Impulse and Continuous Stochastic Control: A Dynamic Programming Approach
1. A predominant approach to impulse stochastic control is that by functional analysis methods (Bensoussan et Lions (1982)). It originated as a generalization of optimal stopping and an appropriate field of applications of quasivaraitional inequalities (=QVI). One understands here impulse actions as deterministic shifts of the state made at sequentially increasing stopping times, usually without a continuous control between them. As to instantaneous iterations of impulse actions, in most works they are excluded either by definition or informally by a cost structure which makes such iterations certainly unprofitable; in others they are ignored while formally not forbidden. Started with diffusion processes, the study was extended later to other cases, among them to piecewise-deterministic processes (=PDP; Costa and Davis (1989)).
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