By Łukasz Delong
Backward stochastic differential equations with jumps can be utilized to unravel difficulties in either finance and insurance.
Part I of this ebook offers the speculation of BSDEs with Lipschitz turbines pushed by means of a Brownian movement and a compensated random degree, with an emphasis on these generated via step tactics and Lévy tactics. It discusses key effects and methods (including numerical algorithms) for BSDEs with jumps and reports filtration-consistent nonlinear expectancies and g-expectations. half I additionally makes a speciality of the mathematical instruments and proofs that are an important for knowing the theory.
Part II investigates actuarial and fiscal functions of BSDEs with jumps. It considers a normal monetary and coverage version and bargains with pricing and hedging of coverage equity-linked claims and asset-liability administration difficulties. It also investigates ideal hedging, superhedging, quadratic optimization, software maximization, indifference pricing, ambiguity threat minimization, no-good-deal pricing and dynamic hazard measures. half III offers another worthy periods of BSDEs and their applications.
This booklet will make BSDEs extra obtainable to people who have an interest in utilising those equations to actuarial and fiscal difficulties. it will likely be useful to scholars and researchers in mathematical finance, danger measures, portfolio optimization in addition to actuarial practitioners.
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Extra info for Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps
For any α > 0 we have the inequality 2 Y (s) − Y (s) f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) ≤ 2 Y (s) − Y (s) f s, Y (s), Z(s), U (s) − f s, Y (s), Z(s), U (s) + 2 Y (s) − Y (s) f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) 2 ≤ 2α Y (s) − Y (s) + + + K α 1 f s, Y (s), Z(s), U (s) − f s, Y (s), Z(s), U (s) α 2 Y (s) − Y (s) + Z(s) − Z (s) 2 2 2 R U (s, z) − U (s, z) Q(s, dz)η(s) . 9), then we are done. Such a choice of α is only possible if ρ is sufficiently large. 5) for large ρ.
14 Let N be the jump measure of a compound Poisson process with jump size distribution q. Consider the stop-loss contract ξ = (J (T ) − K)+ where t ∞ J (t) = 0 0 yN(ds, dy), 0 ≤ t ≤ T , is the compound Poisson process used for modelling insurer’s claims. We assume that the claim size distribution q is supported ∞ on (0, ∞) and satisfies 0 y 2 q(dy) < ∞. In applications we would like to use the Malliavin derivative of ξ . e. (ω, t, z) ∈ Ω × [0, T ] × (0, ∞). Bibliographical Notes Definitions are taken from He et al.
From the practical point of view, the predictable representation yields hedging strategies for financial claims. Let us consider a probability space (Ω, F , P) with a filtration F = (Ft )0≤t≤T . 10) where Z and U are F -predictable processes integrable with respect to W and N˜ . 26 2 Stochastic Calculus This is our second standing assumption, next to (RM) from Sect. 1. 31 in He et al. (1992). 11) where (τn )n≥1 is a sequence of stopping times. 11) we can conclude that Z ∈ H2 (R) and U ∈ H2N (R).
Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps by Łukasz Delong