stochastic differential equations in applicationspersonal.strath.ac.uk › x.mao › talks ›...

Stochastic ModellingWell-known Models

Stochastic verse DeterministicForecasting and Monte Carlo Simulations

Stochastic Differential Equations inApplications

Xuerong Mao FRSE

Department of Mathematics and StatisticsUniversity of Strathclyde

Glasgow, G1 1XH

Xuerong Mao FRSE SDEs



Outline

1 Stochastic Modelling2 Well-known Models

Linear SDE modelsNon-linear SDE models

3 Stochastic verse DeterministicExponential growth modelLogistic ModelSummary

4 Forecasting and Monte Carlo SimulationsMonte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities




One of the important problems in many branches of scienceand industry, e.g. engineering, management, finance, socialscience, is the specification of the stochastic process governingthe behaviour of an underlying quantity. We here use the termunderlying quantity to describe any interested object whosevalue is known at present but is liable to change in the future.Typical examples are

number of cancer cells,number of HIV infected individuals,share price in a company,price of gold, oil or electricity.




Now suppose that at time t the underlying quantity is x(t). Letus consider a small subsequent time interval dt , during whichx(t) changes to x(t) + dx(t). (We use the notation d · for thesmall change in any quantity over this time interval when weintend to consider it as an infinitesimal change.) By definition,the intrinsic growth rate at t is dx(t)/x(t). How might we modelthis rate?




If, given x(t) at time t , the rate of change is deterministic, sayR = R(x(t), t), then

dx(t)x(t)

= R(x(t), t)dt .

This gives the ordinary differential equation (ODE)

dx(t)dt

= R(x(t), t)x(t).




However the rate of change is in general not deterministic as itis often subjective to many factors and uncertainties e.g.system uncertainty, environmental disturbances. To model theuncertainty, we may decompose

dx(t)x(t)

= deterministic change + random change.




The deterministic change may be modeled by

R̄dt = R̄(x(t), t)dt

where R̄ = r̄(x(t), t) is the average rate of change given x(t) attime t . So

dx(t)x(t)

= R̄(x(t), t)dt + random change.

How may we model the random change?




In general, the random change is affected by many factorsindependently. By the well-known central limit theorem thischange can be represented by a normal distribution with meanzero and and variance V 2dt , namely

random change = N(0,V 2dt) = V N(0,dt),

where V = V (x(t), t) is the standard deviation of the rate ofchange given x(t) at time t , and N(0,dt) is a normal distributionwith mean zero and and variance dt . Hence

dx(t)x(t)

= R̄(x(t), t)dt + V (x(t), t)N(0,dt).




A convenient way to model N(0,dt) as a process is to use theBrownian motion B(t) (t ≥ 0) which has the followingproperties:

B(0) = 0,dB(t) = B(t + dt)− B(t) is independent of B(t),dB(t) follows N(0,dt).




The stochastic model can therefore be written as

dx(t)x(t)

= R̄(x(t), t)dt + V (x(t), t)dB(t),

or

dx(t) = R̄(x(t), t)x(t)dt + V (x(t), t)x(t)dB(t)

which is a stochastic differential equation (SDE).





Outline









Exponential growth model

If both R̄ and V are constants, say

R̄(x(t), t) = µ, V (x(t), t) = σ,

then the SDE becomes

dx(t) = µx(t)dt + σx(t)dB(t).

This is a linear SDE. It is known as the geometric Brownianmotion in finance and the exponential growth model in thepopulation theory.





Exponential growth model

IfR̄(x(t), t) =

α(µ− x(t))

x(t), V (x(t), t) = σ,


dx(t) = α(µ− x(t))dt + σx(t)dB(t).

This is known as the mean-reverting process.





Outline









Logistic model

IfR̄(x(t), t) = b + ax(t), V (x(t), t) = σx(t),


dx(t) = x(t)(

[b + ax(t)]dt + σx(t)dB(t)).

