Stochastic Differential Equations

Stochastic Differential Equations

Do not worry about your problems with mathematics, I assure you mine are far greater.

Albert Einstein.

Florian Herzog 2013

Stochastic Differential Equations (SDE)

A ordinary differential equation (ODE)

dx(t)

= f (t, x) , dx(t) = f (t, x)dt ,

(1)

dt

with initial conditions x(0) = x0 can be written in integral form

t

x(t) = x0 + f (s, x(s))ds ,

(2)

0

where x(t) = x(t, x0, t0) is the solution with initial conditions x(t0) = x0. An

example is given as

dx(t)

dt = a(t)x(t) , x(0) = x0 .

(3)

Stochastic Systems, 2013

2

Stochastic Differential Equations (SDE)

When we take the ODE (3) and assume that a(t) is not a deterministic parameter but rather a stochastic parameter, we get a stochastic differential equation (SDE). The stochastic parameter a(t) is given as

a(t) = f (t) + h(t)(t) ,

(4)

where (t) denotes a white noise process.

Thus, we obtain

dX (t)

= f (t)X(t) + h(t)X(t)(t) .

(5)

dt

When we write (5) in the differential form and use dW (t) = (t)dt, where dW (t)

denotes differential form of the Brownian motion,we obtain:

dX(t) = f (t)X(t)dt + h(t)X(t)dW (t) .

(6)

Stochastic Systems, 2013

3

Stochastic Differential Equations (SDE)

In general an SDE is given as

dX(t, ) = f (t, X(t, ))dt + g(t, X(t, ))dW (t, ) ,

(7)

where denotes that X = X(t, ) is a random variable and possesses the initial condition X(0, ) = X0 with probability one. As an example we have already encountered

dY (t, ) = ?(t)dt + (t)dW (t, ) .

Furthermore, f (t, X(t, )) R, g(t, X(t, )) R, and W (t, ) R. Similar as in (2) we may write (7) as integral equation

t

t

X(t, ) = X0 + f (s, X(s, ))ds + g(s, X(s, ))dW (s, ) . (8)

0

0

Stochastic Systems, 2013

4

Stochastic Integrals

For

the

calculation

of

the

stochastic

integral

T

0

g(t, )dW (t, ),

we

assume

that

g(t, ) is only changed at discrete time points ti (i = 1, 2, 3, ..., N - 1), where

0 = t0 < t1 < t2 < . . . < tN-1 < tN < T . We define the integral

T

S = g(t, )dW (t, ) ,

(9)

0

as the Riemann?um

N

(

)

SN () = g(ti-1, ) W (ti, ) - W (ti-1, ) .

(10)

i=1

with N .

Stochastic Systems, 2013

5

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download