Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients

Abstract

We study fuzzy stochastic differential equations driven by multidimensional Brownian motion with solutions of decreasing fuzziness. The drift and diffusion coefficients are random. Under a non-Lipschitz condition, the existence and pathwise uniqueness of solutions to such the equations are proven. The solutions are considered to be fuzzy stochastic processes. The main result is obtained with a help of a sequence of approximate solutions that converge to a desired unique local solution with trajectories having decreasing fuzziness. A parallel assertion for solutions to fuzzy stochastic differential equations of increasing fuzziness is stated as well. We indicate that our considerations of fuzzy stochastic differential equations of decreasing fuzziness can be applied to examine non-Lipschitz set-valued stochastic differential equations with solutions being set-valued stochastic processes.

Keywords

Fuzzy stochastic differential equation fuzzy stochastic integral equation fuzzy stochastic process set-valued stochastic differential equation set-valued stochastic integral equation set-valued stochastic process uncertainty

1 Introduction

Stochastic differential equations constitute a mathematical apparatus for dealing with phenomena whose evolution is governed by random forces. It is known (see e.g. [6 , 39]) that pathwise uniqueness of solution to stochastic differential equation holds if drift and diffusion coefficients are Lipschitz continuous. However, in some applications, these conditions could be very restrictive. In [48] one can find first studies on pathwise uniqueness for stochastic differential equations with weaker requirements imposed on the equations’ coefficients. Nowadays the conditions formulated in [48] are very popular and called the non-Lipschitz conditions. Non-Lipschitz coefficients are studied in many works concerning stochastic differential equations, mentioning only few [1 , 47]. In this paper we shall use conditions of a similar kind in a framework of fuzzy stochastic differential equations.

For a long period of time, stochasticity was considered as the only source of possible uncertainty. However, during last fifty years a new theory of fuzzy sets [50 –52] was developed to deal with a different type of uncertainty than randomness. The fuzzy sets allow for a more flexible modeling of vagueness, imprecision and ignorance in such areas like artificial intelligence, information processing or decision theory. They can also be recognized if we have incomplete or vague information on parameters of a considered dynamic system. It is clear that fuzzy sets are useful to model uncertainty that is not stochastic in its nature but is connected with an imprecision of human knowledge. Teaming up the randomness with the fuzziness gives a perspective to deal with various problems, e.g. sensor signal processing [2], neural networks [3, 15], reliability analysis [13 , 49], T-S fuzzy systems [12, 45], petroleum contamination [11], mechanical structures [16 , 38], water distribution networks [41], statistical inference [42] or option pricing [53].

Coexistence of stochastic uncertainty and fuzziness in dynamic systems motivate to look for some mathematical tools which could be appropriate in description of evolution of such systems. The random fuzzy differential equations [17 –20] and the fuzzy stochastic differential equations [21 –34] can be adequate in modeling of the dynamics of real-world phenomena which are subjected to two kinds of uncertainties: randomness and fuzziness. Some potential applications of such new equations in financial mathematics, population models and control systems can be found in [19 , 31]. Nevertheless the theoretical studies need to be developed as they assist practical applications. It is enough to mention that no numerical procedure of finding an approximate solution is possible if there is no theoretical guarantee that the analytical solution exists. Therefore building a theory of the new discipline of fuzzy stochastic differential equations is of a first importance.

In this paper we develop the new theory of fuzzy stochastic differential equations of decreasing fuzziness, initiated by the author in [32 –34], in a direction to obtain existence of a unique solution under conditions that are weaker than Lipschitz condition used in [32 –34]. Hence, the drift and the diffusion coefficients are considered to be random and non-Lipschitzian. Going into some formal generalities, we mainly consider the following initial value problem for fuzzy stochastic differential equation of decreasing fuzziness

$\begin{matrix} dx (t) \oplus [(- 1) ⊙ f (t, x (t)) dt \\ \oplus 〈 (- 1) g (t, x (t)) d B (t) 〉] = 〈 0 〉, x (0) = x_{0}, \end{matrix}$ where the coefficients f and g satisfy non-Lipschitz conditions. A main result of the paper states that under non-Lipschitz conditions the fuzzy stochastic differential equations of decreasing fuzziness possess unique local solutions. In general, one cannot expect existence of unique global solutions here, since approximate solutions which determine a solution are defined with aid of fuzzy Hukuhara differences and these latter may not exist globally. Such a problem does not appear in fuzzy stochastic differential equations of increasing fuzziness [21 –31] and there is no obstacle to obtain global solutions. We indicate this fact in this paper by formulating an assertion for solutions to non-Lipschitz fuzzy stochastic differential equations of increasing fuzziness whose form is as follows $dx (t) = f (t, x (t)) dt \oplus 〈 g (t, x (t)) d B (t) 〉, x (0) = x_{0} .$

Also we draw attention of the reader to the fact that all results established for non-Lipschitz fuzzy stochastic differential equations of decreasing fuzziness can be applied in examinations of set-valued stochastic differential equations of decreasing diameter, and those of increasing diameter, too.

