. Because of convergence in probability,in general, the integrals in (3′) and (3) seem to be different. However, if we as-sume in (3′) convergence in the mean squared (for L2–processes on Ω × [0, T ])or convergence with probability 1 then this implies the corresponding conver-gence in (3) and the integrals agree. Because of the decomposition theorem forsemimartingales this may be applied to the classical situation of Ito calculus:

4.4 Proposition. If Z is a semimartingale and Y an adapted caglad process thenthe integrals in (3) and (3′) are determined in the (ucp)–sense and agree with theusual Ito–integral

(I)

t−∫0+

Y dZ .

A proof for (3′) and the Ito–integral is given in [17], Proposition 1.1. Since ituses stopping times and convergence in the mean square it may immediately beadapted to the “average” convergence in (3).

As a further special case we will consider the following extension of the tracecorrected Skorohod integral known from Malliavin calculus. Let Z := W be theWiener process and X ∈ L1,2. Then 1(0,t)X is Skorohod integrable for any t ∈

25

(0, 1). As usual we write δ(1(0,t)X) for the Skorohod integral. Moreover, the

corresponding derivative DsX(t) satisfies E1∫0

1∫0

DsX(t)2 ds dt < ∞. (For more

details see, e.g. [14], [15].) The next result extends Theorem 5.3.7 in part I aswell as the corresponding results in [16].

4.5 Theorem. If for ε→ 0,

ε

1∫0

uε−1

(·)∫0

1

u

s+u∫s

DrXs dr ds du

converges in L2 on Ω× [0, 1] to a random process denoted by Tr(DX)+ then

ε

1∫0

uε−1

(·)∫0

X(s)W (s+ u)−W (s)

uds du

converges in the same sense to

(·)∫0

XdW = δ(1(0,·)X) + Tr(DX)+(·) .

Proof. (This short version was suggested by a referee.)Using the rules of Malliavin calculus we get

1

u

t∫0

X(s)(W (s+ u)−W (s)

)ds =

1

u

t∫0

s+u∫s

Xs dWr ds+1

u

t∫0

s+u∫s

DrXs dr ds .

The averages of the second summand converge to Tr(DX)+(t) by assumption.According to the Fubini theorem for the Skorohod integral (which follows fromthe definition of δ as dual operator to the derivative) we obtain for the firstsummand the expression

t∫0

1

u

(t−u)∧r∫0∨(r−u)

X(s) ds dW (r) .

Straigtforward calculations show that for u 0 the process under this Skorohodintegral is asymptotically L1,2-equivalent to

Yu(r) :=1

u

r∫r−u∨0

X(s)ds

26

which converges in L1,2 as u 0 to X(r). This implies the L2–convergence ofthe above integral to the Skorohod integral of X. The corresponding convergenceof the averages w.r.t. u is a consequence.

Remark. Note that the type of averaging in the limit does not play a role as longas it is assumed for the trace Tr(D)+. In a forthcoming paper we will show howother kinds of averaging known from analysis fit into our model.The resulting integral is a forward integral which, in general, does not agree withthe Stratonovich integral, where the symmetric trace has to be added to theSkorohod integral instead of Tr(DX)+.Under the sharper trace condition

l. i.m.u0

E1∫

0

(1

u

s+u∫s

DrX(s) dr −Ds+X(s)

)2

ds = 0

one obtains the result of Russo and Vallois [16]

l. i.m.u0

t∫0

X(s)W (s+ u)−W (s)

udu = δ(1(0,t)X) +

t∫0

Ds+X(s) ds .

The pointwise trace condition in Asch and Potthoff [2] also implies that of The-orem 4.5 and the right–hand side agrees with their definition of the integral.

5 Processes with generalized quadratic varia-

tion and Ito formula

Let D be the set of cadlag functions on [0, T ].

Definition. A process Z ∈ D admits a generalized quadratic variation process(bracket) [Z](t) if

[Z](t) := limε0

(ucp)

ε

1∫0

uε−1

t∫0

1

u

(Zt−(s+ u)−Zt−(s)

)2ds du+

(Z(t)−Z(t−)

)2(4)

exists. The covariation process [Y, Z] of Y, Z ∈ D with generalized brackets isdefined similarly, where ( . . . )2 is replaced by the corresponding product.

Remark. [Y, Z](t) if exists is a cadlag process of bounded variation and [Z](t)is non–decreasing. Moreover, [Y, Z](t) − [Y, Z](t−) =

(Y (t) − Y (t−)

)(Z(t) −

Z(t−)). (Russo and Vallois [17] use again limits without averaging.) Note that

semimartingales or, more generally, Dirichlet processes are special examples andthe bracket agrees with that defined in the corresponding theory. In particular, wemay consider fractional Brownian motion BH with Hurst exponent 1/2 < H < 1as a Dirichlet process with [BH ] ≡ 0.

27

An immediate consequence of the definition of [Z] is the following.