This is the well-known Logistic model in population.





Square root process

IfR̄(x(t), t) = µ, V (x(t), t) =

σ√x(t)

,

then the SDE becomes the well-known square root process

dx(t) = µx(t)dt + σ√

x(t)dB(t).

This is used widely in engineering and finance.





Mean-reverting square root process

IfR̄(x(t), t) =

α(µ− x(t))

x(t), V (x(t), t) =

σ√x(t)

,


dx(t) = α(µ− x(t))dt + σ√

x(t)dB(t).

This is the mean-reverting square root process used widely infinance and population.





Theta process

IfR̄(x(t), t) = µ, V (x(t), t) = σ(x(t))θ−1,


dx(t) = µx(t)dt + σ(x(t))θdB(t),

which is known as the theta process.




Exponential growth modelLogistic ModelSummary

Outline









In the classical theory of population dynamics, it is assumedthat the grow rate is constant µ. Thus

dx(t)x(t)

= µdt ,

which is often written as the familiar ordinary differentialequation (ODE)

dx(t)dt

= µx(t).

This linear ODE can be solved exactly to give exponentialgrowth (or decay) in the population, i.e.

x(t) = x(0)eµt ,

where x(0) is the initial population at time t = 0.





We observe:

If µ > 0, x(t)→∞ exponentially, i.e. the population willgrow exponentially fast.If µ < 0, x(t)→ 0 exponentially, that is the population willbecome extinct.If µ = 0, x(t) = x(0) for all t , namely the population isstationary.





However, if we take the uncertainty into account as explainedbefore, we may have

dx(t)x(t)

= rdt + σdB(t).

This is often written as the linear SDE

dx(t) = µx(t)dt + σx(t)dB(t).

It has the explicit solution

x(t) = x(0) exp[(µ− 0.5σ2)t + σB(t)

].





Recall the properties of the Brownian motion

lim supt→∞

B(t)√2t log log t

= 1 a.s.

andlim inft→∞

B(t)√2t log log t

= −1 a.s.





If µ > 0.5σ2, x(t)→∞ exponentially with probability 1, i.e.the population will grow exponentially fast.If µ < 0.5σ2, x(t)→ 0 exponentially with probability 1, thatis the population will become extinct.

In particular, this includes the case of 0 < µ < 0.5σ2 wherethe population will grow exponentially in the correspondingODE model but it will become extinct in the SDE model.This reveals the important feature - noise may make apopulation extinct. (Comment on stochastic stabilisation.)

If µ = 0.5σ2, lim supt→∞ x(t) =∞ while lim inft→∞ x(t) = 0with probability 1.





ExampleThe solution of the ODE

dx(t)dt

= x(t)

obeyslim

t→∞x(t) =∞.

However, The solution of the SDE

dx(t) = x(t)dt + 2x(t)dB(t)

obeyslim

t→∞x(t) = 0 a.s.





0 5 10 15

010

0020

0030

0040

00

t

X(t)

or x

(t)

true solnEM soln

0 5 10 15

050

150

250

350

t

X(t)

or x

(t)

true solnEM soln

0 5 10 15

050

0010

000

1500

0

t

X(t)

or x

(t)

true solnEM soln

0 5 10 15

010

2030

4050

t

X(t)

or x

(t)

true solnEM soln

x

x





Outline









The logistic model for single-species population dynamics isgiven by the ODE

dx(t)dt

= x(t)[b + ax(t)]. (3.1)





If a < 0 and b > 0, the equation has the global solution

x(t) =b

−a + e−bt (b + ax0)/x0(t ≥ 0) ,

which is not only positive and bounded but also

limt→∞

x(t) =b|a|.

If a > 0, whilst retaining b > 0, then the equation has onlythe local solution

x(t) =b

−a + e−bt (b + ax0)/x0(0 ≤ t < T ) ,

which explodes to infinity at the finite time

T = −1b

log(

ax0

b + ax0

).





