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Malliavin Calculus: An Overview

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What is Malliavin Calculus?

Malliavin calculus, also called the stochastic calculus of variations, extends calculus of variations from deterministic functions to stochastic (random) processes. More specifically, it differentiates random variables defined on a Gaussian probability space (usually a Wiener space) with respect to underlying noise [1]. These processes have infinite variation, so are not differentiable with traditional calculus tools. In addition, the calculus can integrate random variables using integration by parts.

Malliavin calculus was originally developed by Paul Malliavin in 1976, who used it to to study the smoothness of solution densities of stochastic differential equations. The calculus was further used to investigate regularity properties of the law of Wiener functionals and solutions of stochastic differential equations [2], with further contributions by Bismut, Stroock, Watanabe and others. It is widely used in mathematical finance and financial engineering as well as the study of stochastic partial differential equations.

A Useful but Challenging Calculus

The major downside to Malliavin calculus is that is considered by many to be highly technical and theoretical. This should come as no surprise, as it took centuries for mathematicians even to consider the idea that the integral of random process could possibly exist (in fact, a Lebesgue integral of ∫ f dg cannot exist if g is of infinite variation). Therefore, special tools beyond the “usual” calculus are required, which means a large investment of time to understand how the calculus works. For example, the central tool of Malliavin calculus is the following integration by parts formula [3]:

Suppose that (Ω, F, P) is a probability space and F, G: Ω → R are integrable random variables. The integration by parts formula IP (F; G) holds true if there is some random variable H(F; G) such that
Malliavin calculus

Where Cc is the space of infinitely differentiable functions with compact support.

The integration by parts formula IPk (F; G) holds true if there is an integrable random variable Hk(F; G) so that
malliavin calculus integration by parts

In order to grasp the fundamentals, you have to have some knowledge of calculus (particularly differential equations) and predicate logic as well as a background in financial engineering or other practical application.


[1] Hairer, M. (2021). Introduction to Malliavin Calculus. Retrieved August 27, 2021 from:
[2] Nualart, D. (1995). The Malliavin Calculus and Related Topics, Springer-Verlag.
[3] Bally, V. (2003). An elementary introduction to Malliavin calculus. [Research Report] RR-4718, INRIA.

Stephanie Glen. "Malliavin Calculus: An Overview" From Calculus for the rest of us!

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