5.1 Proposition. Any Z ∈ D admitting a generalized bracket lies with probability1 in the function space W

1/2−2,∞ .

(From now on we will often omit the phrase “ with probability 1” if it is clearfrom the context.)

Most of the properties obtained by Russo and Vallois for their generalized bracketremain valid for our notion:

5.2 Proposition. (Cf. [17], Proposition 1.2)Let Y, Z ∈ D be processes with generalized bracket [Y, Z] and define [Y, Z]ε by

[Y, Z]ε(t) = ε

1∫0

uε−1

t∫0

1

u

(Yt−(s+ u)− Yt−(s)

)(Zt−(s+ u)− Zt−(s)

)ds du

+(Y (t)− Y (t−)

)(Z(t)− Z(t−)

).

Then we have

limε0

(ucp)

t∫0

X d[Y, Z]ε =

t∫0

X d[Y, Z]

for any caglad process X on [0, T ].

5.3 Proposition. (Cf. [17], Proposition 2.1)Let Y, Z be continuous processes with generalized brackets [Y ], [Z] and [Y, Z] andF,G be random C1–functions on R. Then F (Y ) and G(Z) admit a mutual bracketgiven by [

F (Y ), G(Z)]

=

(·)∫0

F ′(Y )G′(Z) d[Y, Z] .

5.4 Theorem. Suppose that Z is continuous, admits a generalized bracket andX is representable as

X(t) =

t∫0

Y dZ

for some caglad process Y .

(i) If Y ∈ W β2,∞ with β > 1/2 then X admits a generalized bracket given by

[X](t) =

t∫0

Y 2 d[Z] .

28

(ii) The equation in (i) remains valid if Y ∈ W 1/2−2,∞ and the stochastic integral

converges in the strong sense that

limε0

supγ>0

1

Γ(γ)‖Xε −X‖2

W1/2−γ/22,∞

= 0

where

Xε(t) =1

Γ(ε)limδ0

∞∫δ

uε−1

t∫0

Y (s)1

u

(Zt−(s+ u)− Zt−(s)

)ds du

Remark. Since Z ∈ W1/2−2,∞ the stochastic integral in (i) may pathwise be inter-

preted in the sense of definition (1). Recall that Y is here continuous.

Proof. (i) The equalities

X(s+ u)−X(s) =

s+u∫s

Y dZ and X(t−)−X(s) =

t∫s

Y dZ,

imply Xt−(s+ u)−Xt−(s) =s+u∫s

Y dZt−.

Hence,

ε

1∫0

uε−2

t∫0

(Xt−(s+ u)−Xt−(s)

)2ds du = ε

1∫0

uε−2

t∫0

( s+u∫s

Y dZt−

)2

ds du

= ε

1∫0

uε−1 1

u

t∫0

( s+u∫s

Y (s) dZt−(r) +

s+u∫s

(Y (r)− Y (s)) dZt−(r)

)2

ds du .

Below we will show that

limu0

1

u

t∫0

( s+u∫s

(Y (r)− Y (s)

)dZt−(r)

)2

ds = 0

uniformly in t with probability 1. Therefore the limit as ε 0 of the aboveexpression equals

limε0

(ucp)

ε

1∫0

uε−2

t∫0

Y (s)2(Zt−(s+ u)− Zt−(s)

)2ds du =

t∫0

Y 2 d[Z]

29

according to Proposition 5.2 which leads to the asserted equality.In order to complete the proof of (i) we choose an arbitrary α ∈ (1/2, β) andestimate using the Cauchy–Schwarz inequality and Theorem 1.1 as follows:(

1

u

t∫0

( s+u∫s

(Y (r)− Y (s)) dZt−(r)

)2

ds

)1/2

=

(1

u

(t−u)+∫0

( s+u∫s

Dαs+ Ys+(r) D1−α

(s+u)− Z(s+u)−(r) dr

)2

ds

+1

u

t∫(t−u)+

( t∫s

Dαs+ Ys+(r) D1−α

t− Zt−(r) dr

)2

ds

)1/2

≤ const

(1

u

t∫0

(s+u)∧t∫s

(Dαs+ Ys+(r))2 dr ds

)1/2

≤ const

(1

u

t∫0

(s+u)∧t∫s

(Y (r)− Y (s))2

(r − s)2βdr ds

)1/2

+ const

(1

u

t∫0

(s+u)∧t∫s

(s+u)∧t∫s

(Y (r)− Y (v))2

|r − v|2β+1dr dv ds

)1/2

=: const S1 + const S2 .

For S1 we get

S21 =

t∫0

(s+u)∧t∫s

(Y (r)− Y (s)

)2u(r − s)2β

dr ds ≤T∫

0

(s+u)∧T∫s

(Y (r)− Y (s)

)2(r − s)2β+1

dr ds

which goes to zero as u 0 since

T∫0

T∫0

(Y (r)− Y (s)

)2|r − s|2β+1

dr ds <∞ .

