Minimizing shortfall risk and applications to finance and insurance problems

• 成果类型: Article
• 署名作者: Pham, H
• 署名单位: Sorbonne Universite; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 143-172
• 关键词: OPTIMAL CONSUMPTION; Contingent claims; probability; time

摘要

We consider a controlled process governed by X-x,X-theta = x + integraltheta dS + H-theta, where S is a seznimartingale, Theta the set of control processes theta is a convex subset of L (S) and {H-theta : theta is an element of Theta} is a concave family of adapted processes with finite variation. We study the problem of minimizing the shortfall risk defined as the expectation of the shortfall (B - X-T(x, theta)) + weighted by some loss function, where B is a given nonnegative measurable random variable. Such a criterion has been introduced by Follmer and Leukert [Finance Stoch. 4 (1999) 117-146] motivated by a hedging problem in an incomplete financial market context: Theta = L(S) and H-theta - 0. Using change of measures and optional decomposition under constraints, we state an existence result to this optimization problem and show some qualitative properties of the associated value function. A verification theorem in terms of a dual control problem is established which is used to obtain a quantitative description of the solution. Finally, we give some applications to hedging problems in constrained portfolios, large investor and reinsurance models.