On solutions to set-valued and fuzzy stochastic differential equations

Abstract

We study existence and uniqueness of solutions to nonlinear set-valued stochastic differential equations driven by multidimensional Brownian motion. The conditions imposed on the equation׳s coefficients are non-Lipschitz. The drift coefficient is set-valued and diffusion coefficient is single-valued, both coefficients are random. The approach used in this paper allows the solutions to be set-valued stochastic processes. The set-valued results are then extended for the parallel studies of nonlinear fuzzy stochastic differential equations with solutions being fuzzy stochastic processes.

Introduction

Differential equations are mathematical tools useful in describing many nonlinear dynamical phenomena in e.g. physics, engineering, economics, or biology. Modeling with the ordinary differential equations $\dot{x}\left(t\right)=f(t,x(t\left)\right)$ is appropriate when the knowledge of the considered system is complete and the structure of the system is precisely described. In particular, all the parameters of the system, its initial values and the functional relationships governing the system׳s dynamics should be given as single values of the states space and hence rid of all imprecision. However, in mathematical modeling one often encounters some level of uncertainty. For example, ordinary differential equations require a precise knowledge on the initial values, yet the measurement equipment used in practice is unlikely to yield such information with the required level of precision: the measurements are given within a given range and hence not presented as a single value but as a set of values. Imprecision can also appear in description of the functional relationships characterizing the considered nonlinear system. The dynamical system can have velocities that are not uniquely determined by its state and that can be described by sets of feasible velocities. Such an approach leads to the application of set-valued mappings in modeling nonlinear dynamics in the presence of uncertainty. Set-valued differential equations $X\prime \left(t\right)=F(t,X(t\left)\right)$, where both X and F are set-valued mappings, are very popular tools used in mathematical modeling of the nonlinear dynamical systems with imprecision or incomplete information, see [1], [2], [3], [4]. These equations were formulated in [5] and studied in e.g. [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Currently this topic of research forms an independent branch of the set-valued analysis. The monograph [22] presents extensive studies concerning set-valued differential equations.

The theory of fuzzy differential equations extends the theory of set-valued differential equations. It has been proposed in [23]. The states of nonlinear dynamical systems described with these equations are mathematically described in the form of the fuzzy sets. The notion of a fuzzy set was introduced as an extension of the notion of a set, see [24]. This allows the mathematical modeling of impreciseness, vagueness, ambiguity or fuzziness also in the context of control theory [25]. The fuzzy differential equations are mathematical tools used in modeling dynamical phenomena with ambiguous or fuzzy states. The monograph [26] presents an extensive study of these equations including two methods of defining them. The first definition, along the lines of [23], uses the notion of a fuzzy derivative. A solution to a fuzzy differential equation is then a differentiable mapping with values in the fuzzy sets of ${\mathbb{R}}^d$. In the second definition, introduced in [27], the previously mentioned derivative is not used. Instead, the fuzzy differential equation is treated as a family of differential inclusions that are generated by the right-hand side of the fuzzy equation. There are many papers analyzing the fuzzy differential equations, notably [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41] presenting the results related to these equations defined in any of the two described ways.

The uncertainty considered above results from ambiguity, or more generally – fuzziness. It means nonstatistical inexactness that is due to subjectivity and imprecision of human knowledge rather than to the occurrence of random events. However, very often a second source of uncertainty should be taken into account in mathematical modeling real-world phenomena. This other uncertainty is produced by an influence of random factors and stochastic noises on the considered system. It is connected with uncertainty in prediction of the outcome of an experiment. It breaks the law of causality and the probabilistic methods are applied in its analysis. A need of incorporating two different types of uncertainties, i.e. fuzziness and randomness, into some mathematical models is explained in e.g. the studies of random fuzzy differential equations [42], [43], [44], [45], stochastic fuzzy cellular neural networks [46], sliding mode control for stochastic systems [47], adaptive fuzzy output feedback control for stochastic nonlinear systems [48], nonlinear stochastic Takagi–Sugeno fuzzy systems [49], passivity-based resilient adaptive control scheme for Takagi–Sugeno fuzzy stochastic systems with Markovian switching [50], stochastic fuzzy Markovian jumping neural networks [51] and fuzzy stochastic differential equations [52], [53], [54], [55], [56], [57]. The latter topic is very current. The fuzzy stochastic differential equations find their applications in terms of mathematical models of nonlinear dynamical systems subjected to two kinds of uncertainties: fuzziness and randomness. They were applied in modeling short-term interest rate, dynamics of stock price and in modeling population growth [54].

In this paper we consider the set-valued stochastic differential equations which are understood as the following integral equations:

$$X\left(t\right)=X_0+{\int }_0^{t}F(s,X(s\left)\right)\mathit{ds}+{\int }_0^{t}G(s,X(s\left)\right)\mathit{dB}\left(s\right),$$

where $X_0,F,G$ are random, $X_0,F$ are set-valued, G is single-valued, B denotes a multidimensional Brownian motion. These have solutions in the form of adapted set-valued stochastic processes and generalize the deterministic set-valued differential equations studied extensively in [22]. They could be of interest in the investigations of stochastic systems, where the quantities are not known precisely but they fluctuate within some intervals. Then we extend the set-valued stochastic differential equations to the fuzzy stochastic differential equations. They generalize the deterministic set-valued and fuzzy differential equations and the single-valued stochastic differential equations as well. There are some new mathematical tools in modeling nonlinear dynamics of systems subjected to two kinds of uncertainties: randomness (stochastic uncertainty), vagueness (nonstochastic uncertainty expressed in the language of sets or fuzzy sets), see [52], [53], [54]. Here, the random phenomena evolve in the phase space of fuzzy sets. These equations constitute an interdisciplinary approach in nonlinear dynamical systems and allow us to team up with set-valued, fuzzy and stochastic analysis.

The fuzzy and set-valued stochastic differential equations have been studied with assumptions of Lipschitzian coefficients in [52], [53], [54]. The papers [58], [59] followed them considering some non-Lipschitz coefficients and using the Maruyama successive approximation scheme. In this paper we impose some weaker conditions than those in [58], [59]. Also we exploit the Picard successive approximation sequence. In this way, the class of admissible coefficients in fuzzy and set-valued equations that possess unique solutions is extended. By a theoretical analysis, we show that fuzzy stochastic differential equations possess unique solutions and this is achieved with some non-Lipschitz conditions which are by far the weakest ones used in this new theory of stochastic differential equations. The problem of existence and uniqueness of solutions belongs to the foundations of the theory. In the proofs we use method of successive approximations like in [52], [54]. Then we prove a closeness of solutions to equations having coefficients which do not differ much. Finally, we present an illustration of modeling with a fuzzy stochastic differential equation of a control system. Some simulations of solution sample paths are also included.

The paper is organized as follows. In Section 2 we collect a relevant material concerning set-valued random variables, set-valued stochastic processes and set-valued stochastic Lebesgue–Aumann integral to make the paper self-contained in these subjects. Section 3 is focused on set-valued stochastic differential equations while Section 4 on fuzzy stochastic differential equations. In Section 5 we present an example of application of the fuzzy stochastic differential equations in a real-world problem with control. Section 6 summarizes the contribution and discusses some future research directions.