This book introduces the theory of viscosity solutions and their application to optimal stochastic control for continuous-time Markov processes. The authors employ the dynamic programming approach, where the fundamental equation for the value function becomes a nonlinear partial differential equation of second order, known as the Hamilton-Jacobi-Bellman (HJB) equation. Since the value function often lacks the smoothness required for classical solutions, the theory of viscosity solutions provides a framework for studying these HJB equations and proving continuous dependence on problem data. The text illustrates these concepts with applications in engineering, management science, and financial economics, with the second edition incorporating new material on mathematical finance, risk-sensitive control theory, nonlinear H-infinity control, and differential games.

The first edition of this book was highly recommended by SIAM Review in 1994, described as a fine book likely to become a classic in the area of dynamic programming applied to optimal stochastic control for diffusion processes. The second edition builds upon this foundation, incorporating updated material and applications, particularly in mathematical finance. The inclusion of topics such as risk-sensitive control and differential games suggests a comprehensive treatment for advanced students and researchers. The book's approach through viscosity solutions offers a robust framework for understanding complex control problems.