Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

Abstract The Circular Law under Lindeberg’s condition for the independent blocks of random matrices having zero expectations and double stochastic matrix of covariances of their array is proven.

Abstract The Inverse Tangent Law for the solutions of systems of linear algebraic equations with independent random coefficients is proven under Lindeberg’s condition.

Abstract 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.

Abstract We use the Yosida approximation to find an Itô formula for mild solutions { X x ⁢ ( t ) , t ≥ 0 } {\{X^{x}(t),t\geq 0\}} of SPDEs with Gaussian and non-Gaussian colored noise, with the non-Gaussian noise being defined through a compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in C 1 , 2 ⁢ ( [ 0 , T ] × H ) {C^{1,2}([0,T]\times H)} , as in the case considered for SDEs in [15]. Using this Itô formula, we prove exponential stability and exponential ultimate boundedness properties, in the mean square sense, for mild solutions. We also compare this Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [12], and an Itô formula written in terms of the semigroup of the drift operator [5], which we extend to the non-Gaussian case.

Abstract This paper deals with random motion at non-constant speed with uniformly distributed directions where the direction alternations occur according to renewal epochs of general distribution. We derive the renewal equation for the characteristic function of the transition density of the multidimensional motion. By using the renewal equation, we study the behavior of the transition density near the sphere of its singularity for the two- and four-dimensional cases and variable velocity, and the three-dimensional case for constant velocity. As examples, we have derived the distribution for one-, two- and three-dimensional random motion.

Abstract In this paper, we consider a class of one-dimensional reflected Backward Stochastic Differential Equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. Using a stochastic variational inequality, we characterize its solution.

Abstract Convergence from below of the Esscher premium for an arbitrary insurance risk to the essential supremum of the priced risk when the premium parameter tends to plus infinity is presented in the article. As a corollary of the above mentioned statement, we present convergence of the expected/averaged Esscher transform of an arbitrary random variable to its essential infimum from above when the transformation parameter tends to minus infinity.

Abstract In this note we consider a class of neutral stochastic functional differential equations with finite delays driven simultaneously by a Rosenblatt process and Poisson process in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.