The paper is organized as follows. Section 2 sum-marizes an almost all prerequisite knowledge requ-ired to understand the framework of fuzzy stochastic differential equations. Section 3 contains the studies on fuzzy stochastic differential equations with non-Lipschitz coefficients. Using the method of approximate solutions we prove existence and pathwise uniqueness of solution under non-Lipschitzian conditions. In Section 4 we formulate the results concerning set-valued stochastic differential equations with non-Lipschitz coefficients. They are some consequences of those results presented in Section 3. Hence, we omit the proofs in Section 4.

2 Preliminaries

For a convenience of the reader we collect a background material needed to study the fuzzy stochastic differential equations.

Let $K (R^{d})$ be the family of all nonempty, compact and convex subsets of $ℝ^{d}$ . In $K (R^{d})$ we consider the Hausdorff metric d_H which is defined by $d_{H} (A, B) : = max {sup_{a \in A} inf_{b \in B} ∥ a - b ∥, sup_{b \in B} inf_{a \in A} ∥ a - b ∥},$ where ∥· ∥ denotes a norm in $ℝ^{d}$ . The addition and scalar multiplication in $K (R^{d})$ are defined as usual, i.e., for $A, B \in K (ℝ^{d})$ , $b \in ℝ^{d}$ , $λ \in ℝ$

$\begin{matrix} A + B : = {a + b | a \in A, b \in B}, \\ A + {b} : = {a + b | a \in A}, \\ λ A : = {λ a | a \in A} . \end{matrix}$

Let $(Ω, A, P)$ be a complete probability space and $M (Ω, A; K (ℝ^{d}))$ denote the family of $A$ -measurable set-valued random variables (or set-valued random variables, for short) with values in $K (R^{d})$ , i.e., the mappings $F : Ω \to K (ℝ^{d})$ such that the condition ${ω \in Ω : F (ω) \cap O \neq \emptyset} \in A$ is satisfied for every open set $O \subset ℝ^{d}$ . It is known that $F : Ω \to K (ℝ^{d})$ is a set-valued random variable if, and only if, F is a $A | B_{d_{H}}$ measurable mapping (in classical sense), where $B_{d_{H}}$ denotes the Borel σ-algebra generated by the topology induced by the metric d_H in $K (R^{d})$ . A set-valued random variable $F \in M (Ω, A; K (ℝ^{d}))$ is said to be $L^{p}$ -integrally bounded, p ⩾ 1, if there exists $h \in L^{p} (Ω, A, P; ℝ)$ such that ∥a ∥ ⩽ h (ω) for any a and ω with a ∈ F (ω). It is known (see [7, 8]) that F is $L^{p}$ -integrally bounded if, and only if, ω ↦ |||F (ω) ||| is in $L^{p} (Ω, A, P; ℝ)$ , where $| | | A | | | : = sup_{a \in A} ∥ a ∥$ for $A \in K (ℝ^{d})$ . Let us denote $\begin{matrix} L^{p} (Ω, A, P; K (ℝ^{d})) : = {F \in M (Ω, A; K (ℝ^{d})) : \\ ω \mapsto | | | F (ω) | | | is in L^{p} (Ω, A, P; ℝ)} . \end{matrix}$

The variables $F, G \in L^{p} (Ω, A, P; K (ℝ^{d}))$ are considered to be identical, if F = G holds P-a.e.

