Backward Stochastic Differential Equations with Jumps and by Łukasz Delong

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 speculation of BSDEs with Lipschitz turbines pushed by means of a Brownian movement and a compensated random degree, with an emphasis on these generated by way of step procedures and Lévy procedures. 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 an important for realizing the theory.

Part II investigates actuarial and fiscal functions of BSDEs with jumps. It considers a normal monetary and assurance version and offers 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 possibility measures. half III offers another helpful sessions of BSDEs and their applications.

This e-book will make BSDEs extra available to those that have an interest in making use of those equations to actuarial and fiscal difficulties. it will likely be worthy to scholars and researchers in mathematical finance, danger measures, portfolio optimization in addition to actuarial practitioners.

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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. 22 in He et al.

U f (s)δ(s, z)Q(s, z)η(s)ds, Let us introduce an equivalent probability measure Q by the Radon-Nikodym derivative dQ = M(t), dP Ft 0 ≤ t ≤ T, z f (t)dW (t) + dM(t) = M(t−) R fu (t)δ(t, z)N˜ (dt, dz) . Since the process δ is non-negative and the generator f is non-decreasing in u, the kernel fu (t)δ(t, z) is non-negative. 1 the process M is a square integrable martingale. Hence, M defines an equivalent probability measure. 1, we derive the equation Y¯ (t) = ξ¯ e T t y f (s)ds T + f¯ s, Y (s), Z (s), t T − ¯ Z(s)e s t y f (u)du R U (s, z)δ(s, z)Q(s, dz)η(s) e s t y f (u)du ds dW Q (s) t T − R t U¯ (s, z)e s t y f (u)du N˜ Q (ds, dz), 0 ≤ t ≤ T.

T) ∈ Ω × [0, T ]. 13 Let N be the jump measure of a compound Poisson process with jump size distribution q. Consider a function V : [0, T ] × R → R such that T 2 0 R |V (s, y)| q(dy)ds < ∞ and a Lipschitz continuous function ϕ : R → R. Let T ξ =ϕ 0 We can write ξ = ϕ( obtain Dt,z ξ = ϕ( T 0 T 0 R R V (s, y)N˜ (ds, dy) . V (s,y) d y Υ (ds, dy)). e. (ω, t, z) ∈ Ω × [0, T ] × R \ {0}. 14 Let N be the jump measure of a compound Poisson process with jump size distribution q. Consider the stop-loss contract ξ = (J (T ) − K)+ where t ∞ J (t) = 0 0 yN(ds, dy), 0 ≤ t ≤ T , is the compound Poisson process used for modelling insurer’s claims.

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