In the paper we consider fuzzy stochastic integral equations using the methods of stochastic inclusions. The idea is to consider an associated martingale problem and its solutions in order to obtain a solution to the fuzzy stochastic equation.

In this paper, we shall firstly illustrate why we should discuss the Aumann type set-valued Lebesgue integral of a set-valued stochastic process with respect to time t under the condition that the set-valued stochastic process takes nonempty compact subset of d-dimensional Euclidean space. After recalling some basic results about set-valued stochastic processes, we shall secondly prove that the...

This paper shows the detection of textures by analysing the image by stochastic properties. Different parts of the composed stochastic properties are fused by a decision directed method. The stochastic is mapped on a fuzzy measure and a fuzzy function. The fusion of the so represented stochastic information is achieved by the fuzzy integral. The result of the fuzzy integral is used for a new fu...

This paper gives an elementary introduction to the development of the stochastic integral. I aim to provide some of the foundations for someone who wants to begin the study of stochastic calculus, which is of great importance in the theory of options pricing.

I consider a Langevin equation with field-dependent kernels and investigate supersymmetry of the stochastic generating functional constructed from the Langevin equation. Moreover I describe the stochastic generating functional in terms of a superfield. In the superfield formalism, it becomes clear that the stochastic quantization method with the field-dependent kernel is equivalent to the path-...

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by backward Volterra integral (BSVIE, short). This kind of can cover general discounting (including exponential and non-exponential) situations recursive feature. It known that such time-inconsistent in general. Therefore, instead finding global control, we look time-consistent lo...

We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can b...

In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jum...