Let T > 0, and denote I : = [0, T]. Let the system $(Ω, A, {A_{t}}_{t \in I}, P)$ be a complete, filtered probability space with a filtration ${A_{t}}_{t \in I}$ satisfying the usual hypotheses, i.e., ${A_{t}}_{t \in I}$ is an increasing and right continuous family of sub-σ-algebras of $A$ , and $A_{0}$ contains all P-null sets. We call $X : I \times Ω \to K (ℝ^{d})$ a set-valued stochastic process, if for every t ∈ I a mapping $X (t) : Ω \to K (ℝ^{d})$ is a set-valued random variable. We say that a set-valued stochastic process X is d_H-continuous, if almost all (with respect to the probability measure P) its paths, i.e., the mappings $X (\cdot, ω) : I \to K (ℝ^{d})$ are the d_H-continuous functions. A set-valued stochastic process X is said to be ${A_{t}}_{t \in I}$ -adapted, if for every t ∈ I the set-valued random variable $X (t) : Ω \to K (ℝ^{d})$ is $A_{t}$ -measurable. It is called measurable, if $X : I \times Ω \to K (ℝ^{d})$ is a $B (I) \otimes A$ -measurable set-valued random variable, where $B (I)$ denotes the Borel σ-algebra of subsets of I. If $X : I \times Ω \to K (ℝ^{d})$ is ${A_{t}}_{t \in I}$ -adapted and measurable, then it will be called nonanticipating. Equivalently, X is nonanticipating if, and only if, X is measurable with respect to the σ-algebra $N$ which is defined as follows $N : = {A \in B (I) \otimes A : A^{t} \in A_{t} for every t \in I},$ where A^t = {ω : (t, ω) ∈ A}. A set-valued nonanticipating stochastic process $X : I \times Ω \to K (ℝ^{d})$ is called $L^{p}$ -integrally bounded, if there exists a measurable stochastic process $h : I \times Ω \to ℝ$ such that $𝔼 (\int_{I} | h (s) |^{p} ds) < \infty$ and |||X (t, ω) ||| ⩽ h (t, ω) for a.a. (t, ω) ∈ I × Ω. By $L^{p} (I \times Ω, N; K (R^{d}))$ we denote the set of all equivalence classes (with respect to the equality γ × P-a.e., γ denotes the Lebesgue measure) of nonanticipating and $L^{p}$ -integrally bounded set-valued stochastic processes.

A concept of a fuzzy set generalizes notion of ordinary set [50]. A fuzzy set u in $ℝ^{d}$ is characterized by its membership function (denoted by u again) $u : ℝ^{d} \to [0, 1]$ and u (x) (for each $x \in ℝ^{d}$ ) is interpreted as the degree of membership of x in the fuzzy set u. For fuzzy set $u : ℝ^{d} \to [0, 1]$ one defines so-called α-levels $[u]^{α} : = {a \in ℝ^{d} | u (a) ⩾ α}$ for α ∈ (0, 1] and $[u]^{0} : = cl {a \in ℝ^{d} | u (a) > 0}$ . Let $F (R^{d})$ denote a set of fuzzy sets $u : ℝ^{d} \to [0, 1]$ such that $[u]^{α} \in K (ℝ^{d})$ for every α ∈ [0, 1] and the mapping α ↦ [u] ^α is d_H-continuous on [0, 1]. Note that the set $ℝ^{d}$ can be embedded into $F (R^{d})$ by the embedding $〈 \cdot 〉 : ℝ^{d} \to F (ℝ^{d})$ defined as follows: for $r \in ℝ^{d}$ we have 〈r〉 (x) =1 if x = r, and 〈r〉 (x) =0 if x ¬ = r. Addition u ⊕ v of fuzzy sets $u, v \in F (ℝ^{d})$ and scalar multiplication λ ⊙ u, where $λ \in ℝ$ , $u \in F (ℝ^{d})$ , can be defined levelwise (cf. [4, 10]), i.e. $[u \oplus v]^{α} = [u]^{α} + [v]^{α}, [λ ⊙ u]^{α} = λ [u]^{α} .$

