Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.


Introduction
Deterministic fuzzy differential equations have been developed due to investigations of dynamic systems where the information on parameters of such systems is incomplete or vague.They play an important role in an increasing number of system models in biology [1], engineering [2], civil engineering [3], bioinformatics and computational biology [4], quantum optics and gravity [5], and hydraulic [6,7] and modeling of mechanical systems [8].Many investigations in this area were developed using different approaches for formulations of differential problems in a fuzzy setting (see, e.g., [1,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references therein).Historically, the earliest approach for deterministic fuzzy differential equations was based on a generalization of the Hukuhara derivative of a set-valued function.This was made by Puri and Ralescu in [24] and used by Kaleva in [15,16].Further extensions were developed next also in [25][26][27] where the concept of strongly generalized differentiability was introduced.A different approach was proposed by Hüllermeier in [13] where fuzzy differential equations were interpreted as a family of differential inclusions associated with level sets of their fuzzy right hand sides.Such an approach has been also used next among others in [1,9,12,18,22,28,29] (see also the references therein).A further step is a research concerning stochastic fuzzy differential (or integral) equations which generalize both classical stochastic differential equations and deterministic fuzzy differential equations.In this case, the main problem is a concept of a fuzzy stochastic integral which should cover the notion of the classical stochastic Itô integral.A research concerning stochastic fuzzy differential (or integral) equations driven by a Wiener process has been initiated in different forms in [30][31][32][33], and it can be applied in modeling of phenomenons where two kinds of uncertainties, that is, randomness and fuzziness, are incorporated simultaneously.For applications in stochastic population models, see, for example, [31,34].The aim of this paper is twofold.Firsly, we present a survey of extensions of some of approaches for deterministic fuzzy differential equations to fuzzy stochastic equations driven by semimartingales presented mainly in [35][36][37] and developed further in [38,39].Secondly, we present an analysis of the notion of fuzzy Itô's stochastic integral understood as fuzzy random variable.So, we will start with presentation of some recent results where different

