The Malliavin gradient method for the calibration of stochastic dynamical models

Ewald, C.-O. (2006) The Malliavin gradient method for the calibration of stochastic dynamical models. Applied Mathematics and Computation, 175(2), pp. 1332-1352. (doi:10.1016/j.amc.2005.08.050)

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Publisher's URL: http://dx.doi.org/10.1016/j.amc.2005.08.050

Abstract

We consider a diffusion (X<sub>t</sub>) satisfying the stochastic differential equation dX<sub>t</sub> = β(X<sub>t,</sub> u)dt + σ(X<sub>t,</sub> v)dW<sub>t</sub> where u and v are parameters and consider the problem of minimizing certain functionals of the form L(u,v) := Σ<sup>k</sup><sub>i=1</sub>(E(h<sub>i</sub>(X<sub>t</sub>,))-q<sub>i</sub>)<sup>2</sup> in u and v where t<sub>i</sub> ∈ [0<sub>,</sub> T] are not necessarily distinct time points. For this we combine classical gradient methods with techniques from Malliavin calculus. The proposed technique has a particular advantage to classical techniques in the case when the functions h<sub>i</sub> are not continuous or have singularities. This is the case when the functions h<sub>i</sub> represent certain quantiles, i.e. h<sub>i</sub>(x)≔1<sub>{x⩽pi}</sub> and the problem is to choose the parameters u, v in a way that the stochastic model fits the quantiles best.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Ewald, Professor Christian
Authors: Ewald, C.-O.
Subjects:Q Science > QA Mathematics
College/School:College of Social Sciences > Adam Smith Business School > Economics
Journal Name:Applied Mathematics and Computation
ISSN:0096-3003
Published Online:20 October 2005

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