Averaging principle for fuzzy stochastic diﬀerential equations

This study oﬀers the averaging principle for fuzzy stochastic diﬀerential equations (FSDEs). The solutions to FSDEs can be approximated in the sense of mean square solutions of averaged fuzzy stochastic system under certain assumptions.


Introduction
The averaging method is a powerful tool for investigating the qualitative property of dynamical system in physics as well as in variety of other fields. This method shows a connection between the solutions of averaged systems and the solutions of a standard form [15,16]. Nevertheless, up to now, the averaging principle for fuzzy stochastic differential equations (FSDEs) has not yet been studied in literature. In the present paper, we make the first attempt to study this method for FSDEs.
For crisp stochastic differential equations (SDEs), seminal results on the averaging principle can be found in [5,6,12]. Tan et al. [13] established the averaging method for stochastic differential delay equations (SDDEs) under non-Lipschitz conditions. In [11,14], the authors investigated the averaging principle for SDDEs with jumps and with fractional Brownian motion. Recently, Guo et al. [3] established the averaging method for a class of SDEs with nonlinear terms satisfying the monotone condition, and Luo et al. [7,8] investigated the averaging principle for a class of stochastic fractional differential equations (SFDEs) with time-delays. Ahmed et al. [1] established the averaging principle for Hilfer fractional stochastic delay differential equations with Poisson jumps. On the other hand, FSDEs are utilised in real-world systems where the phenomena is connected to randomness and fuzziness as two types of uncertainty. In [2,4], the authors presented a definition of the fuzzy stochastic Itô integral using a method that allows embedding of a crisp Itô stochastic integral into fuzzy space for building a fuzzy random variable. The present paper aims at extending the averaging principle to FSDEs.
The rest of this paper is organised as follows. Section 2 provides the fundamental tools that are required in upcoming sections. In Section 3, the averaging method for FSDEs under some conditions is investigated. An example is given in Section 4 to illustrate the main result of this paper. Finally, the conclusion is given in Section 5.

Preliminaries
In this section, we introduce some notations, definitions, and preliminary facts which are used in the rest of this paper. Let K(R n ) be the family of nonempty convex and compact subsets of R n . In K(R n ), the distance d H is defined by It is know that K(R n ) is a complete and separable metric space with respect to d H . Let E n be the fuzzy set space of R n , i.e. the set of functions v : be the metric satisfying the following properties Let . : R n −→ E n be an embedding of R n into E n , i.e. for r ∈ R n , one has Let {B(t), t ∈ I := [0, T ]} be an one-dimensional Brownian motion defined on a complete probability space (Ω, A, P) with a filtration {A t } t∈[0,T ] satisfying usual hypotheses. Definition 2.1 (see [10]). By fuzzy stochastic Itô integral we mean the fuzzy random variable t 0 v(s)dB(s) . For every t ∈ I, consider the fuzzy stochastic Itô integral t 0 v(s)dB(s) , which may be interpreted as follows

Main result
Consider the following FSDEs dx(s) = f (t, x(t))dt + g(t, x(t))dB(t) , where f : I × E n −→ E n , g : I × E n −→ R n and x 0 : Ω −→ E n is a fuzzy random variable. Equation (1) is equivalent to the following fuzzy stochastic integral equation We apply conditions on the coefficient functions to ensure that the solution to (1) exists and is unique.
(A1). There exists a constant C 1 > 0 such that ∀ t ∈ I and ∀ x ∈ E n we have (A2). There exists a constant C 2 > 0 such that ∀ t ∈ I and ∀ x, y ∈ E n we have By the work of Malinowski and Michta [10], we know that under the assumptions (A1) and (A2), FSDEs (1) has a unique solution x(t) with the initial data x 0 . Let us consider the standard form of Equation (2) where the initial value x 0 , functions f and g have the same conditions as in Equation (2), and ∈ (0, 0 ) is a small positive parameter with 0 a fixed number. Based on the existence and uniqueness results, Equation (3) also has a unique solution x (t) for every fixed ∈ (0, 0 ) and t ∈ I. We set certain assumptions on the coefficients to see if the solution x (t) can be approximated by a simple process.
Letf : E n −→ E n andg : E n −→ R n be measurable functions satisfying (A1) and (A2), as well as the following additional inequalities: (A3). For x ∈ E n and T ∈ I we have With the above appropriate preparations, we now show that the solution x converges to the solution y of the following averaged FSDEs as −→ 0. Clearly, under similar assumptions as of Equation (3), Equation (4) also has a unique solution y . The main result of this paper is now presented in the form of the following theorem, in which we consider the connections between x and y .

An example
In this section, we give an example to illustrate our main result of this paper. Consider the following FSDEs dX(t) = 4 cos 2 (t)X(t)dt + X(t)dB(t) , The standard form of the above FSDEs is given as Note that f (t, X ) = 4 cos 2 (t)X and g(t, X ) = X . Hence, f (X ) = 1 π π 0 4 cos 2 (t)X dt = 2X andg(X ) = 1 π π 0 g(t, X )dt = X .
Therefore, the averaging form of (7) is The coefficients f (t, X ) and g(t, X ) satisfy the assumptions (A1) − (A2), and hence FSDEs (7) has a unique fuzzy solution. Also, it is observed that the coefficientsf (X ) andg(X ) satisfy the assumption (A3). Therefore, by Theorem 3.1, as −→ 0, the solutions X and Y to Equations (7) and (8) are equivalent in the sense of mean square.

Conclusion
In this work, we have established the averaging principle for FSDEs. We have proved that the solution to the averaged FSDEs converges to that of the standard FSDEs in the sense of mean square. For future researches, we plan to study the averaging principle for fuzzy fractional stochastic differential equations with/without Hilfer fractional derivative.