Singular mean-field control games with applications to optimal harvesting and investment problems

This paper studies singular mean field control problems and singular mean field stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Applications are given to optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and to mean-field singular investment games. In particular, a simple singular mean-field investment game is studied where the Nash equilibrium exists but is not unique.

[1]  R. Pindyck.  Irreversibility, Uncertainty, and Investment , 1990 .

[2]  Robert S. Pindyck,et al.  Irreversibility and the explanation of investment behavior , 1990 .

[3]  T. Kobila.  Solvable Stochastic Investment Problems , 1994 .

[4]  Said Hamadène,et al.  Backward–forward SDE’s and stochastic differential games , 1998 .

[5]  M. P. Johansson,et al.  Malliavin calculus for Levy processes with applications to finance , 2004 .

[6]  M. Royer.  Backward stochastic differential equations with jumps and related non-linear expectations , 2006 .

[7]  K. Ramanan,et al.  The Skorokhod problem in a time-dependent interval , 2007, 0712.2863.

[8]  Bernt Øksendal,et al.  Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps , 2009, SIAM J. Control. Optim..

[9]  Bernt Øksendal,et al.  Portfolio optimization under model uncertainty and BSDE games , 2011 .

[10]  D. Nualart,et al.  Malliavin calculus for backward stochastic differential equations and application to numerical solutions , 2011, 1202.4625.

[11]  Daniel Andersson,et al.  A Maximum Principle for SDEs of Mean-Field Type , 2011 .

[12]  J. Hosking.  A Stochastic Maximum Principle for a Stochastic Differential Game of a Mean-Field Type , 2012 .

[13]  Bernt Øksendal,et al.  A maximum principle for stochastic differential games with g-expectations and partial information , 2012 .

[14]  Bernt Øksendal,et al.  A mean-field stochastic maximum principle via Malliavin calculus , 2012 .

[15]  Bernt Øksendal,et al.  Singular Stochastic Control and Optimal Stopping with Partial Information of Itô-Lévy Processes , 2012, SIAM J. Control. Optim..

[16]  Liangquan Zhang.  The Relaxed Stochastic Maximum Principle in the Mean-field Singular Controls , 2012, 1202.4129.

[17]  M. Quenez,et al.  BSDEs with jumps, optimization and applications to dynamic risk measures , 2013 .

[18]  Bernt Øksendal,et al.  Risk minimization in financial markets modeled by Itô-Lévy processes , 2014, 1402.3131.

[19]  Bernt Øksendal,et al.  Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty , 2014, J. Optim. Theory Appl..

[20]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.