Mean-Variance Hedging on Uncertain Time Horizon in a Market with a Jump

• Published in: Applied mathematics & optimization Vol. 68; no. 3; pp. 413 - 444
• Main Authors: Kharroubi, Idris, Lim, Thomas, Ngoupeyou, Armand
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2013; Springer Nature B.V; Springer Verlag (Germany)
• Subjects: Algebra; Applied mathematics; Brownian motion; Calculus of Variations and Optimal Control; Optimization; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Control; Control systems; Decomposition; DIFFERENTIAL EQUATIONS; Hedging; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; MATHEMATICAL SOLUTIONS; Mathematics; Mathematics and Statistics; Numerical and Computational Physics; Probability; RANDOMNESS; Simulation; STOCHASTIC PROCESSES; Studies; Systems Theory; Theorems; Theoretical; UNCERTAINTY PRINCIPLE; Backward SDE; Progressive enlargement of filtration; Decomposition in the reference filtration; Mean-variance hedging; Random horizon; Jump processes; 60G57; random horizon; 93E20; 60H10; pro-gressive enlargement of filtration; decomposition in the reference filtration AMS subject classifications: 91B30; jump processes
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-013-9213-5

In this work, we study the problem of mean-variance hedging with a random horizon T ∧ τ , where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.