Similarly,

S22 =

1

u

t∫0

(s+u)∧t∫s

(s+u)∧t∫s

(Y (r)− Y (v)

)2|r − v|2β+1

dr dv ds ≤ 2

T∫0

(v+u)∧T∫v

(Y (r)− Y (v)

)2|r − v|2β+1

dr dv

30

which is twice the estimator of S21 .

(ii) In view of Lemma 4.2 (i) we have

Xε(t) =

t∫0

Iε0+ Y dZ .

The processes Iε0+ Y and Z satisfy the conditions of (i) (cf. Theorem 1.1). Con-sequently,

[Xε](t) =

t∫0

(Iε0+ Y )2 d[Z] .

Since Y is bounded and left continuous and d[Z] is a bounded measure, the last

integrals converge uniformly in t tot∫

0

Y 2 d[Z] with probability 1. On the other

hand, by definition of the bracket and continuity of Z,

[Xε](t) = limγ0

(ucp)

1

Γ(γ)‖Xε‖2

W1/2−γ/22 (0,t)

.

The uniform convergence of 1Γ(γ)

‖Xε − X‖2

W1/2−γ/22

to zero as ε 0 admits to

change the limits in ε and γ, so that

limε0

(ucp)

[Xε] = limγ0

ucp)

1

Γ(γ)limε0

(ucp)

‖Xε‖2

W1/2−γ/22 (0,·)

= limγ0

(ucp)

1

Γ(γ)‖X‖2

W1/2−γ/22 (0,·)

.

Again by definition and continuity of X the right–hand side agrees with [X]. (Wealways use that (ucp) is equivalent to uniform convergence with probability 1 ofsubsequences.) Combining this with the above convergence of the left–hand sidewe obtain the assertion.

Similarly as in [17] one can prove the following Ito–type formula for the changeof variables. (The ideas go back to Follmer [6], where the Taylor formula andRiemann sums approximation is used. Russo’s and Vallois’ [17] approach is basedon the integral representation of the remainder in the Taylor expansion. Withslight modifications this may be transformed to average convergence.)

5.5 Theorem. Let Z be a continuous process with generalized bracket [Z]. Thenwe get for any random C1–function F (x, t) on R× [0, T ] with continuous ∂2F

∂x2 and

31

0 ≤ t0 , t ≤ T

F(Z(t), t

)− F

(Z(t0), t0

)=

t∫t0

∂F

∂x

(Z(s), s

)dZ(s) +

t∫t0

∂F

∂t

(Z(s), s

)ds

+1

2

t∫t0

∂2F

∂x2

(Z(s), s

)d[Z](s)

and the stochastic integral is determined by (ucp).

More generally, as an analogue to the classical situation in Ito calculus for semi-martingales we define the following for Z and F as in Theorem 5.5:The process X with

X(t) =

t∫0

Y dZ

for some caglad process Y satisfies the general Ito formula if

F(X(t), t

)− F

(X(t0), t0

)=

t∫t0

∂F

∂x

(X(s), s

)Y (s) dZ(s) +

t∫t0

∂F

∂t

(X(s), s

)ds

+1

2

t∫t0

∂2F

∂x2

(X(s), s

)Y (s)2 d[Z](s).

(5)

Remark.

1) Under the conditions of Theorem 5.4 we obtain from Theorem 5.5

F(X(t), t

)− F

(X(t0), t

)=

t∫t0

∂F

∂x

(X(s), s

)dX(s) +

t∫t0

∂F

∂t

(X(s), s

)ds

+1

2

t∫t0

∂2F

∂x2

(X(s), s

)Y (s)2 d[Z](s) .

Thus, the validity of the general Ito formula in this case is equivalent to

t∫t0

∂F

∂x

(X(s), s

)dX(s) =

t∫t0

∂F

∂x

(X(s), s

)Y (s) dZ(s) .

In the case of deterministic F it is well–known for continuous semimartin-gales and for anticipative integrals X with respect to Z = W under certainconditions. (For the latter cf. [2], [1] and their references.)

32

2) If Y ∈ W 1/2−2,∞ and Z ∈ W β

2 (T−) for some β > 1/2 we may apply the resultsof section 3 to the sample paths of the processes and Theorem 3.1 providesthe general Ito formula with [Z] ≡ 0.

3) We will adopt the general Ito formula (5) and its multidimensional extensionas a main calculation rule in the stochastic calculus based on definitions (3)and (4). In particular, solutions of SDE will be seeked only in the class ofprocesses satisfying this condition.

The higher–dimensional version of Theorem 5.5 reads as follows.

5.6 Theorem. Let Z = (Z1, . . . , Zp) be a continuous Rp–valued process admittinggeneralized brackets [Zj, Zk] and F be a random element of C1(Rp × [0, T ],Rn)with continuous partial derivatives ∂2F

∂xj ∂xk, 1 ≤ j, k ≤ p. Then we have

F(Z(t), t

)− F

(Z(t0), t0

)=

p∑j=1

t∫t0

∂F

∂xj(Z(s), s

)dZj(s) +

t∫t0

∂F

∂t

(Z(s), s

)ds

+1

2

p∑j,k=1

t∫t0

∂2F

∂xj ∂xk(Z(s), s

)d[Zj, Zk](s) .