Let $u, v \in F (ℝ^{d})$ . If there exists $w \in F (ℝ^{d})$ such that u = v ⊕ w then we call w the Hukuhara difference of u and v and we denote it by u ⊖ v. Note that u ⊖ v ¬ = u ⊕ (-1) v. Also u ⊖ v may not exist, but if it exists it is unique. A generalized Hausdorff metric $d_{\infty} : F (ℝ^{d}) \times F (ℝ^{d}) \to [0, \infty)$ is defined by expression $d_{\infty} (u, v) : = sup_{α \in [0, 1]} d_{H} ([u]^{α}, [v]^{α}),$ where d_H is the Hausdorff metric in $K (R^{d})$ . In fact $(F (ℝ^{d}), d_{\infty})$ is a complete metric space, and for every $u, v, w, z \in F (ℝ^{d})$ , $λ \in ℝ$ one has $\begin{matrix} d_{\infty} (u \oplus w, v \oplus w) = d_{\infty} (u, v), \\ d_{\infty} (u \oplus v, w \oplus z) ⩽ d_{\infty} (u, w) + d_{\infty} (v, z), \\ d_{\infty} (λ ⊙ u, λ ⊙ v) = | λ | d_{\infty} (u, v), \\ d_{\infty} (u ⊖ v, 〈 0 〉) = d_{\infty} (u, v), \\ d_{\infty} (u ⊖ v, u ⊖ w) = d_{\infty} (v, w), \\ d_{\infty} (u ⊖ v, w ⊖ z) ⩽ d_{\infty} (u, w) + d_{\infty} (v, z) \end{matrix}$ if the Hukuhara differences exist. For $u \in F (ℝ^{d})$ and α ∈ [0, 1] we denote an indicator of the fuzziness of [u] ^α by $Fuzz ([u]^{α}) : = diam ([u]^{a}) = sup_{a, b \in [u]^{α}} ∥ a - b ∥ .$ It is clear that Fuzz ([u] ^α)< ∞ for every $u \in F (ℝ^{d})$ and every α ∈ [0, 1]. Note that, if u is crisp then Fuzz ([u] ^α) =0 for every α ∈ [0, 1]. If Fuzz ([u] ^α) ⩽ Fuzz ([v] ^α) for every α ∈ [0, 1], we can say that v is at least as fuzzy as u is.

Let $(Ω, A, P)$ be a probability space. A mapping $x : Ω \to F (ℝ^{d})$ is said to be a fuzzy random variable (cf. [40]), if $[x]^{α} : Ω \to K (ℝ^{d})$ is an $A$ -measurable set-valued random variable for all α ∈ [0, 1], here [x] ^α (ω) : = [x (ω)] ^α. We mention that in the framework considered here, this definition is equivalent to $A | B_{d_{\infty}}$ -measurability of $x : Ω \to F (ℝ^{d})$ (see [9]). We will say that a fuzzy random variable $x : Ω \to F (ℝ^{d})$ is said to be $L^{p}$ -integrally bounded, p ⩾ 1, if ω ↦ [x (ω)] ^α belongs to $L^{p} (Ω, A, P; K (R^{d}))$ for every α ∈ [0, 1]. Let $L^{p} (Ω, A, P; F (R^{d}))$ denote the set of all the $L^{p}$ -integrally bounded fuzzy random variables, where we consider $x, y \in L^{p} (Ω, A, P; F (ℝ^{d}))$ as identical if P ([x] ^α = [y] ^α, ∀ α ∈ [0, 1]) =1. In the set $L^{2} (Ω, A, P; F (R^{d}))$ one can define a metric δ₂ by $δ_{2} (x, y) : = {(𝔼 d_{\infty}^{2} (x, y))}^{1 / 2}$ . Then the metric space $(L^{2} (Ω, A, P; F (ℝ^{d})), δ_{2})$ is complete (see [5]).

We call $x : I \times Ω \to F (ℝ^{d})$ a fuzzy stochastic process, if for every t ∈ I the mapping $x (t) : Ω \to F (ℝ^{d})$ is a fuzzy random variable. We say that a fuzzy stochastic process x is d_∞-continuous, if almost all (with respect to the probability measure P) its paths, i.e. the mappings $x (\cdot, ω) : I \to F (ℝ^{d})$ are d_∞-continuous functions. A fuzzy stochastic process x is said to be ${A_{t}}_{t \in I}$ -adapted, if for every α ∈ [0, 1] the set-valued random variable $[x (t)]^{α} : Ω \to K (ℝ^{d})$ is $A_{t}$ -measurable for all t ∈ I. It is called measurable, if $[x]^{α} : I \times Ω \to K (ℝ^{d})$ is a $B (I) \otimes A$ -measurable set-valued random variable for all α ∈ [0, 1]. If $x : I \times Ω \to F (ℝ^{d})$ is ${A_{t}}_{t \in I}$ -adapted and measurable, then it will be called nonanticipating. Equivalently, x is nonanticipating if, and only if, for every α ∈ [0, 1] the set-valued random variable [x] ^α is measurable with respect to the σ-algebra $N$ . A nonanticipating fuzzy stochastic process x is called $L^{p}$ -integrally bounded, if there exists a measurable stochastic process $h : I \times Ω \to ℝ$ such that $𝔼 \int_{I} | h (s) |^{p} ds < \infty$ and ||| [x (t, ω)] ⁰||| ⩽ h (t, ω) for a.a. (t, ω) ∈ I × Ω. By $L^{p} (I \times Ω, N; F (R^{d}))$ we denote the set of nonanticipating and $L^{p}$ -integrally bounded fuzzy stochastic processes $x : I \times Ω \to F (ℝ^{d})$ .

