Optimal cross hedging of Insurance derivatives using quadratic BSDEs

[摘要] ENGLISH ABSTRACT: We consider the utility portfolio optimization problem of an investor whoseactivities are influenced by an exogenous financial risk (like bad weather orenergy shortage) in an incomplete financial market. We work with a fairlygeneral non-Markovian model, allowing stochastic correlations between theunderlying assets. This important problem in finance and insurance is tackledby means of backward stochastic differential equations (BSDEs), which havebeen shown to be powerful tools in stochastic control. To lay stress on theimportance and the omnipresence of BSDEs in stochastic control, we presentthree methods to transform the control problem into a BSDEs. Namely, themartingale optimality principle introduced by Davis, the martingale representationand a method based on Itô-Ventzell's formula. These approaches enableus to work with portfolio constraints described by closed, not necessarily convexsets and to get around the classical duality theory of convex analysis. Thesolution of the optimization problem can then be simply read from the solutionof the BSDE. An interesting feature of each of the different approaches is thatthe generator of the BSDE characterizing the control problem has a quadraticgrowth and depends on the form of the set of constraints. We review somerecent advances on the theory of quadratic BSDEs and its applications. Thereis no general existence result for multidimensional quadratic BSDEs. In theone-dimensional case, existence and uniqueness strongly depend on the formof the terminal condition. Other topics of investigation are measure solutionsof BSDEs, notably measure solutions of BSDE with jumps and numerical approximations.We extend the equivalence result of Ankirchner et al. (2009)between existence of classical solutions and existence of measure solutions tothe case of BSDEs driven by a Poisson process with a bounded terminal condition.We obtain a numerical scheme to approximate measure solutions. Infact, the existing self-contained construction of measure solutions gives riseto a numerical scheme for some classes of Lipschitz BSDEs. Two numericalschemes for quadratic BSDEs introduced in Imkeller et al. (2010) and based,respectively, on the Cole-Hopf transformation and the truncation procedureare implemented and the results are compared.Keywords: BSDE, quadratic growth, measure solutions, martingale theory,numerical scheme, indifference pricing and hedging, non-tradable underlying,defaultable claim, utility maximization.

[发布机构] Stellenbosch University