A closed form solution for a stochastic control problem with quasi-polynomial value function

Details A closed form solution for a stochastic control problem with quasi-polynomial value function A closed form solution for a stochastic control problem with quasi-polynomial value function

2012-04-11

arxiv.org/abs/1204.2498v1

Mathematics

Optimization and Control

Scientific paper

We study a constrained stochastic control problem with jumps; the jump times of the controlled process are given by a Poisson process. The cost functional comprises quadratic components for an absolutely continuous control and the controlled process and an absolute value component for the control of the jump size of the controlled process. We characterize the value function by a quasi-polynomial of degree two, i.e., a "polynomial" whose coefficients depend on the state of the system. This ansatz allows us to derive the solution of the optimization problem via dynamic programming and a novel interpolation method in closed form; the value function is a classical solution of the HJB equation corresponding to the problem. The state space is separated by a time dependent boundary into a continuation region where the optimal jump size of the controlled process is positive and a stopping region where it is zero.

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