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 provides the idea of BSDEs with Lipschitz turbines pushed through a Brownian movement and a compensated random degree, with an emphasis on these generated through step procedures and Lévy tactics. It discusses key effects and strategies (including numerical algorithms) for BSDEs with jumps and reviews filtration-consistent nonlinear expectancies and g-expectations. half I additionally specializes in the mathematical instruments and proofs that are an important for figuring out the theory.
Part II investigates actuarial and monetary functions of BSDEs with jumps. It considers a normal monetary and assurance version and bargains with pricing and hedging of assurance equity-linked claims and asset-liability administration difficulties. It also investigates excellent hedging, superhedging, quadratic optimization, application maximization, indifference pricing, ambiguity chance minimization, no-good-deal pricing and dynamic chance measures. half III provides another necessary sessions of BSDEs and their applications.
This publication will make BSDEs extra available to people who have an interest in utilizing 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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Additional info for Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps
1992). Moreover, given a Brownian motion W and an independent jump process J (a Lévy process or a step process), the weak property of predictable representation holds for (W, J ) and the product of their completed natural filtrations. 22 in He et al. (1992). Hence, by the change of measure we can establish the predictable representation for a Brownian motion and a jump process with a random compensator (depending on W and J ), see Sect. 5. For such a construction we refer to Becherer (2006) and Chap.
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 .
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 .