On a Coupled System of Stochastic Ito^-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.

Surjectivity of near-square random matrices

We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

Some existence results and stability concepts for partial fractional random integral equations with multiple delay

In this paper, we present some results concerning the existence and the stability of solutions for some functional integral equations of Riemann–Liouville fractional order with random effects and multiple delay, by applying a random fixed point theorem with stochastic domain and the measure of noncompactness.