Fuzzy Random Variables and Fuzzy Stochastic Integral
We start with some facts from stochastic analysis needed in the sequel.We recall the notion of a fuzzy stochastic integral with respect to semimartingale integrators studied first in [32] and next used and developed in [35][36][37][38].Let  > 0 and let  = [0, ] or  + .Let (Ω, F, {F  } ∈ , ) be a complete filtered probability space satisfying the usual hypothesis; that is, {F  } ∈ is an increasing and right continuous family of sub--fields of F and F 0 contains all -null sets.Let P denote the smallest -field on  × Ω with respect to which every leftcontinuous and {F  } ∈ -adapted process is measurable.An   -valued stochastic process  is said to be predictable if  is P-measurable.One has P ⊂  ⊗ F, where  denotes the Borel -field on .Let  2, fl  2 (Ω, F, ;   ) for  ≥ 1.By  2 we denote the space of all {F  } ∈ -adapted and càdlàg (i.e., right continuous and with finite left-hand limits) processes ) is a Banach space (see, e.g., [45]).We will use the notation  − fl lim ↗   -a.s.Let  be an {F  } ∈ -adapted and càdlàg process with values in  1 .It is said to be a semimartingale if  =  + , where  is an {F  } ∈ -adapted local martingale and  is an {F  } ∈ -adapted, càdlàg process with finite variation on compact intervals in  (see, e.g., [45] for details).We will assume that  0− =  0 = 0.
Let us consider the class of H 2 -semimartingales, that is, the space of {F  } ∈ -adapted semimartingales with a finite H 2norm: where [, ] denotes the quadratic variation process for a local martingale , while || ⋅ fl ∫ ⋅ 0 |  | represents the total variation of the random measure induced by the paths of the process .Proceeding, similarly as in [32], we will introduce some measure   on the predictable -field P associated with a semimartingale .Since  ∈ H 2 , it follows that  is a square integrable martingale such that  2  = [, ]  for all  ∈ .By the same reason, the process  has a square integrable total variation on .By   denote the Doléans-Dade measure for the martingale ; that is,   is a unique measure on a predictable -field P such that for all  ∈ F  , 0 ≤  < , and  0 ∈ F 0 (see, e.g., [46]).Then for all  ∈ and a measure associated with the process  by the following formula: for every  ∈ P. Then ]  is a finite measure on P. Finally, we define a finite measure   associated with  ∈ H 2 by   fl   + ]  .Let us denote  2 P (  ) fl  2 ( × Ω, P,   ;   ).Then by Proposition 1 in [32] for every  ∈  2 P (  ) and  ∈  there exists a stochastic integral ∫  0     .In order to introduce a notion of fuzzy stochastic integral we begin with some auxiliary facts.Let X be a separable Banach space.By K(X), K  (X), or K   (X) we denote the family of all nonempty closed, all nonempty closed and bounded or nonempty closed bonded, and convex subsets of X, respectively.We will consider the spaces K  (X) or K   (X) with a Hausdorff metric  X :  X (, ) fl max {sup ∈  X (, ) , sup ∈  X (, )} , (6) where  X (, ) fl inf ∈ ‖ − ‖ X and ‖ ⋅ ‖ X is a norm in X.Then (K  (X),  X ) and (K   (X),  X ) are complete metric space (cf.[47]).For  ∈ K(X), we set ‖||‖ X fl  X (, {0}) = sup ∈ ‖‖ X .By a fuzzy set  of a Banach space X we mean a mapping  : X → [0,1].The space of all fuzzy sets of X will be denoted by the symbol F(X).For  ∈ (0, 1] let []  fl { ∈ X : () ≥ } and [] 0 fl cl X { ∈ X : () > 0} where cl X denotes the closure in (X, ‖ ⋅ ‖ X ).In the sequel we deal with the following fuzzy sets: We will use a metric  X in F   (X) described as follows: One can show (cf.[48]) that (F   (X),  X ) is a complete metric space.Other metrics used in the set and the Skorokhod metric where and Λ is the set of strictly increasing continuous functions  : [0, 1] → [0, 1] with (0) = 0 and (1) = 1.Functions   ,  V : [0, 1] → K   (X) are the càdlàg representations for the fuzzy sets , V ∈ F   (X); see Colubi et al. [49] for details.The space (F   (X),  X  ) is separable and noncomplete but (F   (X),  X  ) is a Polish metric space.By B  , we define a borel -field in a metric space (F   (X), ) where  is one of metrics described above.For , V ∈ F   (X) the addition  ⊕ V is defined by Zadeh's extension principle (see [11]).But, due to Lemma 3.4 in [50], it can be also defined levelwise; that is, In what follows, we will use the following version of the theorem of Negoita and Ralescu.
Then by Theorem 1, Proposition 2, and Theorem 3, for every fixed  ∈ , there exists a fuzzy set (say) (, )  ∈ F   ( Using similar methods as in [36] and general properties of stochastic integrals driven by càdlàg semimartingales, one can show the following properties. (c) The mapping is right continuous with finite left-hand limits with respect to the metric   2, .Remark 6.In [38] a slight extension of the notion of fuzzy stochastic integral with respect to the semimartingale  =  +  was proposed.Namely, it was defined as the sum: Although, in general, the fuzzy sets (F) ∫  [35]).Remark 7. In Section 4, we will study the notion of fuzzy stochastic integral driven by the Wiener process and which is understood as a fuzzy-valued random variable.We will show there that such understood fuzzy-valued stochastic integrals may have unbounded in  2 level sets.

Fuzzy Stochastic Differential Equation Driven by a Semimartingale
Below we establish recent result for two different approaches for fuzzy stochastic differential equations.

