Modern Stochastics: Theory and Applications

Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.

In this paper the study of a three-parametric class of Gaussian Volterra processes is continued. This study was started in Part I of the present paper. The class under consideration is a generalization of a fractional Brownian motion that is in fact a one-parametric process depending on Hurst index H. On the one hand, the presence of three parameters gives us a freedom to operate with the processes and we get a wider application possibilities. On the other hand, it leads to the need to apply rather subtle methods, depending on the intervals where the parameters fall. Integration with respect to the processes under consideration is defined, and it is found for which parameters the processes are differentiable. Finally, the Volterra representation is inverted, that is, the representation of the underlying Wiener process via Gaussian Volterra process is found. Therefore, it is shown that for any indices for which Gaussian Volterra process is defined, it generates the same flow of sigma-fields as the underlying Wiener process – the property that has been used many times when considering a fractional Brownian motion.

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

The chaos expansion of a random variable with uniform distribution is given. This decomposition is applied to analyze the behavior of each chaos component of the random variable $\log \zeta $ on the so-called critical line, where ζ is the Riemann zeta function. This analysis gives a better understanding of a famous theorem by Selberg.

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.

Explicit solutions for a class of linear backward stochastic differential equations (BSDE) driven by Gaussian Volterra processes are given. These processes include the multifractional Brownian motion and the multifractional Ornstein-Uhlenbeck process. By an Itô formula, proven in the context of Malliavin calculus, the BSDE is associated to a linear second order partial differential equation with terminal condition whose solution is given by a Feynman-Kac type formula.

In this paper we establish the existence and the uniqueness of the solution of a special class of BSDEs for Lévy processes in the case of a Lipschitz generator of sublinear growth. We then study a related problem of logarithmic utility maximization of the terminal wealth in the filtration generated by an arbitrary Lévy process.

We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein–Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.

In this paper the fractional Cox–Ingersoll–Ross process on ${\mathbb{R}_{+}}$ for $H<1/2$ is defined as a square of a pointwise limit of the processes ${Y_{\varepsilon }}$, satisfying the SDE of the form $d{Y_{\varepsilon }}(t)=(\frac{k}{{Y_{\varepsilon }}(t){1_{\{{Y_{\varepsilon }}(t)>0\}}}+\varepsilon }-a{Y_{\varepsilon }}(t))dt+\sigma d{B^{H}}(t)$, as $\varepsilon \downarrow 0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox–Ingersoll–Ross process are obtained.