Classical solutions to reaction–diffusion systems for hedging problems with interacting Itô and point processes

Abstract

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Itô and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuation problems for contingent claims with a recursive payoff structure.

About the Pricing Equations in Finance

“…Note that as opposed to the set-up of Becherer and Schweizer [7] where linear reaction-diffusion systems of parabolic equations are considered in a diffusion model with regimes (but without jumps in X), here, due to the presence of the obstacles in the problem and of the jumps in X, the related reaction-diffusion system (V2) typically does not have classic solutions.…”

Counterparty risk and funding: immersion and beyond

“…By application of the results of Bielecki, Jakubowski, and Niewȩglowski (2012) (or by a direct proof based on Heath and Schweizer (2000, Theorem 1) and Becherer and Schweizer (2005, Corollary 2.3)), X is a (G, Q) homogenous strong Markov process. Recall from (4.4) and (4.2) that…”

“…Becherer and Schweizer (2005)). In this case, an application of the Itô formula related to the (F, P) jump diffusion X t = (m t , k t ) yields the following functional representation of µ in (6.14):…”

Asset allocation with contagion and explicit bankruptcy procedures

“…Applying a fixed-point argument, Becherer and Schweizer (2005) show that these equations have a unique solution under local Lipschitz conditions on the coefficients of the equation. These conditions will be satisfied in our applications.…”