First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems

In this paper, we derive first-order Pontryagin optimality conditions for risk-averse stochastic optimal control problems subject to final time inequality constraints, and whose costs are general, possibly non-smooth finite coherent risk measures. Unlike preexisting contributions covering this situation, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. In addition, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian.

[1]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[2]  A. Ruszczynski,et al.  Lectures on Stochastic Programming: Modeling and Theory, Third Edition , 2021 .

[3]  Margaret P. Chapman,et al.  Risk-Sensitive Safety Analysis Using Conditional Value-at-Risk , 2021, IEEE Transactions on Automatic Control.

[4]  H'elene Frankowska,et al.  Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces , 2021, Applied Mathematics & Optimization.

[5]  H. Frankowska,et al.  Necessary conditions for stochastic optimal control problems in infinite dimensions , 2018, Stochastic Processes and their Applications.

[6]  William B. Haskell,et al.  Risk Aware Minimum Principle for Optimal Control of Stochastic Differential Equations , 2018, IEEE Transactions on Automatic Control.

[7]  Victor M. Becerra,et al.  Optimal control , 2008, Scholarpedia.

[8]  Xu Zhang,et al.  Stochastic Optimal Control Problems with Control and Initial-Final States Constraints , 2018, SIAM J. Control. Optim..

[9]  Salah Eddine Choutri,et al.  On relaxed stochastic optimal control for stochastic differential equations driven by G-Brownian motion , 2017, 1702.08735.

[10]  J. L. Gall,et al.  Brownian Motion, Martingales, and Stochastic Calculus , 2016 .

[11]  Haisen Zhang,et al.  First and second order necessary conditions for stochastic optimal controls , 2016, 1603.08274.

[12]  Junyi Guo,et al.  A stochastic maximum principle for processes driven by G‐Brownian motion and applications to finance , 2014, 1402.6793.

[13]  F. Clarke.  Functional Analysis, Calculus of Variations and Optimal Control , 2013 .

[14]  H. M. Soner,et al.  Martingale Representation Theorem for the G-Expectation , 2010, 1001.3802.

[15]  Alexander Shapiro,et al.  Lectures on Stochastic Programming: Modeling and Theory , 2009 .

[16]  S. Peng.  G -Expectation, G -Brownian Motion and Related Stochastic Calculus of Itô Type , 2006, math/0601035.

[17]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[18]  M. Sion.  On general minimax theorems , 1958 .

[19]  Michael Ulbrich,et al.  AN INTERIOR-POINT APPROACH FOR SOLVING RISK-AVERSE 1 PDE-CONSTRAINED OPTIMIZATION PROBLEMS WITH 2 COHERENT RISK MEASURES , 2019 .

[20]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[21]  Jiongmin Yong,et al.  A Variational Formula for Stochastic Controls and Some Applications , 2007 .

[22]  Ivan P. Gavrilyuk,et al.  Variational analysis in Sobolev and BV spaces , 2007, Math. Comput..

[23]  G. Parisi.  Brownian motion , 2005, Nature.

[24]  H. Frankowska,et al.  A stochastic filippov theorem , 1994 .

[25]  H. Frankowska.  The maximum principle for an optimal solution to a differential inclusion with end points constraints , 1987 .