Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
Chapter 5: Stochastic Differential Equations
Differential equations are used to describe the evolution of a system. Stochastic Differential Equations (SDEs) arise when a random noise is introduced into ordinary differential equations (ODEs). In this chapter we define two concepts of solutions of SDEs, the strong and the weak solution.
5.1 Definition of Stochastic Differential Equations
Ordinary Differential Equations
If x( t) is a differentiable function defined for t ? 0, ?( x, t) is a function of x, and t, and the following relation is satisfied for all t, 0 ? t ? T
then x( t) is a solution of the ODE with the initial condition x 0. Usually the requirement that x ?( t) is continuous is added. See also Theorem 1.4.
The above equation can be written in other forms.
and (by continuity of x ?( t))
Before we give a rigorous definition of SDEs, we show how they arise as a randomly perturbed ODEs and give a physical interpretation.
White Noise and SDEs
The White Noise process ?( t) is formally defined as the derivative of the Brownian motion,
It does not exist as a function of t in the usual sense, since a Brownian motion is nowhere differentiable.
If ?( x, t) is the intensity of the noise at point x at time t, then it is agreed that ? T 0 ?( X( t),
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