Bensoussan A., Lions J.L. Applications of variational inequalities in stochastic control

North Holland, 1982. — 577 p.
This book treats second order partial differential equations and unilateral problems, as well as stochastic control and optimal stopping-time problems. It deals with branches of mathematics which may at first sight appear totally different and which have developed along quite independent lines, but which are in fact strongly inter-related and which are capable of cross-fertilising each other. The fundamental link lies in the interpretation of the solutions of certain partial differential equations. This interpretation is an extension of the method of characteristics which allows the solution of a linear first-order hyperbolic equation to be expressed explicitly as a functional defined along the characteristic trajectories. A similar phenomenon arises in the case of parabolic or elliptic equations, but the characteristic trajectories then become stochastic processes. In very general terms, it is absolutely necessary to resort to probabilistic models if we wish to be able to give explicit formulas for the solutions of partial differential equations (or of systems of.such equations).