Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 13

Numerical Solution of Stochastic Differential Equations

 

 

Consider the stochastic differential equation

[13.1]

As we know, in an explicit form such an equation is rarely solvable. It is natural that, as in the case of ordinary differential equations, we need numerical methods enabling us to use computers to solve SDEs. However, what is meant by an approximate solution of an SDE the solution of which is a random process?

We typically proceed as follows. Consider a discretization of a fixed time interval [0, T]:

with fixed time step h = T/N. For all such h, we construct discrete-time random processes , k = 0, 1, 2, …, N, that depend on the values of Brownian motion B at discretization moments, i.e. on B_kh, k = 0, 1, 2, …, N, and that “as well as possible” approximate the exact solution X of the equation as h = T/N → 0. It is convenient to extend the processes to the whole time interval t ∈ [0, T]. For example, we can set

(step process), or

(polygonal line).

How can ...