TY - JOUR
T1 - Maximum Principle for Stochastic Control of SDEs with Measurable Drifts
AU - Menoukeu-Pamen, Olivier
AU - Tangpi, Ludovic
N1 - Funding Information:
Olivier Menoukeu-Pamen and Ludovic Tangpi thank the anonymous referees for helpful comments and suggestions. Menoukeu-Pamen acknowledges financial support by the Alexander von Humboldt Foundation, under the program financed by the German Federal Ministry of Education and Research entitled German Research Chair No 01DG15010. Tangpi acknowledges financial support by the NSF Grant DMS-2005832 and the NSF CAREER Award DMS-2143861.
Publisher Copyright:
© 2023, The Author(s).
PY - 2023/6
Y1 - 2023/6
N2 - 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.
AB - 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.
KW - Ekeland’s variational principle
KW - Singular drifts
KW - Sobolev differentiable flow
KW - Stochastic maximum principle
UR - http://www.scopus.com/inward/record.url?scp=85152532517&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85152532517&partnerID=8YFLogxK
U2 - 10.1007/s10957-023-02209-0
DO - 10.1007/s10957-023-02209-0
M3 - Article
AN - SCOPUS:85152532517
SN - 0022-3239
VL - 197
SP - 1195
EP - 1228
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
IS - 3
ER -