(In the Taylor expansion techniques behind the stochastic integrals can only bedetermined in the sense of (ucp) of the sum of the approximating integrals.)

6 Differential equations driven by fractal func-

tions of order greater than 1/2

In [9] the one–dimensional differential equation

dx(t) = a(x(t), t

)dz(t) + b

(x(t), t

)dt

x(t0) = x0

is investigated, where the fractal noise function z is of Holder continuity of ordergreater than 1/2 and a and b possess certain smoothness properties. This equationbecomes precise via integration:

x(t) = x0 +

t∫t0

a(x(s), s

)dz(s) +

t∫t0

b(x(s), s

)ds

where the first integral may be interpreted in the sence of (1), but also as generalRiemann–Stieltjes integral. The unique Holder continuous (of order greater than1/2) local solution is explicitly represented as

x(t) = h(y(t) + z(t), t

)33

for some functions h and y satisfying certain classical differential equations. Anextension of this approach may be found in the thesis [8]. In the paper [12] ofLyons existence and uniqueness of the local solution of the following more generalequation in Rn is proved by means of Picard’s iteration method:

x(t) = x0 +n∑j=1

t∫0

aj(x(s)

)dzj(s)

where the aj are n–dimensional vector fields in Rn with certain smoothness prop-erties and z(t) =

(z1(t), . . . , zn(t)

)is a continuous vector function with finite

p–variation, p < 2. (Note that such functions are special elements of the Holder

space H1/pp .) There an extension of a result of Young [23] concerning the general

Riemann–Stieltjes integral of such functions is used and the solution is again offinite p–variation. Note that Lyons does not prove a contraction principle.Here we will work in the somewhat different space W β

2,∞ , β > 1/2, which willensure the contraction principle. We consider the differential equation

dx(t) =l∑

j=1

aj(x(t), ϕ(t)

)dzj(t)

x(t0) = x0

(6)

for some initial values t0 ∈ (0, T ) , x0 ∈ R and a driving function z = (z1, . . . , zl)with zj ∈ W β

2,∞. The additional parameter function ϕ takes values in Rk with

coordinate functions inW β2,∞ and the aj are Rn–valued C1–vector fields on Rn×Rk

such that all n+ k partial derivatives are locally Lipschitz in the first n variables.We seek a local solution of (6) in the space W β

2,∞(t1, t2) for certain t0 ∈ (t1, t2) ⊂(0, T ). Again we interpret (6) via integration according to (1). For t > t0 we canapply the contraction theorem 2.4 and Picard’s iteration method. Similarly, fort < t0 equation (6) means

x(t) = x0 −l∑

j=1

t0∫t

aj(x(s), ϕ(s)

)dzj(s) .

The integralt0∫t

aj(x(s), s

)dzj(s) =:

t0∫t

f dg may be understood as

b∫a

1(t,t0)f dg = (−1)αl∫

a

Dαa+ (1(t,t0)f)(s)D1−α

b− gb−(s) ds

34

for any 0 ≤ a ≤ t and almost all b > t0 (where gb− ∈ W β2 (b−)), 1/2 < 1− α < β.

According to Theorem 3.1 in part I it agrees with the “backward” integral

(−1)αb∫

a

Dαb− (1(t,t0)f)(s)D1−α

a+ ga+(s) ds

for almost all a (where ga+ ∈ W β2 (a+)(a, b)). Via time reversion with respect to

the starting moment t0 the roles of a+ and b− in this integral may be exchanged,so that the contraction theorem is also applicable to the backward integral. Thisleads to the following.

6.1 Theorem. Under the above conditions there exists some interval (t1, t2) con-tainig t0 such that equation (6) has a solution in W β

2,∞(t1, t2). It may be deter-mined by means of Picard’s iteration method which is contractive. The solutionis unique on the maximal interval of definition.

Remark. In particular, we may choose k = 1, l = m + 1, ϕ(t) = zm+1(t) =t, am+1 = b and obtain the equation in Rn

dx(t) =m∑j=1

aj(x(t), t

)dzj(t) + b

(x(t), t

)dt

x(t0) = x0 .

Here the first sum may be interpreted as a fractal noise term added to a classicalnon–autonomous ordinary differential equation. If the vector fields a1, . . . , amcommute then the solution may again be represented as a smooth function ofthe noise z = (z1, . . . , zm) and a smooth function y solving an ODE (cf. section 7).

In general, the solution x = xϕ of (6) depends continuously on the parameter

function ϕ ∈ W β2,∞(t1, t2) with ϕ(t0) = y0.

