The Annals of Applied Probability

An explicit solution for an optimal stopping/optimal control problem which models an asset sale

Vicky Henderson and David Hobson

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In this article we study an optimal stopping/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.

We formulate this problem as one involving the choice of a stopping time and a martingale. We conjecture the form of the solution and verify that the candidate solution is equal to the value function.

The interesting features of the solution are that it 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.

Article information

Ann. Appl. Probab., Volume 18, Number 5 (2008), 1681-1705.

First available in Project Euclid: 30 October 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91A60: Probabilistic games; gambling [See also 60G40]
Secondary: 60G44: Martingales with continuous parameter 91B28 93E20: Optimal stochastic control

Optimal stopping singular control utility maximization incomplete market local time gambling


Henderson, Vicky; Hobson, David. An explicit solution for an optimal stopping/optimal control problem which models an asset sale. Ann. Appl. Probab. 18 (2008), no. 5, 1681--1705. doi:10.1214/07-AAP511.

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  • [1] Dixit, A. K. and Pindyck, R. S. (1994). Investment under Uncertainty. Princeton Univ. Press.
  • [2] Evans, J. D., Henderson, V. and Hobson, D. (2007). Optimal timing for an asset sale in an incomplete market. Math. Finance. To appear.
  • [3] Henderson, V. (2007). Valuing the option to invest in an incomplete market. Math. Financ. Econ. 1 103–128.
  • [4] Jacka, S. D. (1991). Optimal stopping and best constants for Doob-like inequalities. I. The case p=1. Ann. Probab. 19 1798–1821.
  • [5] Karatzas, I. and Kou, S. G. (1998). Hedging American contingent claims with constrained portfolios. Finance Stoch. 2 215–258.
  • [6] Karatzas, I. and Ocone, D. (2002). A leavable bounded-velocity stochastic control problem. Stochastic Process. Appl. 99 31–51.
  • [7] Karatzas, I. and Sudderth, W. D. (1999). Control and stopping of a diffusion process on an interval. Ann. Appl. Probab. 9 188–196.
  • [8] Karatzas, I. and Wang, H. (2000). Utility maximization with discretionary stopping. SIAM J. Control Optim. 39 306–329 (electronic).
  • [9] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econom. Statist. 51 247–257.
  • [10] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley, Chichester.
  • [11] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus. Reprint of the second (1994) edition. Cambridge Univ. Press, Cambridge.
  • [12] Samuelson, P. A. (1965). Rational theory of warrant pricing. With an appendix by H. P. McKean, A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Management Review 6 13–31 and 32–39.