Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations.
We study convergence to the invariant measure for a class of semilinear stochastic evolution equations driven by Lévy noise, including the case of cylindrical noise. For a certain class of equations we prove the exponential rate of convergence in the norm of total variation. Our general result is applied to a number of specific equations driven by cylindrical symmetric α-stable noise and/or cylindrical Wiener noise. We also consider the case of a "singular" Wiener process with unbounded covariance...
With the use of exponential martingales and the Girsanov theorem we show how to calculate bond prices in a large variety of square root processes. We clarify and correct several errors that abound in financial literature concerning these processes. The most important topics are linear risk premia, the Longstaff double square model, and calculations concerning correlated CIR processes.
Arbitrage-free prices of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the -scheme in time and a wavelet Galerkin method with degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the linear...
Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the...