Fuzzy Stochastic Differential Equations Driven by a Fuzzy Brownian Motion

In our previous work, we study fuzzy Itô integrals driven by a fuzzy Brownian motion. In this article, we continue this study. The purpose of this paper is to study the weak uniqueness of fuzzy stochastic differential equations taking into account fuzzy Brownian motion. For instance, we construct the fuzzy stochastic differential equation driven by a fuzzy Brownian motion. To define and prove our results, we use the fuzzification, the alpha cut method and the Hausdorff distance between two fuzzy quantities. Some results are to our cre-dit in this article like the instance, we construct the fuzzy stochastic differ-rential equation driven by fuzzy Brownian motion. Furthermore, we develop fuzzy Itô calculus driven by a fuzzy Brownian motion. Our result complement existing ones in that the fuzzy version of Brownian motion is taken into account.


Introduction
In our previous research, we have dealt with Fuzzy Itô Integral Driven by a Fuzzy Brownian Motion and in the present work, we develop Fuzzy stochastic differential equations driven by a fuzzy Brownian motion [1].
Historically, the earliest approach for deterministic fuzzy differential equations x ′ ∈    verifying (2.1) exists, we say that f  is Hukuhara generalized differentiable (gH-differentiable) at 0 x . Definition 2.2. (Bede and Stefanini [14] x  α all differentiable in 0 x . We say that Definition 2.3 (Hinge point, Stefanini and Bede [14]) We say that the point partially derivable functions with respect to x. We say that 1) ( ) , ii gH Definition 2.7. (Adapted fuzzy process, Malinowski [5])

3) For all
In the differential form, (3.1) is denoted by Progressively, we can now present a one-dimensional fuzzy Itô process. Now, let us give the fuzzy version of the integration by parts using generalised Hukuhara (Bede and Stefanini [14]).
Proof. We know that Then, from the classical integration by parts, we obtain Due to the definition of Hukuhara difference and from this inequality By expressing the above integral in a sum of little integrals, for all Thus the equality (3.3) is obtained from the equality of fuzzy mumbers.

Fuzzy Itô Formula
The most important result in the fuzzy stochastic integration is the fuzzy Itô formula. The Itô formula is also named variables change formula.

Fuzzy Itô Formula Driven by a Fuzzy Brownian Motion
The lower and the upper bounds of these α-cuts are given by In the same way, we could take B t B t . From the classical Itô formula of the function Thus, the equations from (4.5) to (4.10) by taking the minimum and the maximum in (4.11) give for all which establishes by definition, the equality sought.

Fuzzy Itô Formula of a Fuzzy Itô Process
Let us start this part by the following theorem. Let  t X Fuzzy d'Itô process such that is also a fuzzy Itô process and where ( ) . These i l are abtained in this manner Without loss of generalities, we can deal with the case of the second order generalized Hukuhara differential of type (1) Hence, for all By taking the maximum and the minimum to every terms from the Equation Because all the derivatives are taken in the sense of generalized Hukuara and due to the definition 2.4, we have for every

Thus we obtain for every
Proof. From the definition of D, we have for all We have expanded out and have assumed that

Fuzzy Stochastic Differential Equations
Notice that the notion of fuzzy differential equations defined in an uncertain framework has already been the subject of several investigations (see Buckley and Feuring [17] and Puri and Ralescu [2]). In the same way, we will define in this section, the notion of fuzzy stochastic differential equations driven by a fuzzy Brownian motion. Using the Hukuhara differential, we have the following definition.
Definition 5.1. A fuzzy stochastic differential Equation (FSDE), on a filtered probability space ≥ Ω   t P and driven by a fuzzy Brownian motion is given by where  F and  G are borel-measurable fuzzy stochastic functions defined In the integral form, the fuzzy stochastic differential Equation (FSDE) driven by a fuzzy Brownian motion (5.1) is denoted The solution of (5.1) is defined like that: P a e , that is We say that t X  is a strong fuzzy solution wether t X  is adapted to the canonical fuzzy filtration There is a weak unicity since two solutions of the fuzzy stochastic differential equation have necessarily the same law.
Proof. Due to the Doob inequality and the classical Itô integral isometry we obtain from the hypothesis (5.3) And from the hypothesis (5.4), we get   sup max , d