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Multi-dimensional Diffusion Processes

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Book cover Relative Optimization of Continuous-Time and Continuous-State Stochastic Systems

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

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Abstract

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.

In a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs [1].

Felix Klein

German Mathematician

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Notes

  1. 1.

    In most cases, we choose the word “curve.”

  2. 2.

    With no confusion, we use U to denote both a curve and a function.

  3. 3.

    Meaning the curve itself is smooth, not a curve containing smooth points of a function. In this sense, a semi-smooth curve is better to be called “a semi-smooth-point curve.”

  4. 4.

    Precisely, \(\{dx_1, dx_2 \}\) in this lemma are the coordinates with the smooth quadrants; they are different from the original coordinates of (5.1). For notational simplicity, instead of using \(\{ dx'_1 , dx'_2 \}\), we keep the same notation.

  5. 5.

    The results in Lemmas 5.7 and 5.8 are intuitively clear, but their proofs are technically cumbersome. So we just provide an intuitive explanation, and the detailed proofs have to be worked out.

  6. 6.

    The provenance of this quote is not clear. I first saw it in Øksendal’s book [24], according to “Quote Investigator,” the earliest instance of this quote is by Earl C. Kelley in “The Workshop Way of Learning,” Page 2, Harper and Brothers, New York, 1951.

  7. 7.

    Note that here the “\(\varDelta X(t)\)” is due to \(\varDelta x\), not \(\varDelta t\).

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Authors and Affiliations

  1. Department of Automation, Shanghai Jiao Tong University, Shanghai, China

    Xi-Ren Cao

  2. Professor Emeritus, Department of Electrical and Computer Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong

    Xi-Ren Cao

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Correspondence to Xi-Ren Cao .

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Cao, XR. (2020). Multi-dimensional Diffusion Processes. In: Relative Optimization of Continuous-Time and Continuous-State Stochastic Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-41846-5_5

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