Backward Stochastic Differential Equations and Viscosity Solutions of Systems of Semilinear Parabolic and Elliptic PDEs of Second Order

The aim of this set of lectures is to present the theory of backward stochastic differential equations, in short BSDEs, and its connections with viscosity solutions of systems of semi– linear second order partial differential equations of parabolic and elliptic type, in short PDEs. Linear BSDEs have appeared long time ago, both as the equations for the adjoint process in stochastic control, as well as the model behind the Black & Scholes formula for the pricing and hedging of options in mathematical finance. These linear BSDEs can be solved more or less explicitly (see the proof of theorem 1.6). However, the first published paper on nonlinear BSDEs, [37], appeared only in 1990. Since then, the interest for BSDEs has increased regularly, due to the connections of this subject with mathematical finance, stochastic control, and partial differential equations. We refer the interested reader to El Karoui, Peng, Quenez [18], El Karoui, Quenez [19], the reference therein and in particular the work of Duffie and his co–authors [11], [12], [13] and [14] for developments on the use of BSDEs as models in mathematical finance, as well as the connection of BSDEs with stochastic control. BSDEs is also an efficient tool for constructing Γ–martingales on manifolds, with rescribed limit, see Darling [9]. The present notes develop the theory of BSDEs, and its connections with PDEs. We have concentrated our presentation on the connection with viscosity solutions of PDEs. Also this approach is appealing, it is not the unique possible presentation. We have developped both the parabolic and the elliptic cases, the latter being presented in the two cases of systems of equations in IR, and equations in a bounded set with Dirichlet boundary conditions. We have left out the case of equations with Neumann boundary conditions, which is thoroughly exposed in Pardoux, Zhang [44], and the study of coupled forward–backward SDEs and its