6.2 Theorem. Let y0 ∈ Rk and 0 < C < 1 ≤ K be given. Suppose thatthe conditions of Theorem 6.1 are fulfilled and the vector fields aj have locallyLipschitz partial derivatives. Then there is a sufficiently small interval (t1, t2)containing t0 such that for any two parameter functions ϕ, ψ with ϕ(t0) = ψ(t0) =y0 and ‖ϕ − y0‖Wβ

2,∞(t1,t2) ≤ K , ‖ψ − y0‖Wβ2,∞(t1,t2) ≤ K the solutions xϕ and xψ

of (6) exist on (t1, t2) and satisfy

‖xϕ − xψ‖Wβ2,∞(t0,t)

< C ‖ϕ− ψ‖Wβ2,∞(t0,t)

.

Proof. Without loss of generality we may assume that K = 1. (Otherwise con-sider a(x,Ky) instead of a(x, y).) Replacing in Theorem 2.4 the functions (f, h, ϕ)

35

by (ϕ, ψ, f) we obtain for a sufficiently small interval (t1, t2) 3 t0 the operator

Af ϕ := x0 +l∑

j=1

(·)∫t0

aj(f(s), ϕ(s)

)dzj(s)

for arbitrary f ∈ W β2,∞(t1, t2) with f(t0) = x0 and ‖f − x0‖Wβ

2,∞(t1,t2) ≤ 1 satisfies

‖Afφ− Afψ‖Wβ2,∞(t1,t2) ≤

C

1 + C‖ϕ− ψ‖Wβ

2,∞(t1,t2) .

By the original version of Theorem 2.4 (for (f, h, ψ)) we may choose t1, t2 so that

‖Afϕ− Ahφ‖Wβ2,∞(t1,t2) ≤

C

1 + C‖f − h‖Wβ

2,∞(t1,t2)

for arbitrary h ∈ W β2,∞(t1, t2) with h(t0) = x0 and ‖h− x0‖Wβ

2,∞(t1,t2) ≤ 1.

Picard’s iteration method provides the solutions xϕ and xψ and the corresponding

approximations in the n–th step x(n)ϕ , x

(n)ψ which fulfill the conditions on f and

h as above. For brevity we will omit in the resulting norm estimations thesubscription W β

2,∞(t1, t2) and note that

‖xϕ − xψ‖ = limn→∞

‖x(n)ϕ − x

(n)ψ ‖ .

The above arguments yield

‖x(n)ϕ − x

(n)ψ ‖ = ‖A

x(n−1)ϕ

ϕ− Ax(n−1)ψ

ψ‖

≤ ‖Ax(n−1)ϕ

ϕ− Ax(n−1)ψ

ϕ‖+ ‖Ax(n−1)ψ

ϕ− Ax(n−1)ψ

ψ‖

≤ C

1 + C

∥∥x(n−1)ϕ − x

(n−1)ψ

∥∥+C

1 + C‖ϕ− ψ‖ .

By induction we get

‖x(n)ϕ − x

(n)ψ ‖ ≤

n−1∑i=1

(C

1 + C

)i‖ϕ− ψ‖ ≤

∞∑i=0

(C

1 + C

)i‖ϕ− ψ‖

= C ‖ϕ− ψ‖ .

Hence,‖xϕ − xψ‖ ≤ C ‖ϕ− ψ‖ .

36

7 Stochastic differential equations with fractal

noise

7.1 The case of commuting vector fields

We now return to the random case and apply the results of sections 4 and 5 toSDE.Let Z1, . . . , Zm be one–dimensional continuous random processes on [0, T ] withgeneralized covariation processes of the form

[Zj, Zk](t) =

t∫0

qjk(s) ds

for some continuous random functions qjk. Denote Z(t0) =: Z0. We consider thestochastic differential equation in Rn

dX(t) =m∑j=1

aj(X(t), t

)dZj(t) + b

(X(t), t

)dt

X(t0) = X0

(7)

for certain random vector fields a1, . . . , am, b and an arbitrary random initial vec-tor X0.

Definition. A solution of (7) is a continuous random process X = (X1, . . . , Xn)admitting generalized covariation processes [Xj, Xk] which satisfies the multidi-mensional version of the Ito formula (5) with respect to its coordinatewise integralrepresentation

X(t) = X0 +m∑j=1

t∫t0

aj(X(s), s

)dZj(s) +

t∫t0

b(X(s), s

)ds

where the noise term is defined by convergence of the sum of the approximatingintegrals with respect to the Zj via (3).

We first consider the case m = 1, i.e., [Z](t) =t∫

0

q(s) ds and

dX(t) = a(X(t), t

)dt+ b

(X(t), t

)dt

X(t0) = X0 .(8)

Here we will construct a pathwise solution basing on the Ito formula. For therandom vector fields a and b we assume the following conditions (with probability1):

37

(C1) a ∈ C1 (Rn × [0, T ],Rn), all partial derivatives are locally Lipschitz inx ∈ Rn

(C2) b ∈ C (Rn × [0, T ],Rn) , b(x, t) is locally Lipschitz in x ∈ Rn.