Once again, the growth rate b here is not a constant but astochastic process. Therefore, bdt should be replaced by

bdt + N(0, v2dt) = bdt + vN(0,dt) = bdt + vdB(t),

where v2 is the variance of the noise intensity. Hence the ODEevolves to an SDE

dx(t) = x(t)(

[b + ax(t)]dt + vdB(t)). (3.2)





The variance may or may not depend on the state x(t). We firstconsider the latter, say

v = σx(t).

Then the SDE (3.2) becomes

dx(t) = x(t)(

[b + ax(t)]dt + σx(t)dB(t)). (3.3)

How is this SDE different from its corresponding ODE?





Significant Difference between ODE (3.1) and SDE(3.3)

ODE (3.1): The solution explodes to infinity at a finite timeif a > 0 and b > 0.SDE (3.3): With probability one, the solution will no longerexplode in a finite time, even in the case when a > 0 andb > 0, as long as σ 6= 0. Moreover, the stochasticpopulation system is persistent.





Example

dx(t)dt

= x(t)[1 + x(t)], t ≥ 0, x(0) = x0 > 0

has the solution

x(t) =1

−1 + e−t (1 + x0)/x0(0 ≤ t < T ) ,

which explodes to infinity at the finite time

T = log(

1 + x0

x0

).

However, the SDE

dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)]

will never explode as long as σ 6= 0.Xuerong Mao FRSE SDEs




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60

80

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(b)�

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Note on the graphs:In graph (a) the solid curve shows a stochastic trajectorygenerated by the Euler scheme for time step ∆t = 10−7 andσ = 0.25 for a one-dimensional system (3.3) with a = b = 1.The corresponding deterministic trajectory is shown by thedot-dashed curve. In Graph (b) σ = 1.0.





Key Point:When a > 0 and ε = 0 the solution explodes at the finite timet = T ; whilst conversely, no matter how small ε > 0, thesolution will not explode in a finite time. In other words,

noise may suppress explosion.





If the variance is independent of the state x(t), namelyv = σ = const ., then the SDE (3.2) becomes

dx(t) = x(t)(

[b + ax(t)]dt + σdB(t)). (3.4)

How is this SDE different from others?





For this SDE, we require a < 0 in order to have noexplosion.If b < 0.5σ2 then limt→∞ x(t) = 0 with probability 1,namely the noise makes the population extinct.If b > 0.5σ2, the population is persistent and

limT→∞

1T

∫ T

0x(t)dt =

b − 0.5σ2

|a|a.s.





Outline









Noise changes the behaviour of population systemssignificantly.Noise may suppress the potential population explosion.Noise may make the population extinct.Noise may make the population persistent.




Monte Carlo simulationsTest problemEM method for nonlinear SDEsEM method for financial quantities

Outline









Most of SDEs used in practice do not have explicit solutions.How can we use these SDEs to forecast?One of the important techniques is the method of Monte Carlosimulations. There are two main motivations for suchsimulations:

using a Monte Carlo approach to compute the expectedvalue of a function of the underlying underlying quantity, forexample to value a bond or to find the expected payoff ofan option;generating time series in order to test parameterestimation algorithms.





Outline









Question: Can we trust the Monte Carlo simulations?

Test problemThe linear SDE

dX (t) = 2X (t)dt + X (t)dB(t), X (0) = 1

has the explicit solution

x(t) = exp(1.5t + B(t)).

The Monte Carlo simulation can be carried out based on theEuler-Maruyama (EM) method

x(0) = 1, x(i + 1) = x(i)[1 + 2∆ + ∆Bi ], i ≥ 0,

where ∆Bi = B((i + 1)∆)− B(i∆) ∼ N(0,∆).





