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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: sb30@columbia.edu
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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 

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