For convenience, we use the notation $x \overset{P . 1}{=} y$ which is an abbreviation of P (x = y ) = 1, where x, y are random elements. Also we will write $x (t) \overset{I P . 1}{=} y (t)$ instead of P (x (t) = y (t) ∀ t ∈ I ) = 1, where x, y are the stochastic processes. Similar notations will be used in the case of inequalities.

Let $(Ω, A, {A_{t}}_{t \in I}, P)$ be a complete, filtered probability space with a filtration ${A_{t}}_{t \in I}$ satisfying the usual hypotheses. Let $x \in L^{p} (I \times Ω, N; F (ℝ^{d}))$ , p ⩾ 1. For such the process x we can define (see [21 –23]) the fuzzy stochastic Lebesgue–Aumann integral which is a fuzzy random variable $Ω ∋ ω \mapsto \int_{I} x (s, ω) ds \in F (ℝ^{d}) .$

Then $\int_{0}^{t} x (s) ds$ (from now on we do not write the argument ω) is understood as ∫_I1_[0,t] (s) x (s) ds. For the fuzzy stochastic Lebesgue–Aumann integral we have the following properties (see [21 –23]).

Proposition 2.1. Let p ⩾ 1. If $x, y \in L^{p} (I \times Ω, N; F (ℝ^{d}))$ then

$I \times Ω ∋ (t, ω) \mapsto \int_{0}^{t} x (s, ω) ds \in F (ℝ^{d})$ belongs to $L^{p} (I \times Ω, N; F (R^{d}))$ ,

the fuzzy process $(t, ω) \mapsto \int_{0}^{t} x (s, ω) ds$ is d_∞-continuous,

$\begin{matrix} sup_{u \in [0, t]} & d_{\infty}^{p} (\int_{0}^{u} x (s) ds, \int_{0}^{u} y (s) ds) \\ \overset{I P . 1}{⩽} t^{p - 1} \int_{0}^{t} d_{\infty}^{p} (x (s), y (s)) ds, \end{matrix}$

for every t ∈ I it holds $\begin{matrix} 𝔼 sup_{u \in [0, t]} & d_{\infty}^{p} (\int_{0}^{u} x (s) ds, \int_{0}^{u} y (s) ds) \\ ⩽ t^{p - 1} 𝔼 \int_{0}^{t} d_{\infty}^{p} (x (s), y (s)) ds . \end{matrix}$

In the studies of the fuzzy stochastic differential equations we need also a notion of a fuzzy stochastic Itô integral which should be a fuzzy random variable. As we mentioned e.g. in [20 –22] it is not possible to define such the integral in such a fashion that it is not a crisp random variable. Hence, we consider the diffusion part of the fuzzy stochastic differential equation as the crisp stochastic Itô integral whose values are embedded into $F (R^{d})$ . A formal mathematical setting is then as follows. Let {B (t)} _t∈I be a one-dimensional ${A_{t}}_{t \in I}$ -Brownian motion defined on a complete probability space $(Ω, A, P)$ with a filtration ${A_{t}}_{t \in I}$ satisfying usual hypotheses. Let ∫_Ix (s) dB (s) denote the crisp stochastic Itô integral (see e.g. [6 , 35,39]) for $x \in L^{2} (I \times Ω, N; ℝ^{d})$ . Then by a fuzzy stochastic Itô integral we mean the fuzzy random variable of the form $Ω ∋ ω \mapsto 〈 \int_{I} x (s) dB (s) (ω) 〉 \in F (ℝ^{d}) .$

For every t ∈ I one can consider the fuzzy stochastic Itô integral $〈 \int_{0}^{t} x (s) dB (s) 〉$ . The latter integral means $〈 \int_{0}^{t} x (s) dB (s) 〉 = 〈 \int_{I} 1_{[0, t]} (s) x (s) dB (s) 〉$ .