We first consider pathwise the auxiliary partial differential equation on Rn×R×[0, T ]

∂h

∂z(y, z, t) = a

(h(y, z, t), t

)h(Y0, Z0, t0) = X0

(9)

where Y0 is a arbitrary random vector in Rn. Picard’s iteration method providesw.p.1 a (non–unique) local solution h ∈ C1 in a neighborhood of (Y0, Z0, t0) withpartial derivatives being Lipschitz in y and

det

(∂h

∂y(y, z, t)

)6= 0 .

Moreover,∂2h

∂z2(y, z, t) =

n∑j=1

∂a

∂xj(h(y, z, t), t

)aj(h(y, z, t), t

)We seek the solution X of (8) in the form

X(t) = h(Y (t), Z(t), t

)(10)

for a random C1–process Y in Rn with Y (t0) = Y0 to be determined (in depen-dence on the choice of h). Applying the Ito formula to F (z, t) := h

(Y (t), z, t

)we

obtain

dX(t) =∂h

∂z

(Y (t), Z(t), t

)dZ(t) +

n∑k=1

∂h

∂yk(Y (t), Z(t), t

)Y k(t) dt

+∂h

∂t

(Y (t), Z(t), t

)dt+

1

2

∂2h

∂z2

(Y (t), Z(t), t

)q(t) dt

= a(X(t), t

)dZ(t) +

n∑k=1

∂h

∂yk(Y (t), Z(t), t

)Y k(t) dt

+∂h

∂t

(Y (t), Z(t), t

)dt+

1

2

n∑j=1

∂a

∂xj(h(Y (t), Z(t), t), t

)aj(h(Y (t), Z(t), t), t

)q(t) dt.

38

Comparing the coefficients we are lead to the following ordinary differential equa-tion for Y (in matrix representation):

Y (t) =

(∂h

∂y

(Y (t), Z(t), t

))−1 [b(h(Y (t), Z(t), t), t

)− ∂h

∂t

(Y (t), Z(t), t

)− 1

2q(t)

(∂a

∂x

(h(Y (t), Z(t), t), t

))a(h(Y (t), Z(t), t), t

)]Y (t0) = Y0 .

(11)

The unique local solution Y (t) may be determined via Picard’s iteration method,which is contractive.

7.1.1 Theorem. (Existence and uniqueness)

(i) Under the conditions (C1) and (C2) any representation of the form (10)with h fulfilling (9) as above and Y determined by (11) provides a solutionof the SDE (8).

(ii) If X is an arbitrary solution of (8) in the sense of the above definition thenit agrees with any of the representations in (i) on the common interval ofdefinition.

Proof. (i) follows immediately from the above construction applying the Ito for-mula (s. Theorem 5.5).(ii) Take h

(Y (t), Z(t), t

)as in (i) and let X(t) be another local solution of (8).

The mapping(y, z, t) →

(h(y, z, t), z, t

)is invertible in a neighborhood of (Y0, Z0, t0). Let

(u(x, z, t), z, t

)be the inverse

mapping, i.e.,u(h(y, z, t), z, t

)= y .

Then we get the matrix equality(∂u

∂x(x, z, t)

)=

(∂h

∂y

(u(x, z, t), z, t

))−1

39

and furthermore,

∂u

∂z(x, z, t) = −

n∑k=1

∂u

∂xk(x, z, t) ak(x, t)

∂u

∂t(x, z, t) = −

n∑k=1

∂u

∂xk(x, z, t)

∂hk

∂t

(u(x, z, t), z, t

)∂2u

∂z2(x, z, t) = −

n∑j,k=1

∂2u

∂xj∂xk(x, z, t) aj(x, t) ak(x, t)

−n∑k=1

∂2u

∂z ∂xk(x, z, t) ak(x, t)−

n∑j,k=1

∂u

∂xk(x, z, t)

∂ak

∂xj(x, t) aj(x, t) .

We now will apply the higher–dimensional version of the Ito formula (5) to thefunction u(x, z, t) and the process

(X(t), Z(t)

)given in the integral representation

X(t) = X0 +

t∫t0

a(X(s), s

)dZ(s) +

t∫t0

b(X(s), s

)ds

Z(t) = Z0 +

t∫t0

dZ(s)

40

in a neighborhood of t0. Since u(X0, Z0, t0) = Y0 this yields

u(X(t), Z(t), t

)− Y0 =

t∫t0

∂u

∂z

(X(s), Z(s), s

)dZ(s)

+n∑k=1

t∫t0

∂u

∂xk(X(s), Z(s), s

)ak(X(s), s

)dZ(s)

+1

2

n∑j,k=1

t∫t0

∂2u

∂xj ∂xk(X(s), Z(s), s

)aj(X(s), s

)ak(X(s), s

)q(s) ds

+1

2

n∑k=1

t∫t0

∂2u

∂z ∂xk(X(s), Z(s), s

)ak(X(s), s

)q(s) ds

+1

2

n∑k=1

t∫t0

∂2u

∂z2

(X(s), Z(s), s

)q(s) ds+

t∫t0

∂u

∂t

(X(s), Z(s), s

)ds

+n∑k=1

t∫t0

∂u

∂xk(X(s), Z(s), s

)bk(X(s), s

)ds .

