Controlled Stochastic Differential Equations
In this chapter, we introduce the stochastic control problem and discuss various classes of controls, approximations to these classes and questions concerning the existence of an optimal control. We work with one particular type of cost functional for simplicity in the development. In Section 1, ordinary admissible controls are introduced and it is shown why they might not be adequate for our needs. Section 2 introduces the notion of “relaxed control” for deterministic problems and shows how to use them to prove the existence of an optimal control. It is also shown that these “generalized” or relaxed controls can be approximated by piecewise constant ordinary controls.
KeywordsStochastic Differential Equation Weak Convergence Wiener Process Weak Sense Convergent Subsequence
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