Controlled Variance and Jumps

Abstract

In the models dealt with in Chapters 10–12, such as (10.1.1), (11.1.1) and their time varying forms in Chapter 12, neither the noise coefficient σ(⋅) nor the jump coefficient q(⋅) depended on the control. However, the control dependent forms are treated by methods which are very similar to those used in Chapters 10–12, and identical convergence results are obtainable under the natural extensions of the conditions used in those chapters. Local consistency remains the primary requirement. Recall that relaxed controls were introduced owing to the issue of closure: A bounded sequence of solutions x ⁿ (⋅) under ordinary controls u ⁿ (⋅) (for either an ordinary or a stochastic differential equation) would not necessarily have a subsequence which converged to a limit process which was a solution to the equation driven by an ordinary control. But, if the controls (whether ordinary or relaxed) were represented as relaxed controls, then the sequence of (solutions, controls) was compact, so that we could extract a convergent subsequence and the limit solution was driven by the limit relaxed control. While the introduction of relaxed controls enlarged the problem, it did not affect the infimum of the costs or the numerical method. It was used purely for mathematical purposes, and not as a practical control.