We consider a stationary sequence $$(X_n)$$ constructed by multiple stochastic integral and an infinite-measure conservative dynamical system. The random measure defining the is non-Gaussian infinitely divisible has finite variance. Some additional assumptions on system give rise to parameter $$\beta \in (0,1)$$ quantifying conservativity of This $$ together with order determines decay rate cov...

We propose a multi-scale stochastic volatility model in which a fast mean-reverting factor of volatility is built on top of the Heston stochastic volatility model. A singular pertubative expansion is then used to obtain an approximation for European option prices. The resulting pricing formulas are semi-analytic, in the sense that they can be expressed as integrals. Difficulties associated with...

Using the multiple stochastic integrals we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one parameter and two parameter cases. When the drift is zero, we show that in the one-parameter case the solution in an exponential, thus positive, function while in the two-parameter settings t...

The equations satisfied by the gene frequency for various cases of dominance are converted to the type of stochastic differential equation associated with a diffusion process. Using the physical approach to stochastic integrals, the solutions of the corresponding Fokker-Planck equations are obtained. Conditions for quasifixation are investigated and observed frequencies of the gene medionigra o...

We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion index H≥1 2. show convergence these schemes at first and second order. The processes obtained in limit case are Rosenblatt process if H>3 4 standard Brownian otherwise. These results under assumption that integrand is a “controlled” process. provide many examples such processes, particul...