We consider a general convex stochastic control model. Our main interest concerns monotonicity results and bounds for the value functions and for optimal policies. In particular, we show how the value functions depend on the transition kernels and we present conditions for a lower bound of an optimal policy. Our approach is based on convex stochastic orderings of probability measures. We derive several interesting sufficient conditions of these ordering concepts, where we make also use of the Blackwell ordering. The structural results are illustrated by partially observed control models and Bayesian information models.Key Words: Convex stochastic control models, Monotonicity results, Bounds, Convex stochastic orderings and Blackwell ordering.
I IntroductionWe consider a discrete-time stochastic control model. Throughout this paper it is assumed that the value functions are convex. Such convex stochastic control models arise e.g. in investment and portfolio theory, resource management and production planning (for bibliographies and further applications see Heyman/ Sobel (1984) and Hinderer (1984)). In partially observed control models and Bayesian information models, the value functions are always convex (cf. Rieder (1991)). Hence these models under uncertainty are important examples of convex stochastic control models.Our main interest concerns structural properties of the value functions and of optimal policies. In particular, we are interested in monotonicity results and bounds. These results can be used to derive efficient algorithms and good approximations. They are also useful for a sensitivity analysis of stochastic models. In this paper we concentrate on the subset of convex value functions. Analoguous results can be derived in the same way for concave value functions and some other classes of functions.In section 2 we introduce a general convex stochastic control model. Partially observed control models and Bayesian information models are formulated as important special cases. Convex stochastic orderings of probability measures are defined and studied in detail in section 3. We present several properties and 0340-9422/94/39 : 2/187 -207 $2.50 9 1994 Physica-Verlag, Heidelberg
The purpose of this article is to analyze and compare two standard portfolio insurance methods: Option-based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI). Various stochastic dominance criteria up to third order are considered. We derive parameter conditions implying the second-and third-order stochastic dominance of the CPPI strategy. In particular, restrictions on the CPPI multiplier resulting from the spread between the implied volatility and the empirical volatility are analyzed.
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