Backward Stochastic Differential Equations with Jumps and by Łukasz Delong

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 offers the idea of BSDEs with Lipschitz turbines pushed by way of a Brownian movement and a compensated random degree, with an emphasis on these generated through step strategies and Lévy strategies. It discusses key effects and methods (including numerical algorithms) for BSDEs with jumps and reviews 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 knowing the theory.

Part II investigates actuarial and monetary purposes of BSDEs with jumps. It considers a common monetary and assurance version and offers with pricing and hedging of assurance equity-linked claims and asset-liability administration difficulties. It also investigates excellent hedging, superhedging, quadratic optimization, software maximization, indifference pricing, ambiguity threat minimization, no-good-deal pricing and dynamic chance measures. half III offers another helpful periods of BSDEs and their applications.

This publication will make BSDEs extra obtainable to people who have an interest in using those equations to actuarial and monetary difficulties. it will likely be invaluable to scholars and researchers in mathematical finance, possibility measures, portfolio optimization in addition to actuarial practitioners.

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

Example text

We choose sufficiently large ρ and α such that α > K and α + the estimate E eρt Y (t) − Y (t) 2 T +E K α = ρ. We obtain 2 eρs Z(s) − Z (s) ds t T +E t 2 R eρs U (s, z) − U (s, z) Q(s, dz)η(s)ds ≤ Kˆ E eρT ξ − ξ 2 T +E eρs Y (s) − Y (s) t · f s, Y (s), Z(s), U (s) − f s, Y (s), Z(s), U (s) ds , 0 ≤ t ≤ T. 1 are very useful in the study of BSDEs, and they are often applied in this book. 1). 2 and the predictable representation property. Next, we show convergence of the sequence by using the a priori estimates.

5) holds for any ρ. 13) which we use in the next part of this proof. 3. 6). 8) we get T 2 eρt Y (t) − Y (t) + ρ t T 2 eρs Y (s) − Y (s) ds + t 2 eρs Z(s) − Z (s) ds 44 3 T + 2 =e eρs U (s, z) − U (s, z) N (ds, dz) R t ρT 2 ξ −ξ T −2 Backward Stochastic Differential Equations—The General Case eρs Y (s) − Y (s) t · −f s, Y (s), Z(s), U (s) + f s, Y (s), Z (s), U (s) ds T −2 eρs Y (s−) − Y (s−) Z(s) − Z (s) dW (s) t T −2 R t eρs Y (s−) − Y (s−) U (s, z) − U (s, z) N˜ (ds, dz), 0 ≤ t ≤ T. 14) It is straightforward to derive the following estimate 2 sup eρt Y (t) − Y (t) t∈[0,T ] ≤ eρT ξ − ξ T +2 2 eρs Y (s) − Y (s) 0 · f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) ds T + 2 sup t∈[0,T ] T + 2 sup t∈[0,T ] ≤ eρT ξ − ξ T +2 eρs Y (s−) − Y (s−) Z(s) − Z (s) dW (s) t R t eρs Y (s−) − Y (s−) U (s, z) − U (s, z) N˜ (ds, dz) 2 eρs Y (s) − Y (s) 0 · f s, Y (s), Z(s), U (s) − f s, Y (s), Z (s), U (s) ds t + 4 sup t∈[0,T ] t + 4 sup t∈[0,T ] eρs Y (s−) − Y (s−) Z(s) − Z (s) dW (s) 0 0 R eρs Y (s−) − Y (s−) U (s, z) − U (s, z) N˜ (ds, dz) .

We point out that we introduce the predictable representation property (PR) as an assumption. In general, the predictable representation property does not have to hold. However, in our case it is possible to construct a probability space (Ω, F , P) in such a way that any F -local martingale has the predictable representation. 49 in He et al. (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.

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