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An explicit solution for an optimal stopping/optimal control problem which models an asset sale. (English) Zbl 1165.60021

The paper deals with an mixed optimal stopping and optimal control problem which models the decision facing a risk-averse agent over when to sell an asset. The market is incomplete so that the asset exposure cannot be hedged. In addition to the decision over when to sell, the agent has to choose a control strategy which corresponds to a feasible wealth process. The problem is formulated as one involving the choice of a stopping time and a martingale. Let $U\left(·,·,·\right)$ be payoff function in the optimization problem which form depends of utility function and it is a function of time, control process $X$ and an exogeneous Markov process $Y$. The set of feasible strategies is $𝒳=\left\{X:𝔼{X}_{0}=x;\phantom{\rule{0.277778em}{0ex}}{X}_{u}+{Y}_{u}\ge 0,\phantom{\rule{0.277778em}{0ex}}u\le \tau ;X\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{càdlàg}\phantom{\rule{4.pt}{0ex}}\text{martingale}\phantom{\rule{4.pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}{\left[X,Y\right]}_{t}\equiv 0\right\}$. The generic problem is to find ${V}_{☆}={sup}_{\tau ,\phantom{\rule{0.277778em}{0ex}}X\in 𝒳}𝔼\left[U\left(\tau ,{X}_{\tau },{Y}_{\tau }\right)\right]$ [see I. Karatzas and H. Wang, SIAM J. Control Optim. 39, 306–329 (2000; Zbl 0963.93079) for terminology of such optimization problem]. The form of the solution has been proposed and verified. The solution is available in a very explicit form, that for some parameter values the optimal strategy is more sophisticated than might originally be expected, and that although the setup is based on continuous diffusions, the optimal martingale may involve a jump process.

One interpretation of the solution is that it is optimal for the risk-averse agent to gamble. The elements in the model which are necessary for the main conclusion that gambling can be beneficial are that the market is incomplete, that the real asset is indivisible, and that the asset sale is irreversible[see A. K. Dixit and R. S. Pindyck, Investment under Uncertainty. Princeton Univ. Press. (1994) for real option problems and F.T. Bruss and T. S. Ferguson, Ann. Appl. Probab. 12, No. 4, 1202–1226 (2002; Zbl 1005.60054), D. Łebek and K. Szajowski, Bull. Belg. Math. Soc. - Simon Stevin 14, No. 1, 143–155 (2007; Zbl 1138.60317)] for high risk investments].

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 91A60 Probabilistic games; gambling 60G44 Martingales with continuous parameter 91B28 Finance etc. (MSC2000) 93E20 Optimal stochastic control (systems)