Explicit solution of a nonlinear filtering problem for Lévy processes with application to finance
type: Item
instance of: scholarly article
title: Explicit solution of a nonlinear filtering problem for Lévy processes with application to finance
zbMATH Open document ID: 1053.60040
author: Thilo Meyer-Brandis
author: F. Proske
published in: Applied Mathematics and Optimization
publication date: 2004-11-05T00:00:00Z
review text: Consider the dynamics of a signal process \(X\) satisfying the stochastic differential equation (SDE) \[ dX_t = b(X_t) dt + \sigma(X_t) dB^X_t \] driven by standard Wiener process \(B^X\), where \(X\) is partially observed by the observation process \(Y\) governed by the SDE \[ dY_t = h(t,Y_t) dt + dB^Y_t + \int_{R_0} \zeta N_\lambda(dt,d\zeta) \] driven by standard Wiener process \(B^Y\) and an integer-valued random measure \(N_\lambda\) with predictable compensator \[ \mu(dt,d\zeta,\omega) = \lambda(t,X_t,\zeta) \,dt\, \nu(d\zeta) \] for a Lévy measure \(\nu\) and intensity function \(\lambda\). Suppose also that \(B^X\), \(B^Y\) and \(N\) are independent processes. The authors study the related nonlinear filtering problem to find the least squares-estimate to the transformed process \(f(X_t)\) under given information of \(\sigma\)-algebra \(F^Y_t\) based on the observation process \(Y\), where \(f\) is Borel-measurable. This means that their aim is to determine an explicit solution of the optimal filter \(E [ f(X_t) | F^Y_t ]\). They show how ``white noise calculus'' for Lévy processes can be used to find explicit solutions. For this purpose, they solve the related Zakai equation for the unnormalized conditional density (recall that this equation is a Lévy-driven linear stochastic partial differential equation). Note that this Zakai equation has a unique strong \(L^p\)-solution. Eventually, the authors illustrate how their results can be used in dynamic modelling of exponential-type asset prices in mathematical finance theory.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 91B24
Mathematics Subject Classification ID: 91B70
zbMATH DE Number: 2113875
zbMATH Keywords: nonlinear filtering
zbMATH Keywords: Lévy processes
zbMATH Keywords: Stochastic partial differential equations
zbMATH Keywords: white noise analysis
zbMATH Keywords: Zakai equation
zbMATH Keywords: explicit \(L^p\)-solution
zbMATH Keywords: optimal filter
zbMATH Keywords: Asset pricing theory in finance
MaRDI profile type: Publication
full work available at URL: s00245-004-0798-6
OpenAlex ID: W2032755050
DOI: 10.1007/S00245-004-0798-6
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description: scientific article; zbMATH DE number 2113875