Substituting the above expressions for the derivatives we get

u(X(t), Z(t), t

)= Y0 +

n∑k=1

t∫t0

[∂u

∂xk(X(s), Z(s), s

)bk(X(s), s

)

− ∂u

∂xk(X(s), Z(s), s

) ∂hk∂t

(u(X(s), Z(s), s

), s)− 1

2

k∑j=1

∂u

∂xk(X(s), Z(s), s

)∂ak

∂xj(X(s), s

)aj(X(s), s

)q(s)

]ds

i.e.,

d

dtu(X(t), Z(t), t

)=

(∂u

∂x

(X(t), Z(t), t

))[b(X(t), t

)−∂h∂t

(u(X(t), Z(t), t

))−

1

2q(t)

(∂a

∂x

(X(t), t

))a(X(t), t

)].

41

Regarding the above matrix equality for(∂u∂x

)we infer that Y (t) :=

u(X(t), Z(t), t

)satisfies the ODE (11):

˙Y (t) =

(∂h

∂y

(Y (t), Z(t), t

))−1 [b(h(Y (t), Z(t), t

), t)− ∂h

∂t

(Y (t), Z(t), t

)− 1

2q(t)

∂a

∂x

(h(Y (t), Z(t), t

), t)a(h(Y (t), Z(t), t), t

)]Y (t0) = Y0 .

By uniqueness of the solution of (11) we obtain Y (t) = Y (t), i.e.,u(X(t), Z(t), t

)= Y (t), which implies X(t) = h

(Y (t), Z(t), t

)in a neighborhood

of t0. Since t0 may be replaced by an arbitrary t with det(∂h∂y

(Y (t), Z(t), t))6= 0

the assertion follows.

7.1.2 Theorem. For Z ∈ W β2,∞ with β > 1/2 the results of Theorem 7.1.1

remain valid if the condition (C1) is weakened to a ∈ C1(Rn × [0, T ],Rn) w.p.1.

Proof. First note that in this case the quadratic variation process vanishes, i.e.,q ≡ 0. We may use Theorem 3.2 for the corresponding Ito formula. The rest isthe same as in the previous proof.

We next will consider two special cases:

1) If we choose for h the unique solution of

∂h

∂z(y, z, t) = a

(h(y, z, t), t

)h(y, Z0, t) = y

then we obtain the approach suggested by Doss [4] and Sussman [19] for thespecial case of non–random time autonomous vector fields within classicalIto calculus.

2) If det(∂a∂x

(X0, t0))6= 0 we can take

h(y, z, t) = h(y + z, t)

with

∂h

∂z(z, t) = a

(h(z, t), t

)h(Z0, t0) = X0

For n = 1 this has been treated in [9].

42

We now turn to the case m > 1 with commuting random vector fields aj. Herewe replace condition (C1) by the following:

(C1′) aj ∈ C1(Rn × [0, T ],Rn), all partial derivatives are locally Lipschitz in x andJaj, akK ≡ 0, 1 ≤ j, k ≤ m, where J·, ·K denotes the Lie bracket of the vec-tor fields with respect to the argument x ∈ Rn which is determined almosteverywhere.

The PDE (9) takes now the higher–dimensional form on Rn × Rm × [0, T ]

∂h

∂zj(y, z, t) = aj

(h(y, z, t), t

), j = 1, . . . ,m

h(Y0, Z0, t0) = X0 .

(9′)

The commutativity property of the vector fields aj guarantees the existence of aC1–solution in a neightborhood of (Y0, Z0, t0) with

det

(∂h

∂y(y, z, t)

)6= 0

and ∂h∂t

(y, z, t) being Lipschitz in y. The remaining procedure is the same asbefore: The local solution of (7) is unique and may be represented in the form

X(t) = h(Y (t), Z(t), t

)(10′)

where the random C1–process Y is uniquely determined by the following exten-sion of the ODE (9):

Y (t) =

(∂h

∂y

(Y (t), Z(t), t

))−1[b(h(Y (t), Z(t), t

), t)− ∂h

∂t

(Y (t), Z(t), t

)− 1

2

m∑j,k=1

qjk(t)

(∂aj∂x

(h(Y (t), Z(t), t

), t))

ak

(h(Y (t), Z(t), t

), t)]

Y (t0) = Y0 .

(11′)

Remark. The problem of existence of a global solution may be reduced to thesame question for the differential equations (9′) and (11′), i.e., to correspondinggrowth conditions on the vector fields.

