In the present paper, measure differential equations involving the distributional Henstock-Kurzweil integral are investigated. Theorems on the existence and structure of the set of solutions are established by using Schauder$^prime s$ fixed point theorem and Vidossich theorem. Two examples of the main results paper are presented. The new results are generalizations of some previous results in t...

We consider the controllability problem for continuity equation, corresponding to neural ordinary differential equations (ODEs), which describes how a probability measure is pushedforward by flow. show that controlled equation has very strong properties. Particularly, given solution of bounded Lipschitz vector field defines trajectory on set measures. For this trajectory, we there exist piecewi...

although elzaki transform is stronger than sumudu and laplace transforms to solve the ordinary differential equations withnon-constant coefficients, but this method does not lead to finding the answer of some differential equations. in this paper, a method is introduced to find that a differential equation by elzaki transform can be ‎solved?‎

We consider the stochastic model of water pollution, which mathematically can be written with a stochastic partial differential equation driven by Poisson measure noise. We use a stochastic particle Markov chain method to produce an implementable approximate solution. Our main result is the annealed law of large numbers establishing convergence in probability of our Markov chains to the solutio...

In this note, we provide a non trivial example of differential equation driven by a fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, whose solution admits a smooth density with respect to Lebesgue’s measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris type lemma in the roug...

Abstract. We study the asymptotic behavior of solutions to stochastic evolution equations with monotone drift and multiplicative Poisson noise in the variational setting, thus covering a large class of (fully) nonlinear partial differential equations perturbed by jump noise. In particular, we provide sufficient conditions for the existence, ergodicity, and uniqueness of invariant measures. Furt...

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuoussemimartingale. This result generalizes Dupire’s forward equation to a large class of non-Markovian models with jumps and allows to retrieve various forward equations previously obtained for ...

Before the proof, we detail the assumptions needed for Theorem 1. For pSGLD, its associated Stochastic Differential Equation (SDE) has an invariant measure ρ(θ), the posterior average is defined as: φ̄ , ∫ X φ(θ)ρ(θ)dθ for some test function φ(θ) of interest. Given samples (θt)t=1 from pSGLD, we use the sample average φ̂ to approximate φ̄. In the analysis, we define a functional ψ that solves the ...