By Łukasz Delong

Backward stochastic differential equations with jumps can be utilized to resolve difficulties in either finance and insurance.

Part I of this ebook provides the idea of BSDEs with Lipschitz turbines pushed via a Brownian movement and a compensated random degree, with an emphasis on these generated by means of step techniques and Lévy methods. It discusses key effects and methods (including numerical algorithms) for BSDEs with jumps and experiences filtration-consistent nonlinear expectancies and g-expectations. half I additionally makes a speciality of the mathematical instruments and proofs that are the most important for figuring out the theory.

Part II investigates actuarial and monetary purposes of BSDEs with jumps. It considers a common 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, application maximization, indifference pricing, ambiguity chance minimization, no-good-deal pricing and dynamic hazard measures. half III provides another invaluable periods of BSDEs and their applications.

This booklet will make BSDEs extra obtainable to those that have an interest in utilising those equations to actuarial and monetary difficulties. it is going to be precious to scholars and researchers in mathematical finance, hazard measures, portfolio optimization in addition to actuarial practitioners.

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Extra resources for Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps

Example text

3 Stochastic Integration 23 where we integrate with respect to the compensator of a random measure N . t Then ( 0 R V (s, z)N˜ (ds, dz), 0 ≤ t ≤ T ) is a càdlàg local martingale and t ( 0 R V (s, z)N (ds, dz), 0 ≤ t ≤ T ) is a càdlàg process. Let N be the jump measure of a càdlàg process J . We also have the property t R 0 E[ V (s, z)N (ds, dz) = V s, J (s) 1 J (s)=0 (s), 0 ≤ t ≤ T. s∈(0,t] Notice that if V is a non-negative predictable process satisfying T 0 R V (t, z)Q(t, dz)η(t)dt] < ∞, then T E R 0 T V (t, z)N(dt, dz) = E R 0 V (t, z)Q(t, dz)η(t)dt .

12) we finally get Y −Y 2 √ ≤ K˜ (2 + ρT )E eρT ξ − ξ 2 2 ds . 6). 4. 7). 17) we derive Y −Y 2 S2 ≤ Kˆ E eρT ξ − ξ T +E 2 eρs f s, Y (s), Z(s), U (s) − f s, Y (s), Z(s), U (s) 2 ds 0 T +E eρs f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) 2 ds . 13). In Sect. 1. 9) we get E eρt Y (t) − Y (t) 2 T + ρE 2 eρs Y (s) − Y (s) ds t T +E 2 eρs Z(s) − Z (s) ds t T +E 2 R t ≤ E eρT ξ − ξ T + 2E eρs U (s, z) − U (s, z) Q(s, dz)η(s)ds 2 eρs Y (s) − Y (s) t · f s, Y (s), Z(s), U (s) − f s, Y (s), Z(s), U (s) ds T + 2E eρs Y (s) − Y (s) t · f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) ds .

The Girsanov-Meyer theorem now yields that t 0 R V m (s, z) N (ds, dz) − 1 + κ(s, z) Q(s, dz)η(s)ds , 0 ≤ t ≤ T, is a Q-local martingale. Let (τk )k≥1 be a localizing sequence of stopping times for t m ˜Q 0 R V (s, z)N (ds, dz), let (τn )n≥1 be a localizing sequence of stopping times t for 0 R V (s, z)N (ds, dz), and let τ be a stopping time. We have 30 2 τk ∧τn ∧τ EQ R 0 V m (s, z)N (ds, dz) τk ∧τn ∧τ = EQ Stochastic Calculus R 0 V m (s, z) 1 + κ(s, z) Q(s, dz)η(s)ds . Taking the limit k → ∞, m → ∞ and applying the Lebesgue monotone convergence theorem, we show τn ∧τ EQ R 0 V (s, z)N (ds, dz) τn ∧τ = EQ 0 R V (s, z) 1 + κ(s, z) Q(s, dz)η(s)ds .

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