Deterministic Systems and Stochastic Input

This chapter examines stochastic problems defined by 1 D [ X ( x , t ) ] = Y ( x , t ) , t ≥ 0 , x ∈ D ⊂ ℝ n , $$D\left[ {X\left( {x,t} \right)} \right] = Y\left( {x,t} \right),\;t \geqslant 0,\;x \in D \subset {\mathbb{R}^n},$$ where n ≥ 1 is an integer, x and t denote space and time parameters, respectively, V can be an algebraic, differential, or integral operator with deterministic coefficients that may or may not depend on time, Y is the random input, and X denotes the output. It is common in applications to concentrate on values of X at a finite number of points X k ∈ D rather then all points of D. Systems described by the evolution in time of A’ at a finite number of points and at all points in D are called discrete and continuous respectively. The focus of this chapter is on discrete systems since they are used extensively in applications and are simpler to analyze than continuous systems. The vector X(t) ∈ ℝd collecting the processes X(xk, t), called the state vector defines the evolution of a discrete system. The mapping from Y to X can be with or without memory. An extensive discussion on memoryless transformations of random processes can be found in [79] (Chapters 3, 4, and 5) and is not presented here.