3 Fuzzy stochastic differential equations of decreasing fuzziness

Let 0< T < ∞, I = [0, T] and let $(Ω, A, P)$ be a complete probability space with a filtration ${A_{t}}_{t \in I}$ satisfying usual hypotheses. By $B = {B (t)}_{t \in I}$ we denote an m-dimensional ${A_{t}}$ -Brownian motion defined on $(Ω, A, {A_{t}}_{t \in I}, P)$ , $m \in ℕ$ . The process $B$ is defined as follows $B = (B^{1}, B^{2}, \dots, B^{m})^{'}$ , where B¹ = {B¹ (t)} _t∈I, B² = {B² (t)} _t∈I, …, B^m = {B^m (t)} _t∈I are the independent, one-dimensional ${A_{t}}_{t \in I}$ -Brownian motions, and the symbol ^′ denotes transposition. In this part of the paper we shall consider the fuzzy stochastic differential equations of decreasing fuzziness driven by m-dimensional Brownian motion and with random coefficients. Such the equations were introduced for the first time in [32 –34] and studied with imposition of Lipschitz coefficients. These equations can be written in a symbolic differential form as:

$dx (t) \oplus [(- 1) ⊙ f (t, x (t)) dt$ (3.1) $\oplus 〈 (- 1) g (t, x (t)) d B (t) 〉] \overset{I P . 1}{=} 〈 0 〉, x (0) \overset{P . 1}{=} x_{0},$ with $f : I \times Ω \times F (ℝ^{d}) \to F (ℝ^{d})$ , $g : I \times Ω \times F (ℝ^{d}) \to ℝ^{d} \times ℝ^{m}$ , and $x_{0} : Ω \to F (ℝ^{d})$ being a fuzzy random variable. Since g = (g¹, g², …, g^m) where $g^{k} : I \times Ω \times F (ℝ^{d}) \to ℝ^{d}$ , we can rewrite (3.1) as follows $dx (t) \oplus [(- 1) ⊙ f (t, x (t)) dt$ (3.2) $\begin{matrix} \oplus & 〈 \sum_{k = 1}^{m} (- 1) g^{k} (t, x (t)) {dB}^{k} (t) 〉] \overset{I P . 1}{=} 〈 0 〉, \\ x (0) \overset{P . 1}{=} x_{0} . \end{matrix}$

If x₀, f, g^k’s were single-valued and single-defined mappings then equations of the form (3.1) would represent the classical stochastic differential equations whose solutions are $ℝ^{d}$ -valued stochastic processes. In this sense the theory of fuzzy stochastic differe-ntial equations extends the theory of classical single-valued stochastic differential equations.

In general, for the equations (3.1) we cannot expect global solutions existing on entire interval I. Instead, we should only consider local solutions. Therefore we consider $\tilde{T} \in (0, T]$ and $\tilde{I} = [0, \tilde{T}]$ . For the definition of a local solution we assume that $\tilde{T} < T$ . If $\tilde{T} = T$ , then the definition given below describes notion of the global solution to (3.1).

Definition 3.1. By a local solution to equation (3.1) we mean a fuzzy stochastic process $x : \tilde{I} \times Ω \to F (ℝ^{d})$ such that

$x \in L^{2} (\tilde{I} \times Ω, N; F (ℝ^{d}))$ ,

x is d_∞-continuous,

it holds $x (t) \overset{\tilde{I} P . 1}{=} x_{0} ⊖ [(- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds$ (3.3) $\oplus 〈 \sum_{k = 1}^{m} (- 1) \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉] .$

The second term of the right-hand side of (3.3) is the fuzzy stochastic Lebesgue integral, while the third one is the sum of classical $ℝ^{d}$ -valued stochastic Itô integrals which is embedded into $F (R^{d})$ .

Definition 3.2. A local solution $x : \tilde{I} \times Ω \to F (ℝ^{d})$ to (3.1) is said to be unique, if $d_{\infty} (x (t), y (t)) \overset{\tilde{I} P . 1}{=} 0$ , where $y : \tilde{I} \times Ω \to F (ℝ^{d})$ is any other local solution to (3.1) existing on $\tilde{I} \times Ω$ .

Remark 3.3. It is clear, reading Definition 3.1, that the initial fuzzy random variable x₀ has a direct influence on existence of solution (as well as on whether the solutions are global or local) to (3.1), since existence of Hukuhara differences in (3.3) depends straight on x₀. Hence, the existence of a solution to (3.1) with the data x₀, f, g does not mean (in general) existence of the solution to (3.1) with the data y₀, f, g where y₀ differs from x₀ on a set of positive probability.

