Abstract In this paper, we investigate doubly reflected generalized backward stochastic differential equations with two reflecting right-continuous with left-limited barriers in a general filtration supporting a Brownian motion and an independent integer-valued random measure. We establish the existence and uniqueness of the solution when the barriers and their left limits are completely separated without assuming Mokobodski’s hypothesis or the regularity condition.

Abstract In this paper, we investigate the existence and uniqueness of the solution for a system of nonlinear, fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our study is conducted within the framework of a separable real Hilbert space, and we employ a continuation method to establish the proof.

Abstract The purpose of this work is to use the mixed fractional Brownian motion with parameter H ∈ ( 0 , 1 2 ) {H\in(0,\frac{1}{2})} as a modeling tool in financial mathematic, specifically in stochastic volatility modeling. We employ high frequency data from Oxford-Man Institute of Quantitative Finance Realized Library to estimate the smoothness of log-volatility. We show through empirical experiments that the measured smoothness of log-volatility is similar to that of mixed fractional Brownian motion when 0.09 ≤ H ≤ 0.2 {0.09\leq H\leq 0.2} . We propose to model the log-volatility by a mixed Fractional Brownian motion with H < 1 2 {H<\frac{1}{2}} and we construct a stationary mixed fractional Ornstein–Uhlenbeck process d ⁢ X t = - λ ⁢ ( X t - μ ) ⁢ d ⁢ t + γ ⁢ d ⁢ M t H , t ∈ [ 0 , T ] {dX_{t}=-\lambda(X_{t}-\mu)dt+\gamma dM_{t}^{H},t\in[0,T]} as a stationary model of log-volatility X t {X_{t}} and we show that it tends in distribution to the mixed fractional Brownian motion when λ → 0 {\lambda\rightarrow 0} .

Abstract In this work we present an easy and direct construction of a Stochastic integral respect a cylindrical martingale-Lévy process. To do this, we apply the radonification technique. The radonification technique has been very useful to define an genuine stochastic process starting from a cylindrical process combine with Hilbert–Schmidt operators. With this work we present a self-contained construction, easy to follow to understand this new integration theories that are getting strong nowadays.

Abstract The limit theorem for the estimator G 59 {G_{59}} for the regularized solution of the Kolmogorov–Wiener filter is proved.

Abstract The canonical equation K 107 {K_{107}} is founded for random zebra-structured matrices.

Abstract In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 {\frac{1}{2}<\alpha<1} . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our result is provided.

Abstract We prove a Wegner estimate for alloy-type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set (a particular class of sets of positive measure). The proof exploits on one hand recently proven unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes and on the other hand the well developed machinery for proving Wegner estimates. We obtain a Wegner estimate with optimal volume dependence at all energies, and localization near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.