Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T12:28:14.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic differential portfolio games

Part of: Markov processes Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Sid Browne
Sid Browne*
Affiliation:
Columbia University and Goldman Sachs & Co.
*
∗Postal address: 402 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study stochastic dynamic investment games in continuous time between two investors (players) who have available two different, but possibly correlated, investment opportunities. There is a single payoff function which depends on both investors’ wealth processes. One player chooses a dynamic portfolio strategy in order to maximize this expected payoff, while his opponent is simultaneously choosing a dynamic portfolio strategy so as to minimize the same quantity. This leads to a stochastic differential game with controlled drift and variance. For the most part, we consider games with payoffs that depend on the achievement of relative performance goals and/or shortfalls. We provide conditions under which a game with a general payoff function has an achievable value, and give an explicit representation for the value and resulting equilibrium portfolio strategies in that case. It is shown that non-perfect correlation is required to rule out trivial solutions. We then use this general result explicitly to solve a variety of specific games. For example, we solve a probability maximizing game, where each investor is trying to maximize the probability of beating the other's return by a given predetermined percentage. We also consider objectives related to the minimization or maximization of the expected time until one investor's return beats the other investor's return by a given percentage. Our results allow a new interpretation of the market price of risk in a Black-Scholes world. Games with discounting are also discussed, as are games of fixed duration related to utility maximization.

Keywords

Stochastic differential gamesportfolio theorystochastic controldiffusionsmartingales

MSC classification

Secondary: 93E20: Optimal stochastic control 60G40: Stopping times; optimal stopping problems; gambling theory 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 126 - 147
Copyright
Copyright © 2000 by The Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bell, R. M., and Cover, T. M. (1980). Competitive optimality of logarithmic investment. Math. Operat. Res. 5, 161–166.CrossRefGoogle Scholar
Browne, S. (1995). Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Operat. Res. 20, 937–958.Google Scholar
Browne, S. (1997). Survival and growth with a fixed liability: optimal portfolios in continuous time. Math. Operat. Res. 22, 468–493.CrossRefGoogle Scholar
Browne, S. (1998). The rate of return from proportional portfolio strategies. Adv. Appl. Prob. 30, 216–238.Google Scholar
Browne, S. (1999). Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3, 275–294.CrossRefGoogle Scholar
Dubins, L. E., and Savage, L. J. (1965, repr. 1976). How to Gamble If You Must: Inequalities for Stochastic Processes. Dover, New York.Google Scholar
Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press.Google Scholar
Elliott, R. (1976). The existence of value in stochastic differential games. SIAM J. Contr. and Opt. 14, 85–94.CrossRefGoogle Scholar
Fleming, W. H., and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, New York.Google Scholar
Fleming, W. H., and Souganides, P. E. (1989). On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38, 293–314.Google Scholar
Friedman, A. (1976). Stochastic Differential Equations and Applications, Vol 2. Academic Press, New York.Google Scholar
Heath, D., Orey, S., Pestien, V. C., and Sudderth, W. D. (1987). Minimizing or maximizing the expected time to reach zero. SIAM J. Contr. and Opt. 25, 195–205.Google Scholar
Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, New York.Google Scholar
Maitra, A. P., and Sudderth, W. D. (1996). Discrete Gambling and Stochastic Games. Springer, New York.Google Scholar
Mazumdar, M., and Radner, R. (1991). Linear models of economic survival under production uncertainty. Econ. Theory 1, 13–30.CrossRefGoogle Scholar
Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Econ. Theory 3, 373–413.CrossRefGoogle Scholar
Merton, R. (1990). Continuous Time Finance. Blackwell, Oxford.Google Scholar
Nilakantan, L. (1993). Continuous-Time Stochastic Games. Ph.D. dissertation, University of Minnesota, MN.Google Scholar
Orey, S., Pestien, V. C., and Sudderth, W. D. (1987). Reaching zero rapidly. SIAM J. Contr. and Opt. 25, 1253–1265.CrossRefGoogle Scholar
Pestien, V. C., and Sudderth, W. D. (1985). Continuous-time red and black: how to control a diffusion to a goal. Math. Operat. Res. 10, 599–611.Google Scholar
Pestien, V. C., and Sudderth, W. D. (1988). Continuous-time casino problems. Math. Operat. Res. 13, 364–376.Google Scholar