7.2 The case of nilpotent Lie algebras generated by thevector fields

In [22] Yamato extended the Doss approach for Ito or Stratonovitch SDE to thecase where the Lie algebra generated by the vector fields a1, . . . , am is nilpotent

43

of order p > 1. This method also works for our anticipative integrals withtime dependent random vector fields provided that the iterated integrals of theprocesses Z1, . . . , Zm up to order p satisfy the Ito formula for the functionsunder consideration. For simplicity we will demonstrate the main ideas on thecase p = 2 (see also Ikeda and Watanabe [7], chapter III, 2, example 2.2 forStratonovitch SDE). We suppose the following:

(C1′′) aj ∈ C1(Rn × [0, T ],Rn) ,∂aj∂t

(x, t) and∂2aj∂xi ∂xk

(x, t) are locally Lipschitz in xand Jaj, Jal, amKK ≡ 0 , j, l,m = 1, . . .m, i, k = 1, . . . n, where the Lie bracketis again taken with respect to the space argument x.

The condition (C2) is as before.

If the vector fields aj do not commute then the PDE (9′) is not integrable. There-fore the space Rm will be enlarged to RM withM = m+m(m−1)/2. The elementsare denoted by

z = (z1, . . . , zm, z12, . . . , zik, . . . , zm−1m)

with ordered pairs i < k. (9′) is replaced by a system of PDE on R×RM × [0, T ]with values in Rn:

∂h

∂zj(y, z, t) = aj

(h(y, z, t), t

)−

j−1∑i=1

ziJai, ajK(h(y, z, t), t

), j = 1, . . . ,m

∂h

∂zjk(y, z, t) = Jaj, akK

(h(y, z, t), t

), 1 ≤ j < k ≤ m

h(Y0, Z0, t0) = X0 ,

(9′′)

whose vector fields denoted by aj and ajk, respectively, commute in view of thealgebraic assumption on the primary vector fields aj. As before we take anyC1–solution h with

det

(∂h

∂y(y, z, t0)

)6= 0

and ∂h∂t

(y, z, t) being Lipschitz continuous in y in a neighborhood of(Y0, (Z0, 0), t0

)and seek the solution of (7) in the form

X(t) = h(Y (t), Z(t), t

)(10′′)

with Zjk(t) :=t∫t0

Zj dZk , 1 ≤ j < k ≤ m.

44

The random process Y (t) is determined by the ODE

Y (t) =

(∂h

∂y

(Y (t), Z(t), t

))−1[b(h(Y (t), Z(t), t

), t)− ∂h

∂t

(Y (t), Z(t), t

)− 1

2

m∑j,k=1

qjk(t)

(∂aj∂x

(h(Y (t), Z(t), t

), t))

ak

(h(Y (t), Z(t), t

), t)

− 1

2

m∑i=1

∑1≤j<k≤m

Zj(t) qik(t)

(∂ai∂x

(h(Y (t), Z(t), t

), t))

ajk

(h(Y (t), Z(t), t

), t)

− 1

2

∑1≤i<j≤m

m∑k=1

Zi(t) qjk(t)

(∂aij∂x

(h(Y (t), Z(t), t

), t))

ak

(h(Y (t), Z(t), t

), t)

− 1

2

∑1≤i<j≤m

∑1≤k<l≤m

Zi(t)Zk(t) qjl(t)

(∂aik∂x

(h(Y (t), Z(t), t

), t))

akl

(h(Y (t), Z(t), t

), t)]

Y (t0) = Y0 .

(11′′)

In order to make this precise we assume the following

(C3) The stochastic integrals Zij(t) =t∫t0

Zi dZj are determined and possess the

covariation properties[Zi, Zjk

]=

(·)∫0

Zj(s) qik(s) ds[Zij, Zkl

]=

(·)∫0

Zi(s)Zk(s) qjl(s) ds.

Moreover, the processes Zij satisfy the Ito formula for the random functionsh considered in (9′′), i.e., the differentials dZ ij may be replaced by Zi dZj.

Remark. Clearly, the general problem consists in checking the last condition in(C3). Note that by Theorem 3.2 it is fulfilled for all vector fields (independentlyof the structure of the Lie algebra) if the processes Zj are elements of W β

2,∞ forsome β > 1/2.

The rest is completely analogous to the proof of Theorem 7.2.1: Differentiating(10′′) according to the Ito rule and using the expressions in (9′′) and (11′′) for the

45

derivatives one obtains that X(t) is indeed a local solution of (7). The proof ofuniqueness in the sense of our solution concept by means of the inverse mappingtheorem is similar to the case m = 1.

Finally, the case p > 2 is a straightforward extension, though the algebraic partis much harder: Here the iterated integrals of the processes Z1, . . . , Zm up toorder p are involved. The problem again consists in blowing up the vector fieldsa1, . . . , am in order to get an integrable system of PDE for determining h suchthat the use of the iterated integrals as additional arguments leads throughthe Ito formula for h to the coefficients aj at dZj. An algebraic procedure fordetermining such a differential system is fully described in Yamato [22]. Theremainig part is before. Again, this approach does only work if the iteratedintegrals satisfy the Ito formula for the random functions under consideration.

46

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