Price systems for markets with transaction costs and control problems for some finance problems

In a market with transaction costs, the price of a derivative can be expressed in terms of (preconsistent) price systems (after Kusuoka (1995)). In this paper, we consider a market with binomial model for stock price and discuss how to generate the price systems. From this, the price formula of a derivative can be reformulated as a stochastic control problem. Then the dynamic programming approach can be used to calculate the price. We also discuss optimization of expected utility using price systems.Comment: Published at http://dx.doi.org/10.1214/074921706000001094 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Differential games of inf-sup type and Issacs equations

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Asymptotics of the probability minimizing a "down-side" risk

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.