Malliavin calculus

Ebook "Stochastic calculus of variations in mathematical finance" demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options.

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The aim of this work is to study the tail distribution of the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion. We first prove the existence and uniqueness of the solution.

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This paper considers a generalization of fractional Bessel type process. It is also a type of singular stochastic differential equations driven by fractional Brownian motion which has been studied by some authors.

7p viirenerosenfeld 02-06-2022 8 1   Download

This paper considersa generalization of fractional Bessel type process. It is also a type of singular stochastic differential equations driven by fractional Brownian motion which were studied by some authors. Undersome assumptions of coefficients, this equation has a unique positive solution.

7p nguaconbaynhay12 13-06-2021 18 1   Download

Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief, in constructing and exploiting natural differentiable structures on abstract probability spaces; in other words, Stochastic Calculus of Variations proceeds from a merging of differential calculus and probability theory. As optimization under a random environment is at the heart of mathematical finance, and as differential calculus is of paramount importance for the search of extrema, it is not surprising that Stochastic Calculus of Variations appears in mathematical finance.

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The mathematical theory now known as Malliavin calculus was first introduced by Paul Malliavin in [157] as an infinite-dimensional integration by parts technique. The purpose of this calculus was to prove the results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motion. For several years this was the only known application. Therefore, since this theory was considered quite complicated by many, Malliavin calculus remained a relatively unknown theory also among mathematicians for some time.

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