Risk-sensitive optimal stopping with unbounded terminal cost function

In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller–Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.

[1]  Marcin Pitera,et al.  Long-Run Risk-Sensitive Impulse Control , 2019, SIAM J. Control. Optim..

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

[3]  Olivier Menoukeu-Pamen,et al.  Viscosity Solution for Optimal Stopping Problems of Feller Processes , 2018, 1803.03832.

[4]  N. Bäuerle,et al.  Risk-sensitive stopping problems for continuous-time Markov chains , 2016, 1610.05922.

[5]  J. Norris.  Appendix: probability and measure , 1997 .

[6]  Optimal stopping time problem in a general framework , 2012 .

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

[8]  Lukasz Stettner,et al.  Penalty Method for Finite Horizon Stopping Problems , 2011, SIAM J. Control. Optim..

[9]  Albert N. Shiryaev,et al.  Optimal Stopping Rules , 1980, International Encyclopedia of Statistical Science.

[10]  Lukasz Stettner,et al.  Finite Horizon Optimal Stopping of Time-Discontinuous Functionals with Applications to Impulse Control with Delay , 2010, SIAM J. Control. Optim..

[11]  R. Carmona,et al.  OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS , 2008 .

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

[13]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal Stopping and Free-Boundary Problems , 2006 .

[14]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

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

[16]  Bruno Bassan,et al.  Regularity of the value function and viscosity solutions in optimal stopping problems for general Markov processes , 2002 .

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

[18]  Daniel Hernández-Hernández,et al.  Risk Sensitive Markov Decision Processes , 1997 .

[19]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[20]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

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

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

[23]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour IX-1979 , 1981 .

[24]  N. Karoui.  Les Aspects Probabilistes Du Controle Stochastique , 1981 .

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

[26]  A. G. Fakeev.  Optimal Stopping of a Markov Process , 1971 .

[27]  A. G. Fakeev.  Optimal Stopping Rules for Stochastic Processes with Continuous Parameter , 1970 .

[28]  H. D. Miller,et al.  The Theory Of Stochastic Processes , 1977, The Mathematical Gazette.

[29]  W. Rudin.  Principles of mathematical analysis , 1964 .