A Direct Approach.
For further considerations, we assume that the -field F is separable with respect to the probability measure  and  is a continuous semimartingale.
Remark 15.Using similar methods as in [35], one can prove existence and uniqueness results to fuzzy stochastic integral equation (with respect to continuous H 2 -semimartingale  =  + ) or to its -component version.Such results were established in [38].Further considerations for fuzzy stochastic equations in the metric space (F   ( 2, ),   2, ) setup and under weaker assumptions than Lipschitz type can be found in [39].
To proceed further, we assume the following conditions: (h1)  is a given {F  } ∈ -adapted, H 2 -semimartingale with a decomposition  =  + , where  is an {F  } ∈adapted increasing predictable process.
(h2)   is absolutely continuous with respect to the product measure  ⊗  on the -field P.
Then we have the following result (see [37]).
Remark 19.In [32] similar approach was used to fuzzy stochastic differential equations driven by the Wiener process.The idea implemented in this case was based on martingale problem approach for the existence of weak solutions to the system of associated stochastic integral inclusions.Thus the existence of fuzzy solutions could be achieved under weaker conditions than Lipschitz continuity imposed on the right-hand side of a fuzzy stochastic differential equation.But also in this case its deterministic counterpart reduces to the one described in Remark 18.

Fuzzy-Valued Stochastic Equations with Fuzzy Stochastic Solutions
The approaches presented above, that is, direct and based on stochastic inclusions, are different.It is easy to note that they lead to the different notions of fuzzy-valued solutions.In the first one the fuzzy solution is meant as a continuous fuzzyvalued mapping  :  → F   ( 2, ) satisfying (21), while in the second approach by a solution to (34) we mean an element in F  ( 2 ) described in Definition 16.In a series of papers [33,34,[40][41][42][43][44] another concept of the notion of fuzzy stochastic integral equation was proposed.Generally speaking, it is understood as the following relation:  (  )) understood as random set for every  ∈ .Then (see, e.g., [33,40]) under the classical Lipschitz type assumption, there exists a unique continuous stochastic fuzzy-valued process  being a solution to (40).Unfortunately, in the series of papers mentioned above, only such diffusion case was studied.Therefore one can ask if it is possible to consider fuzzy stochastic differential equations with more general fuzzy-valued diffusion terms and with solutions being fuzzy-valued stochastic processes.Unfortunately, avoiding such trivial diffusion cases the answer is negative in general.As it is shown below this problem is strictly connected with the lack of integrally boundedness in general of set-valued stochastic integrals treated as set-valued random variables even for integrally bounded or bounded integrands.Suppose that for a fuzzy stochastic process ℎ we can define a fuzzy-valued Itô stochastic integral which is assumed to be a fuzzy-valued random variable: Then for every  ∈ [0, 1] we have a set-valued random mapping  In order to have an extension (beyond the crisp case) of a fuzzy stochastic integral as a fuzzy random variable, it should coincide in a crisp case with ordinary (single-valued) Itô's stochastic integral.Thus in the case when ℎ is such that [ℎ(, )]  = {(, )},  ∈ [0,1], for some   -valued predictable (or nonanticipating in this case) and square integrable stochastic process , it follows that   () is defined as in Section 2 and it is a setvalued stochastic integral with respect to a semimartingale which is now a Wiener process.Assume now that ℎ is such that the sets [ℎ(, )]  are not singletons for all  ∈ [0, 1] and (, ) ∈  × Ω.Then it follows that the sets ∫  0 [ℎ()]  () need not be singletons as well.Hence they need not be decomposable subsets of the space  2,  fl  2 (Ω, F  , ;   ) (see [32,53] for details) while the set S 2 for all  ∈ [0,1], where cl  2,  dec() denotes a closed decomposable hull of a given set  ⊂  2,  (see, e.g., [54] for details).Now, let us consider a special one-dimensional case and the mapping ℎ :  × Ω → F   ( 1 ) defined by ℎ(, ) fl  for every (, ) ∈  × Ω, where  ∈ F   ( 1 ) is such that () = ( + 1)I [−1,0) () + (1 − )I [0,1] () for  ∈  1 .Hence [ℎ(, )]  = [−1 + , 1 − ] for  ∈ [0, 1] and (, ) ∈  × Ω.Then by virtue of Corollary 3.9 (as well as Corollary 3.11 and Remark 3.12) in [55] for such chosen mapping ℎ, it follows that the set cl  2,1  dec(∫  0 [ℎ()]  ()) is unbounded as a subset of the space  2,1  for every  ∈ [0, 1).Therefore by (47) for every  ∈ [0, 1) the set S