Multi-dimensional Diffusion Processes

Part of the Communications and Control Engineering book series (CCE)


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.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of AutomationShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Professor Emeritus, Department of Electrical and Computer EngineeringThe Hong Kong University of Science and TechnologyKowloonHong Kong

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