Finite Horizon Optimal Stopping of Time-Discontinuous Functionals with Applications to Impulse Control with Delay

We study finite horizon optimal stopping problems for continuous-time Feller-Markov processes. The functional depends on time, state, and external parameters and may exhibit discontinuities with respect to the time variable. Both left- and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, on the initial state of the process, and on the stopping horizon. We construct $\varepsilon$-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag, and execution delay.

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

[2]  Damien Lamberton.  Optimal stopping with irregular reward functions , 2009 .

[3]  Bernt Øksendal,et al.  Optimal Stochastic Impulse Control with Delayed Reaction , 2008 .

[4]  Lepeltier,et al.  A PROBABILISTIC APPROACH TO THE REDUITE IN OPTIMAL STOPPING , 2008 .

[5]  L. Stettner,et al.  Impulsive Control of Portfolios , 2007 .

[6]  H. Pham,et al.  Impulse control problem on finite horizon with execution delay , 2007, math/0703769.

[7]  E. Bayraktar,et al.  The Effects of Implementation Delay on Decision-Making Under Uncertainty , 2007, math/0703833.

[8]  Bruno Bassan,et al.  Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions , 2002 .

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

[10]  Bernt Øksendal,et al.  Optimal Consumption and Portfolio with Both Fixed and Proportional Transaction Costs , 2001, SIAM J. Control. Optim..

[11]  Agnès Sulem,et al.  Explicit Solution of Inventory Problems with Delivery Lags , 1995, Math. Oper. Res..

[12]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[13]  Wendell H. Fleming,et al.  Advances in Filtering and Optimal Stochastic Control , 1982 .

[14]  Optimal stopping of controlled Markov processes , 1982 .

[15]  Convex inequalities in stochastic control , 1981 .

[16]  J. Zabczyk,et al.  Strong envelopes of stochastic processes and a penalty method , 1981 .

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

[18]  J. Menaldi.  On the Optimal Stopping Time Problem for Degenerate Diffusions , 1980 .

[19]  Avner Friedman,et al.  Optimal Stopping Problems in Stochastic Control , 1979 .

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

[21]  J. Bismut,et al.  Temps d'arrÊt optimal, théorie générale des processus et processus de Markov , 1977 .

[22]  J. Mertens.  Strongly supermedian functions and optimal stopping , 1973 .

[23]  V. Matskyavichyus.  Passing to the limit in optimal stopping problems for Markov processes , 1973 .

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