Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 9

Linear Stochastic Differential Equations

 

 

9.1. Explicit solution of a linear SDE

DEFINITION 9.1.– A linear stochastic differential equation (LSDE) is an equation of the form

[9.1]

 

where a_i and b_i (i = 1, 2) are non-random functions bounded on every finite interval [0, T](e.g. continuous); it is easy to check that the coefficients of an LSDE satisfy the Lipschitz conditions, and thus the equation has a unique solution. If a_i and b_i are constants, then the LSDE is called autonomous; if a2 = b₂ = 0, then it is called homogeneous.

Denote

PROPOSITION 9.2.– The random process Φ_t, t 0, is a solution of the homogeneous LSDE

[9.2]

 

with the initial condition Y₀ = 1. The process Φ is called the fundamental solution of equation [9.2]

Proof. Denote

 

 

Thus, Φ_t = e^Zt, t 0. Then, using Itô's formula, we get:

 

The initial condition Φ₀ = 1 is clearly satisfied.

PROPOSITION 9.3.– The ...