0.0 0.5 1.0 1.5 2.0

24

68

1012

t

X(t)

or x

(t)

true solnEM soln

0.0 0.5 1.0 1.5 2.0

05

1015

2025

t

X(t)

or x

(t)

true solnEM soln

0.0 0.5 1.0 1.5 2.0

020

6010

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0

t

X(t)

or x

(t)

true solnEM soln

0.0 0.5 1.0 1.5 2.0

24

68

10

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X(t)

or x

(t)

true solnEM soln

x

x





Outline









Typically, let us consider the square root process

dS(t) = rS(t)dt + σ√

S(t)dB(t), 0 ≤ t ≤ T .

A numerical method, e.g. the Euler–Maruyama (EM) methodapplied to it may break down due to negative values beingsupplied to the square root function. A natural fix is to replacethe SDE by the equivalent, but computationally safer, problem

dS(t) = rS(t)dt + σ√|S(t)|dB(t), 0 ≤ t ≤ T .





Discrete EM approximation

Given a stepsize ∆ > 0, the EM method applied to the SDEsets s0 = S(0) and computes approximations sn ≈ S(tn), wheretn = n∆, according to

sn+1 = sn(1 + r∆) + σ√|sn|∆Bn,

where ∆Bn = B(tn+1)− B(tn).





Continuous-time EM approximation

s(t) := s0 + r∫ t

0s̄(u))du + σ

∫ t

0

√|s̄(u)|dB(u),

where the “step function” s̄(t) is defined by

s̄(t) := sn, for t ∈ [tn, tn+1).

Note that s(t) and s̄(t) coincide with the discrete solution at thegridpoints; s̄(tn) = s(tn) = sn.





The ability of the EM method to approximate the true solution isguaranteed by the ability of either s(t) or s̄(t) to approximateS(t) which is described by:

Theorem

lim∆→0

E(

sup0≤t≤T

|s(t)−S(t)|2)

= lim∆→0

E(

sup0≤t≤T

|s̄(t)−S(t)|2)

= 0.





Outline









Bond

If S(t) models short-term interest rate dynamics, it is pertinentto consider the expected payoff

β := E exp

(−∫ T

0S(t)dt

)from a bond. A natural approximation based on the EM methodis

β∆ := E exp

(−∆

N−1∑n=0

|sn|

), where N = T/∆.

Theorem

lim∆→0|β − β∆| = 0.





European call option

A European call option with the exercise price K at expiry timeT pays S(T )− K if S(T ) > K otherwise 0.

TheoremLet r be the risk-free interest rate and define

C = e−rT E[(S(T )− K )+

],

C∆ = e−rT E[(s̄(T )− K )+

].

Thenlim

∆→0|C − C∆| = 0.





Example - the Black-Scholes model

Consider a BS model

dS(t) = 0.05S(t)dt + 0.03S(t)dB(t), S(0) = 10

and a European call option with the exercise price K = 10.05 atexpiry time T = 1, where 0.05 is the risk-free interest rate and0.03 is the volatility. By the well-known Black-Scholes formulaon the option, we can compute the value of a European calloption at time zero is

C = 0.4487318.

On the other hand, we can let ∆ = 0.001, simulate 1000 pathsof the SDE, compute the mean payoff at T = 1, discounting itby e−0.05, we get the estimated option value

C∆ = 0.4454196





To be more reliable, we can carry out such simulation, say 10times, to get 10 estimated values:

0.4454196, 0.4611569, 0.4512847, 0.4490462,⋃.4294038,

0.4618921, 0.4556195, 0.4559547, 0.4399189, 0.4489594.

Their mean valueC̄∆ = 0.4498656

gives a better estimation for C.





Up-and-out call option

An up-and-out call option at expiry time T pays the Europeanvalue with the exercise price K if S(t) never exceeded the fixedbarrier, c, and pays zero otherwise.

TheoremDefine

V = E[(S(T )− K )+I{0≤S(t)≤c, 0≤t≤T}

],

V∆ = E[(s̄(T )− K )+I{0≤s̄(t)≤c, 0≤t≤T}

].

Thenlim

∆→0|V − V∆| = 0.


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