Long-Run Risk-Sensitive Impulse Control

In this paper we consider long-run risk sensitive average cost impulse control applied to a continuous-time Feller-Markov process. Using the probabilistic approach, we show how to get a solution to a suitable continuous-time Bellman equation and link it with the impulse control problem. The optimal strategy for the underlying problem is constructed as a limit of dyadic impulse strategies by exploiting regularity properties of the linked risk sensitive optimal stopping value functions. In particular, this shows that the discretized setting could be used to approximate near optimal strategies for the underlying continuous time control problem, which facilitates the usage of the standard approximation tools. For completeness, we present examples of processes that could be embedded into our framework.

[1]  Maurice Robin,et al.  On Some Ergodic Impulse Control Problems with Constraint , 2018, SIAM J. Control. Optim..

[2]  Ł. Stettner,et al.  Risk sensitive optimal stopping , 2019, 1912.02486.

[3]  J. Menaldi,et al.  Ergodic control of reflected diffusions with jumps , 1989 .

[4]  Tao Chen,et al.  Adaptive Robust Control under Model Uncertainty , 2017, SIAM J. Control. Optim..

[5]  Lukasz Stettner,et al.  Zero-Sum Markov Games with Impulse Controls , 2020, SIAM J. Control. Optim..

[6]  Łukasz Stettner,et al.  On impulsive control with long run average cost criterion , 1983 .

[7]  S. Pliska,et al.  Risk-Sensitive Dynamic Asset Management , 1999 .

[8]  A Remark on Impulse Control Problems with Risk-sensitive Criteria , 2007 .

[9]  L. Stettner,et al.  On Risk-Sensitive Ergodic Impulsive Control of Markov Processes , 2002 .

[10]  Hideo Nagai.  Risk-sensitive quasi-variational inequalities for optimal investment with general transaction costs , 2006, Asymptot. Anal..

[11]  P. Whittle.  Risk-Sensitive Optimal Control , 1990 .

[12]  Tomasz R. Bielecki,et al.  Economic Properties of the Risk Sensitive Criterion for Portfolio Management , 2003 .

[13]  Thomas Parisini,et al.  Neural Approximations for Optimal Control and Decision , 2019 .

[14]  Lukasz Stettner,et al.  Risk-Sensitive Control of Discrete-Time Markov Processes with Infinite Horizon , 1999, SIAM J. Control. Optim..

[15]  Marcin Pitera,et al.  Long-run risk sensitive dyadic impulse control , 2019 .

[16]  Sören Christensen,et al.  A General Verification Result for Stochastic Impulse Control Problems , 2017, SIAM J. Control. Optim..

[17]  Hideo Nagai,et al.  Optimal Strategies for Risk-Sensitive Portfolio Optimization Problems for General Factor Models , 2002, SIAM J. Control. Optim..

[18]  Xin Guo,et al.  Impulse Control of Multidimensional Jump Diffusions , 2009, SIAM J. Control. Optim..

[19]  Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs , 2007 .

[20]  M. Robin,et al.  Contrôle impulsionnel des processus de Markov , 1978 .

[21]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[22]  Albert N. Shiryaev,et al.  Optimal Stopping Rules , 2011, International Encyclopedia of Statistical Science.

[23]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[24]  P. Moerbeke.  On optimal stopping and free boundary problems , 1973, Advances in Applied Probability.

[25]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[26]  Ibtissam Hdhiri,et al.  Risk sensitive impulse control of non-Markovian processes , 2011, Math. Methods Oper. Res..

[27]  Mark H. A. Davis,et al.  Risk-Sensitive Investment Management , 2014 .

[28]  U. Rieder,et al.  Markov Decision Processes with Applications to Finance , 2011 .

[29]  F. Dufour,et al.  Impulsive Control for Continuous-Time Markov Decision Processes: A Linear Programming Approach , 2014, 1402.6106.

[30]  Jose Luis Menaldi,et al.  Second Order Elliptic Integro-Differential Problems , 2002 .

[31]  Ralph Neuneier,et al.  Risk-Sensitive Reinforcement Learning , 1998, Machine Learning.

[32]  Alain Bensoussan,et al.  Impulse Control and Quasi-Variational Inequalities , 1984 .

[33]  Marcin Pitera,et al.  Long run risk sensitive portfolio with general factors , 2015, Math. Methods Oper. Res..

[34]  Maurice Robin,et al.  On Some Impulse Control Problems with Constraint , 2017, SIAM J. Control. Optim..

[35]  Lukasz Stettner,et al.  Impulse Control Maximizing Average Cost per Unit Time: A Nonuniformly Ergodic Case , 2016, SIAM J. Control. Optim..

[36]  M. Robin.  On Some Impulse Control Problems with Long Run Average Cost , 1981 .