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 methods (including numerical algorithms) for BSDEs with jumps and reports filtration-consistent nonlinear expectancies and g-expectations. half I additionally specializes in the mathematical instruments and proofs that are an important for realizing the theory.

Part II investigates actuarial and monetary purposes of BSDEs with jumps. It considers a basic 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 chance minimization, no-good-deal pricing and dynamic danger measures. half III provides another worthy sessions of BSDEs and their applications.

This e-book will make BSDEs extra available to those that have an interest in utilizing 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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Additional resources for Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps

Sample text

We introduce the processes Yˆ (t) = Y (t)e t 0 α(s)ds , ˆ = Z(t)e Z(t) t 0 α(s)ds , 0 ≤ t ≤ T. 3 Examples of Linear and Nonlinear BSDEs Without Jumps 59 or t Yˆ (t) = Yˆ (0) + Q ˆ (s), Z(s)dW 0 ≤ t ≤ T. s. 25) we next deduce that Yˆ is a Q-martingale. Hence, we obtain the representation Yˆ (t) = EQ [Yˆ (T )|FtW ]. The process Zˆ is now derived from the predictable representation of the Q-martingale Yˆ . We will observe that linear BSDEs arise when we investigate pricing and hedging problems in complete markets and when we deal with quadratic pricing and hedging in incomplete markets.

4 in He et al. (1992). 17) 0 ≤ t ≤ T , z ∈ R. 1. We have to impose stronger assumptions on (φ, κ) so that the local martingale M is a true martingale. In this book we use the following proposition. 1 Let M := (M(t), 0 ≤ t ≤ T ) be the stochastic exponential defined by dM(t) = φ(t)dW (t) + M(t−) R κ(t, z)N˜ (dt, dz), where φ and κ are predictable processes such that M(0) = 1, 28 2 2 φ(t) ≤ K, R κ(t, z) > −1, κ(t, z) Q(t, dz)η(t) ≤ K, Stochastic Calculus 0 ≤ t ≤ T, 0 ≤ t ≤ T , z ∈ R. The process M is a square integrable, positive martingale.

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. 1 Assume that (A1)–(A3) hold. 1) has a unique solution (Y, Z, U ) ∈ S2 (R) × H2 (R) × H2N (R). Proof 1.

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