Explicit solution of a nonlinear filtering problem for Lévy processes with application to finance
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.
Stochastic partial differential equations
scientific article; zbMATH DE number 2113875
scientific article