Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-20T03:31:43.666Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal strategies in a risk selection investment model

Part of: Operations research and management science Special processes Stochastic processes

Published online by Cambridge University Press:  19 February 2016

David Assaf ,
Yuliy Baryshnikov  and
Wolfgang Stadje
David Assaf*
Affiliation:
University of Jerusalem
Yuliy Baryshnikov*
Affiliation:
Universität Osnabrück
Wolfgang Stadje*
Affiliation:
Universität Osnabrück
*
∗ Postal address: Department of Statistics, Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel.
∗∗ Postal address: Fachbereich Mathematik Informatik, Universität Osnabrück, 49069 Osnabrück, Germany.
∗∗ Postal address: Fachbereich Mathematik Informatik, Universität Osnabrück, 49069 Osnabrück, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the following stochastic investment model: opportunities occur randomly over time, following a renewal process with mean interarrival time d, and at each of them the decision-maker can choose a distribution for an instantaneous net gain (or loss) from the set of all probability measures that have some prespecified expected value e and for which his maximum possible loss does not exceed his current capital. Between the investments he spends money at some constant rate. The objective is to avoid bankruptcy as long as possible. For the case e>d we characterize a strategy maximizing the probability that ruin never occurs. It is proved that the optimal value function is a concave function of the initial capital and uniquely determined as the solution of a fixed point equation for some intricate operator. In general, two-point distributions suffice; furthermore, we show that the cautious strategy of always taking the deterministic amount e is optimal if the interarrival times are hyperexponential, and, in the case of bounded interarrival times, is optimal ‘from some point on’, i.e. whenever the current capital exceeds a certain threshold. In the case e = 0 we consider a class of natural objective functions for which the optimal strategies are non-stationary and can be explicitly determined.

Keywords

Stochastic investment modelcontrolled risk processrisk selectionruin probabilityminimal expected ruin timedynamic programmingoptimal strategycautious strategy

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 90B50: Management decision making, including multiple objectives 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 518 - 539
Copyright
Copyright © Applied Probability Trust 2000 

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

[1] Browne, S. (1990). Maximizing the expected time to ruin for a company operating N distinct funds with a ‘superclaims’ process. Insur. Math. Econ. 9, 33–37.Google Scholar
[2] 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
[3] Browne, S. (1997). Survival and growth with a liability: optimal portfolio strategies in continuous time. Math. Operat. Res. 22, 468–493.Google Scholar
[4] Browne, S. and Whitt, W. (1996). Portfolio choice and the Bayesian Kelly criterion. Adv. Appl. Prob. 28, 1145–1176.Google Scholar
[5] Duffie, D. (1996). Dynamic Asset Pricing Theory, 2nd edn. Princeton University Press, Princeton, NJ.Google Scholar
[6] Dynkin, E. B. and Yushkevich, A. A. (1979). Controlled Markov Processes. Springer, New York.Google Scholar
[7] Karatzas, I. (1997). Lectures on the Mathematics of Finance. CRM Monograph Series Vol. 8, American Mathematical Society, Washington, DC.Google Scholar
[8] Kelly, J. L. (1956). A new interpretation of information rate. Bell System Tech. J. 35, 917–926.Google Scholar
[9] MacLean, L. C. and Ziemba, W. T. (1991). Growth-security profiles in capital accumulation under risk. Ann. Operat. Res. 31, 501–510.Google Scholar
[10] Maitra, A. P. and Sudderth, W. D. (1996). Discrete Gambling and Stochastic Games. Springer, New York.Google Scholar
[11] Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modelling. Springer, New York.Google Scholar
[12] Pestien, V. C. and Sudderth, W. D. (1988). Continuous time casino problems. Math. Operat. Res. 13, 364–376.Google Scholar
[13] Prabhu, N. U. (1965). Queues and Inventories. John Wiley, New York.Google Scholar
[14] Puterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, New York.Google Scholar
[15] Ross, S. M. (1974). Dynamic programming and gambling models. Adv. Appl. Prob. 6, 598–606.Google Scholar
[16] Rotando, L. M. and Thorp, E. O. (1992). The Kelly criterion and the stock market. Amer. Math. Monthly 99, 922–931.Google Scholar