70 research outputs found
A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuation
We propose a numerical recipe for risk evaluation defined by a backward
stochastic differential equation. Using dual representation of the risk
measure, we convert the risk valuation to a stochastic control problem where
the control is a certain Radon-Nikodym derivative process. By exploring the
maximum principle, we show that a piecewise-constant dual control provides a
good approximation on a short interval. A dynamic programming algorithm extends
the approximation to a finite time horizon. Finally, we illustrate the
application of the procedure to financial risk management in conjunction with
nested simulation and on an multidimensional portfolio valuation problem
Common Mathematical Foundations of Expected Utility and Dual Utility Theories
We show that the main results of the expected utility and dual utility
theories can be derived in a unified way from two fundamental mathematical
ideas: the separation principle of convex analysis, and integral
representations of continuous linear functionals from functional analysis. Our
analysis reveals the dual character of utility functions. We also derive new
integral representations of dual utility models
Conditional Risk Mappings
We introduce an axiomatic definition of a conditional convex risk mapping. By employing the techniques of conjugate duality we derive properties of conditional risk mappings. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings.Risk, Convex Analysis, Conjugate Duality, Stochastic Optimization, Dynamic Programming, Multi-Stage Programming
Optimization of Risk Measures
We consider optimization problems involving coherent risk measures. We derive necessary and sufficient conditions of optimality for these problems, and we discuss the nature of the nonanticipativity constraints. Next, we introdice dynamic risk measures, and we formulate multistage optimization problems involving these measures. Conditions similar to dynamic programming equations are developed. The theoretical considerations are illustrated with many examples of mean-risk models applied in practice.risk measures, mean-risk models, duality, optimization, dynamic programming
Convexification of Stochastic Ordering
We consider sets defined by the usual stochastic ordering relation and by the second order stochastic dominance relation. Under fairy general assumptions we prove that in the space of integrable random variables the closed convex hull of the first set is equal to the second set.Stochastic Dominance, Stochastic Ordering
Portfolio Optimization With Stochastic Dominance Constraints
We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.portfolio optimization, stochastic dominance, risk, utility functions, duality
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