We investigate functions which are solutions of the equation ″ = − 2 and related ones. Solutions which satisfy also the initial conditions (0) = 0, ′(0) = 1 and (0) = 1, ′(0) = 0 are known as lemniscatic sine and cosine functions respectively. Taking our cue from the theory of elementary...

Set-indexed stochastic analysis and set-indexed stochastic calculus are faced here with a new approach of dimension's reduction. We introduce a new tool (main flow) in order to deal with one-parameter calculus in set-indexed framework. We prove an Ito formula for any Brownian functional where...

This paper studies the comparison theorem of forward-backward differential equations with Poisson jumps(FBSDEJ for short). We use a purely probabilistic approch including stopping time techneque and iteration method to prove comparison theorem. Here we only investigate some classes of FBSDEJs...

We introduce and study a summability method of power series in several variables, and investigate applications to formal solutions of singular perturbation problems and partial differential equations. Doing so, we extend results of Lutz, Miyake and Schäfke, resp. Balser, for the complex heat...

A class of measure-valued branching diffusions with interactive immigration is constructed by solving a stochastic integral equation with Poisson process based on a system of excursion laws of a Dawson-Watanabe superprocess, extending the results of Pitman and Yor (1982) and Shiga (1990)

In this paper, by considering the Adomian decomposition method, explicit solutions are calculated for partial differential equations with initial conditions. The method does not need linearization, weak nonlinearly assumptions or perturbation theory. The decomposition series analytic solution of...

In this paper, by considering the Adomian decomposition method, explicit solutions are calculated for partial differential equations with initial conditions. The method does not need linearization, weak nonlinearly assumptions or perturbation theory. The decomposition series analytic solution of...