Abstract
In this paper, we consider stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. We establish a necessary and sufficient stochastic maximum principle. To achieve this, we first derive an explicit representation of the first variation process (in the Sobolev sense) of the controlled diffusion. Since the drift coefficient is not smooth, the representation is given in terms of the local time of the state process. Then we construct a sequence of optimal control problems with smooth coefficients by an approximation argument. Finally, we use Ekeland’s variational principle to obtain an approximating adjoint process from which we derive the maximum principle by passing to the limit. The work is notably motivated by the optimal consumption problem of investors paying wealth tax.
Original language | English (US) |
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Pages (from-to) | 1195-1228 |
Number of pages | 34 |
Journal | Journal of Optimization Theory and Applications |
Volume | 197 |
Issue number | 3 |
DOIs | |
State | Published - Jun 2023 |
All Science Journal Classification (ASJC) codes
- Control and Optimization
- Applied Mathematics
- Management Science and Operations Research
Keywords
- Ekeland’s variational principle
- Singular drifts
- Sobolev differentiable flow
- Stochastic maximum principle