Multi-dimensional Diffusion Processes

In this chapter, we first discuss the stochastic calculus of multi-dimensional diffusion processes with semi-smooth functions, and we derive the Tanaka formula for multi-dimensional semi-smooth functions with the local time on the semi-smooth curve along its gradient direction. With this formula, we extend the relative optimization approach to stochastic control to multi-dimensional systems. Optimality conditions are derived for systems with semi-smooth value functions and no viscosity solution is involved. This approach provides new insights and motivates the research on stochastic control and stochastic calculus of multi-dimensional systems, in particular, for problems with non-smooth features and degenerate points. The analysis is intuitive and results are preliminary, and hopefully they would motivate new research topics.