Stochastic Integrals of Set-Valued Processes and Fuzzy Processes

Abstract

We define the stochastic integrals of a set-valued process and a fuzzy process with respect to a cylindrical Brownian motion on a Hilbert space. We also give their properties, which are useful for the study of fuzzy stochastic differential equations and stochastic differential inclusions with a set-valued diffusion term.

Representation theorems, set-valued and fuzzy set-valued Ito integral

“…It is easy to prove that {I t, (X) : ∈ (0, 1]} is not increasing and left continuous with respect to . Then by usually method dealt with fuzzy set-valued random variables, we have the same result as [14,Theorem 4.6] but by using new set-valued Ito integral defined in Section 4. Hence, we can define Ito integral for a fuzzy set-valued stochastic process as follows.…”

“…In [14], Kim used the definition of stochastic integral of set-valued stochastic process introduced by Kisielewicz in [16] and discussed its properties. We have to notice that (A) t 0 f (s) dB s is a stochastic process taking set values in…”

Set-Valued Stochastic Lebesque Integral and Representation Theorems

“…It may be useful in the area of stochastic control and mathematical finance. Secondly, we defined a new type Lebesgue integral of a set-valued stochastic process with respect to time t based on the nice works such as Kisielewicz 6 , Kim 15 , M. Kisielewicz, M. Michta and J. Motyl 13,14 . And then we discussed some properties of set-valued Lebesgue integral, especially we proved the presentation theorem of setvalued stochastic integral.…”

“…Kim 15 used the definition of stochastic integral of set-valued stochastic process introduced by Kisielewicz 6 and discussed its properties. We called it Aumann type integral since the idea came from Aumann integral of a set-valued random variable 16 .…”

Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

“…In 1999, Kim and Kim [2] used the definition of stochastic integrals of set-valued stochastic process with respect to the Brownian motion. They called it Aumann ( [3]) type It integrals.…”