Malliavin calculus – Knowledge and References

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Chapter 11: Miscellaneous Topics Used for Engineering Problems

Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017

Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes and is used to model systems that behave randomly. The main types of stochastic calculus are the Itô calculus and its variational relative known as the Malliavin calculus. For technical reasons the integral is the most useful for general classes of processes. We present here the Itoô lemma. Interested readers may pursue references [19], [30], [50] and [54] for detailed study.

Optimal control of mean-field jump-diffusion systems with noisy memory

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Journal Information

Published in International Journal of Control, 2019

Heping Ma, Bin Liu

We have adopted two approaches for deriving necessary and sufficient maximum principles. The first approach is via Malliavin calculus. In the second approach, the necessary and sufficient maximums are obtained without resort to Malliavin calculus. In this section, we will study the connection between the adjoint Equations (3.1) and (4.3). We say that pi, qi, ri, i = 1, 2 are solutions of the two-dimensional time-advanced mean-field BSDE (4.3) if qi, ri, i = 1, 2 are predictable, the estimate holds for i = 1, 2, and pi, qi, ri satisfy Equation (4.3).