Stochastic volatility, jumps and hidden time changes
The paper deals with stochastic processes of the form \(X(t)=\beta(A(t))\), where \((\beta(u),\;u\geq 0)\) is a Brownian motion; \(A(t)\) is an increasing, right continuous process independent of \(\beta(t)\). The authors prove that if \(A(t)\) is a purely discontinuous subordinator, then the joint Laplace transform of \(([X,X]_{t},A(t))\) is given by \[ E\left[\exp\left(\left({\lambda\over 2}[X,X]_{t}+mA(t)\right)\right)\right]= \exp\left(-t\int\limits_{0}^{\infty}\nu_{A}(du)\left(1-{e^{-mu}\over(1+ \lambda u)^{1/2}}\right)\right), \] where \(\nu_{A}(du)\) is the Lévy measure of the process \(A(t)\). For the case \(A(t)=\gamma(t)\), where \(\gamma(t)\) is a standard gamma process, the recovery and stochastic scaling problems are formulated and solved. General results on the recovery problem as well as explicit solutions for some recovery problems are presented. The authors show that in the class of subordinators with completely monotone Lévy densities the gamma process is the only one with a conditional law that is path independent and is a function of the total variation of the composite process. The stochastic scaling problem is formulated and for a variety of subordinators and scaling levels the absence of a solution is shown and explicit solutions in other cases are obtained.
scientific article; zbMATH DE number 1833868