Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions

Craddock, Mark; Lennox, Kelly A.

doi:10.1016/j.jde.2011.09.024

Cited by 27 publications

(24 citation statements)

References 13 publications

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“…A further line of research is related to the fine analysis of the possible symmetries, see, e.g., [11,12,16] and references therein, characterizing the particular problem into considerations. In fact, using the F eynman − Kac formula one can reduce the original stochastic problem into a deterministic (PDE) one, then exploit possible Lie symmetries to simplify the framework.…”

Section: Resultsmentioning

confidence: 99%

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

Marconi¹

2016

Int. J. of Pure and Appl. Math.

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Abstract:The present paper provides a sensitivity analysis of bond options exploiting the probabilistic properties of Malliavin Calculus and the computational benefits of the Polynomial Chaos Expansion. The purpose is to use the integration by parts formula of Malliavin Calculus in order to obtain the Greeks, of a given financial derivative, in a form which is suitable for numerical simulation. In particular, such computations will be performed usign the so called Polynomial Chaos Expansion technique, then comparing obtained results versus those retrieved using both the usual Monte-Carlo approach and the analytical formula.

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Section: Resultsmentioning

confidence: 99%

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

Marconi¹

2016

Int. J. of Pure and Appl. Math.

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“…This relation could give us a natural way to study properties such as symmetries of partial differential equation directly studying what change in the underlying stochastic process whenever we apply a symmetric transformation. It is worth to mention that such an approach has been extensively used during last decades by a number of different points of view, see, e.g., [13] concerning the analysis of nonlinear PDEs, [3,4,8] to what concerns applications in mathematical finance, also in the light of the following papers [6,7] and references therein, and also to simplify certain tipe of computational tasks that arise in the field of multivariate statistics when huge quantities of data have to be taken into account, see, e.g., [10,11] and references therein. It follows that the development of a general theory that relates Lie theory of symmetries for PDEs with the transformation of the stochastic process that the symmetry could bring to it, could potentially bring new methods to better analyse relevant models characterizing heterogeneous fields, as in the case, e.g., of meteorology, finance, biology, etc.…”

Section: Discussionmentioning

confidence: 99%

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

Guida¹

2017

Int. J. of Pure and Appl. Math.

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This paper proposes a different point of view to look at the Nelson's derivatives. In particular we relate the Nelson's forward derivative of a function of a diffusion, with certain properties, to the partial differential equation coming from the Feynman-Kac's theorem. We will see that whenever the Nelson forward derivative of functions of a diffusion process, that satisfies a certain stochastic differential equations, is computed at the initial point of the process and whenever we put it equal to zero, we obtain a partial differential equation. The latter is the same equation coming from the Feynman-Kac's theorem. Finally we will see how we could reduce the stochastic derivatives to the forward Nelson's derivative of a certain process by supposing the existence of a new complex processes related to the backward filtration.

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“…The relationship between Lie symmetry analysis and harmonic analysis and representation theory has been developed over the years, beginning with [5], [4] and [6]. See also [9], [10], [21], [20] and [13].…”

Section: Introduction To the Methodsmentioning

confidence: 99%

New classes of non-convolution integral equations arising from Lie symmetry analysis of hyperbolic PDEs

Craddock

Yakubovich

2017

Journal of Differential Equations

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In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial differential equations. Our analysis leads to new integral equations of nonconvolution type, which can be solved by classical methods. We derive solutions of these integral equations, which in turn lead to solutions of the associated Cauchy problems.

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Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions

Cited by 27 publications

References 13 publications

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

On the Relation Between the Nelson Stochastic Derivative and the Feynman-Kac Theorem

New classes of non-convolution integral equations arising from Lie symmetry analysis of hyperbolic PDEs

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