By Łukasz Delong
Backward stochastic differential equations with jumps can be utilized to resolve difficulties in either finance and insurance.
Part I of this booklet provides the speculation of BSDEs with Lipschitz turbines pushed through a Brownian movement and a compensated random degree, with an emphasis on these generated by way of step procedures 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 specializes in 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 bargains with pricing and hedging of assurance 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 hazard measures. half III offers another invaluable periods of BSDEs and their applications.
This e-book will make BSDEs extra obtainable to those that have an interest in making use of those equations to actuarial and fiscal 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
The process Y has the representation Y (t) = EQ e T t α(s)ds) 0 ≤ t ≤ T, ξ |FtW , and the control process Z is derived from Z(t) = e− t 0 α(s)ds Z (t), 0 ≤ t ≤ T, and the representation e T 0 α(s)ds ξ = EQ e T 0 α(s)ds T ξ + Z (s)dW Q (s), 0 where the equivalent probability measure Q is defined by dQ W F =e dP T T 0 T 0 β(s)dW (s)− 12 β 2 (s)ds . 1. 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.
T) ∈ Ω × [0, T ], for all (y, z, u), (y z , u), (y, z, u ) ∈ R × R × L2Q (R), where δ y,z,u,u : Ω × [0, T ] × R → (−1, ∞) is a predictable proR |δ cess such that the mapping t → bounded in (y, z, u, u ), T (iii) E[ 0 |f (t, 0, 0, 0)|2 dt] < ∞, y,z,u,u (t, x)|2 Q(t, dx)η(t) is uniformly and let (ξ , f ) satisfy (A1)–(A3). 1) with (ξ, f ) and (ξ , f ). If • ξ¯ = ξ − ξ ≥ 0, • f¯(t, y, z, u) = f (t, y, z, u) − f (t, y, z, u) ≥ 0, (t, y, z, u) ∈ [0, T ] × R × R × R, then Y (t) ≥ Y (t), 0 ≤ t ≤ T .
1 the stochastic σ (t) exponential M is a square integrable martingale. Hence, we can define an equivalent probability measure Q by dQ dP |FT = M(T ). 1 we deduce that the dynamics of S under the new measure Q is given by dS(t) = r(t)dt + σ (t)dW Q (t). 1 yield that e− t 0 r(s)ds S(t) is a Q-martingale. 10 Consider a compound Poisson process J with intensity λ and jump size distribution q. Let N denote the corresponding jump measure. Choose a predictable process κ such that |κ(t, z)| < 1, (t, z) ∈ [0, T ] × R.