About the Pricing Equations in Finance

In this article we study a Markovian BSDE and the associated system of partial integro-differential obstacle problems, in a flexible set-up made of a jump-diffusion with regimes. These equations are motivated by numerous applications in financial modeling, whence the title of the paper. This financial motivation is developed in the first part of the paper, which provides a synthetic view of the theory of pricing and hedging financial derivatives, using backward stochastic differential equations (BSDEs) as main tool. In the second part of the paper, we establish the well-posedness of reflected BSDEs with jumps coming out of the pricing and hedging problems exposed in the first part. We first provide a construction of a Markovian model made of a jump-diffusion – like component X interacting with a continuous-time Markov chain – like component N . The jump process N defines the so-called regime of the coefficients of X, whence the name of jump-diffusion with regimes for this model. Motivated by optimal stopping and optimal stopping game problems (pricing equations of American or game contingent claims), we introduce the related reflected and doubly reflected Markovian BSDEs, showing that they are well-posed in the sense that they have unique solutions, which depend continuously on their input data. As an aside, we establish the Markov property of the model. In the third part of the paper we derive the related variational inequality approach. We first introduce the systems of partial integro-differential variational inequalities formally associated to the reflected BSDEs, and we state suitable definitions of viscosity solutions for these problems, accounting for jumps and/or systems of equations. We then show that the state-processes (first components Y ) of the solutions to the reflected BSDEs can be characterized in terms of the value functions of related optimal stopping or game problems, given as viscosity solutions with polynomial growth to related integro-differential obstacle problems. We further establish a comparison principle for semi-continuous viscosity solutions to these problems, which implies in particular the uniqueness of the viscosity solutions. This comparison principle is subsequently used for proving the convergence of stable, monotone and consistent approximation schemes to the value functions. Finally in the last part of the paper we provide various extensions of the results needed for applications in finance to pricing problems involving discrete dividends on ∗This research benefited from the support of the ‘Chaire Risque de crédit’ (Fédération Bancaire Française), of the Europlace Institute of Finance and of Ito33.