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 speculation of BSDEs with Lipschitz turbines pushed by way of a Brownian movement and a compensated random degree, with an emphasis on these generated by means of step strategies and Lévy methods. It discusses key effects and strategies (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 figuring out the theory.
Part II investigates actuarial and fiscal purposes of BSDEs with jumps. It considers a common monetary and assurance version and offers with pricing and hedging of coverage equity-linked claims and asset-liability administration difficulties. It also investigates excellent hedging, superhedging, quadratic optimization, software maximization, indifference pricing, ambiguity danger minimization, no-good-deal pricing and dynamic chance measures. half III provides another valuable periods of BSDEs and their applications.
This publication will make BSDEs extra available to people who have an interest in making use of those equations to actuarial and monetary difficulties. it is going to be important to scholars and researchers in mathematical finance, chance 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
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. Next, we consider the backward stochastic differential equation T Y (t) = ξ + T β(s) Z(s) ds − Z(s)dW (s), t 0 ≤ t ≤ T. 2 Let ξ be an F W -measurable random variable such that ξ ∈ L2 (R), and let β be an F W -predictable, positive, bounded process. 33). The process Y has the representation Y (t) = sup EQ ξ |FtW , Q∈Q 0 ≤ t ≤ T, where Q= Q∼P: dQ W F =e dP T T 0 φ(s)dW (s)− 21 T 0 |φ(s)|2 ds , φ is F W -predictable, φ(t) ≤ β(t) .
Since |φ(t)| ≤ β(t), we have φ(t)z ≤ β(t)|z|, (t, z) ∈ [0, T ] × R, and from the comparison principle we deduce that Y φ (t) ≤ Y (t), 0 ≤ t ≤ T . We get sup |φ(t)|≤β(t) Y φ (t) ≤ Y (t), 0 ≤ t ≤ T.
50 3 Backward Stochastic Differential Equations—The General Case 3. The existence and uniqueness of a solution. 1). Indeed, it is easy to show lim E n→∞ T f s, Y n (s), Z n (s), U n (s) ds sup t∈[0,T ] t 2 T − = 0, f s, Y (s), Z(s), U (s) ds t lim E n→∞ lim E n→∞ T t∈[0,T ] t = 0, Z(s)dW (s) t T sup t∈[0,T ] 2 T Z n (s)dW (s) − sup R t T U n (s, z)N˜ (ds, dz) − R t U (s, z)N˜ (ds, dz) 2 = 0. 7). 7) we can derive some useful norm estimates for the solution (Y, Z, U ). 22) T where C and V are given.
Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications: BSDEs with Jumps by Łukasz Delong