A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation ...

We study the recognition capabilities of Hopfield model with auxiliary hidden layers, which emerge naturally upon a Hubbard-Stratonovich transformation. show that such at zero temperature outperform those original model, due to substantial increase storage capacity and lack defined basin attraction. The modified does not fall abruptly into regime complete confusion when memory load exceeds shar...

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in Stratonovich sense. We apply mapping method to obtain new trigonometric, elliptic, and rational fractional solutions. Because of importance KdV characterizing behavior waves shallow water, obtained solutions are beneficial interpreting certain fascinating phys...

We study a kinetic toy model for spray of particles immersed in an ambient fluid, subject to some additional random forcing given by mixing, space-dependent Markov process. Using the perturbed test function method, we derive hydrodynamic limit system. The law limiting density satisfies stochastic conservation equation Stratonovich form, whose drift and diffusion coe...

We propose two new positive weak second-order approximations for the CIR equation dXt = (a − bXt) dt + σ √ Xt dBt based on splitting, at each step, the equation into the deterministic part dXt = (a − bXt) dt, which is solved exactly, and the stochastic part dXt = σ √ Xt dBt, which is approximated in distribution. The schemes are illustrated by encouraging simulation results. Introduction In thi...

We derive the stochastic version of Magnus expansion for linear systems differential equations (SDEs). The main novelty with respect to related literature is that we consider SDEs in It\^o sense, progressively measurable coefficients, which an explicit It\^o-Stratonovich conversion not available. prove convergence up a stopping time {\tau} and provide novel asymptotic estimate cumulative distri...