Pathwise integration, Malliavin calculus and stochastic control

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics

Abstract

This project is concerned with the development of pathwise integration techniques in the scope of Malliavin calculus. The main applications will be new classifications for stochastic processes, development of new numerical methods and new theoretical developments in stochastic optimal control taking advantage of the new pathwise interpretation paradigm. The project takes inspiration in the marriage between analysis and probability by creating and developing tools specific to one field and applying them to the other. The project leads lastly to applications in path-dependent partial differential equations. These appear naturally within the scope of stochastic optimal control problems involving so-called path-dependent contracts when using the pathwise technique. Addressed is also the deeper issue of giving meaning to martingales from a pathwise point of view.

Publications

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Imkeller P (2019) Differentiability of SDEs with drifts of super-linear growth in Electronic Journal of Probability

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509644/1 01/10/2016 30/09/2021
1783872 Studentship EP/N509644/1 01/10/2016 31/03/2020 William John Salkeld
 
Description We proved and published new Large Deviations Principles results for McKean Vlasov Equations and new Malliavin Differentiability results for general SDEs with random coefficients and a drift satisfying a one-sided lipschitz condition (or which McKean Vlasov Equations are one example).

LDP is a way of measuring the locations on which the law of a stochastic process is concentrated when the random perturbation of the system is small. In particular, the measure dependency in the equations which includes the small probability events where the particle deviates from the expected regions does not cause the process to move to new areas.
https://projecteuclid.org/euclid.aoap/1550566836

Malliavin Differentiability is a property of stochastic processes that allows one to measure the regularity of the law of the process. In the case of one-sided Lipschitz conditions, the standard method for proving Malliavin Differentiability fails and we had to use new methods to deal with non-integrable error terms.
https://projecteuclid.org/euclid.ejp/1549616424

We solve a new representation for the support of McKean-Vlasov Equations. The support represents the collection of paths that the process could be observed to take. The challenge for McKean-Vlasov equations is that the paths are themselves dependent on the law of the solution. To get around this, we construct functional quantizations for the law of Brownian motion as a measure over the (non-reflexive) Banach space of Hölder continuous paths. We obtain a sequence of deterministic finite supported measures that converge to the law of a Brownian motion with explicit rate. The functional quantization results then yield a novel way to build deterministic, finite supported measures that approximate the law of the McKean-Vlasov Equation driven by the Brownian motion which crucially avoid the use of random empirical distributions. These are then used to solve an approximate skeleton process that characterises the support of the McKean-Vlasov Equation.
See https://arxiv.org/abs/1911.01992

We solve the small ball probabilities of Gaussian Rough paths. While many works on Rough Paths study the Large Deviations Principles for Stochastic Processes driven by Gaussian Rough paths, it is a noticeable gap in the literature that Small Ball Probabilities have not been extended to the Rough Path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. Small Ball Probabilities provide microscopic information and are used to establish a locally accurate approximation for a random variable. In particular, we note that the signature map retains the compactness properties of the Gaussian process even though the mapping is not even continuous.
See https://arxiv.org/abs/2002.06447
Exploitation Route These results represent new knowledge for the study of McKean-Vlasov equations which have a wide variety of applications.
Sectors Financial Services, and Management Consultancy,Other