Download Backward Stochastic Differential Equations with Jumps and by Łukasz Delong PDF

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 by way of a Brownian movement and a compensated random degree, with an emphasis on these generated through step techniques and Lévy techniques. It discusses key effects and strategies (including numerical algorithms) for BSDEs with jumps and stories 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 monetary purposes of BSDEs with jumps. It considers a basic monetary and coverage version and bargains with pricing and hedging of assurance equity-linked claims and asset-liability administration difficulties. It also investigates ideal hedging, superhedging, quadratic optimization, software maximization, indifference pricing, ambiguity possibility minimization, no-good-deal pricing and dynamic threat measures. half III provides another necessary periods of BSDEs and their applications.

This ebook will make BSDEs extra available to those that have an interest in making use of those equations to actuarial and monetary difficulties. will probably be worthy to scholars and researchers in mathematical finance, danger 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

Sample text

We prove the representation of the solution. 15), we notice that Q is an equivalent probability measure. 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 .

E. (ω, t, z) ∈ Ω × [0, T ] × R \ {0}. 1. 23). The next two results are taken from Delong and Imkeller (2010b). 5 Consider a finite measure q on R. e. (s, y) ∈ [0, T ] × R, ([0,T ]×R)2 Dt,z ϕ(s, y) q(dy)dsυ(dt, dz) < ∞. e. (ω, t, z) ∈ Ω × [0, T ] × R. 6 Let ϕ : Ω × [0, T ] × R → R be a predictable process which satisfies E[ [0,T ]×R |ϕ(s, y)|2 υ(ds, dy)] < ∞. Then ϕ ∈ L1,2 (R) Moreover, if 0 [0,T ]×R [0,T ]×R ϕ(s, y)Υ (ds, dy) ∈ D T Dt,z if and only if R ϕ(s, y)Υ (ds, dy) ∈ D1,2 (R). e. (ω, t, z) ∈ Ω × [0, T ] × R, and stochastic integral in the Itô sense.

Key results for BSDEs are presented, which are further developed in next chapters. ) ds t T − T Z(s)dW (s) − t t R U (s, z)N˜ (ds, dz), 0 ≤ t ≤ T . 1). The processes Z and U are called control processes. They control an adapted process Y so that Y satisfies the terminal condition. ) ds t Ł. 1007/978-1-4471-5331-3_3, © Springer-Verlag London 2013 37 38 3 T − Backward Stochastic Differential Equations—The General Case T Z(s)dW (s) − t R t U (s, z)N˜ (ds, dz), 0 ≤ t ≤ T. 1), let us have a look at a BSDE with zero generator and a BSDE with generator independent of (Y, Z, U ).

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