The following property of solutions to (3.1) relates to semantics of fuzzy stochastic differential equations as those ones connected with decreasing fuzziness.

Remark 3.4. (Theorem 3.3 [34]). Let x be a local or global solution to (3.1). Then for almost all ω and for every α ∈ [0, 1] the function t ↦ Fuzz ([x (t, ω)] ^α) is nonincreasing.

In [21 –23], [27 –30] we studied the fuzzy stoch-astic differential equations of increasing fuzziness. Their symbolic form was as follows: $dx (t) \overset{I P . 1}{=} f (t, x (t)) dt \oplus 〈 g (t, x (t)) d B (t) 〉,$ (3.4) $x (0) \overset{P . 1}{=} x_{0},$

In this case there are not any obstacles to obtain global solutions $x \in L^{2} (I \times Ω, N; F (ℝ^{d}))$ to (3.4) under suitable assumptions imposed on the nonlinearities f and g. Here, the existence of global solutions can be achieved regardless of initial value x₀. This happens because the solution is defined as a fuzzy stochastic process x satisfying $\begin{matrix} x (t) & \overset{I P . 1}{=} & x_{0} \oplus \int_{0}^{t} f (s, x (s)) ds \\ \oplus & 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉, \end{matrix}$ and this does not require existence of any Hukuhara differences. In [23] we pointed out an assertion concerning the solutions to (3.4). It is connected with a geometric property of α-levels of solution’s values. This property said that values of the solution become fuzzier and fuzzier as time variable t increases.

Remark 3.5. (Theorem 3.8 [23]). If $x : I \times Ω \to F (ℝ^{d})$ is a solution to (3.4), then for almost all ω and for every α ∈ [0, 1] the function I ∋ t ↦ Fuzz ([x (t, ω)] ^α) ∈ [0, ∞) is nondecreasing.

A main goal of this section is to accomplish existence and uniqueness of solutions to the fuzzy stochastic differential equations (3.1) under conditions which are weaker than global Lipschitz and linear growth conditions used in [32 –34]. We propose to consider the following conditions:

$x_{0} \in L^{2} (Ω, A_{0}, P; F (ℝ^{d}))$ ,

the mapping $f : (I \times Ω) \times F (ℝ^{d}) \to F (ℝ^{d})$ is $N \otimes B_{d_{\infty}} | B_{d_{\infty}}$ -measurable and each g^k: $(I \times Ω) \times F (ℝ^{d}) \to ℝ^{d}$ is $N \otimes B_{d_{\infty}} | B (ℝ^{d})$ -measurable,

P-a.e. it holds: $d_{\infty}^{2} (f (t, ω, u), f (t, ω, v)) ⩽ κ (d_{\infty}^{2} (u, v))$ for every t ∈ I for every $u, v \in F (ℝ^{d})$ , $∥ g^{k} (t, ω, u) - g^{k} (t, ω, v) ∥^{2} ⩽ κ (d_{\infty}^{2} (u, v))$ for every k ∈ {1, 2, …, m} for every t ∈ I for every $u, v \in F (ℝ^{d})$ , where $κ : ℝ_{+} \to ℝ_{+}$ is a continuous, concave, nondecreasing function such that κ (0) =0, κ (u) >0 for u > 0 and $\int_{0^{+}} \frac{du}{κ (u)} = \infty$ ,

there exists a constant C > 0 such that P-a.e. it holds $\begin{matrix} d_{\infty}^{2} (f (t, ω, 〈 0 〉), 〈 0 〉) ⩽ C for every t \in I, \\ ∥ g^{k} (t, ω, 〈 0 〉) ∥^{2} ⩽ C \end{matrix}$ for every k ∈ {1, 2, …, m} and for every ∀t ∈ I,

there exists $\tilde{T} \in (0, T]$ such that the sequence of the fuzzy mappings $x_{n} : \tilde{I} \times Ω \to F (ℝ^{d})$ ( $\tilde{I} = [0, \tilde{T}]$ ) is well defined (i.e., the fuzzy Hukuhara differences do exist), where $\begin{matrix} x_{0} (t) \overset{\tilde{I} P . 1}{=} x_{0}, and for n = 1, 2, \dots \\ x_{n} (t) \overset{\tilde{I} P . 1}{=} x_{0} ⊖ [(- 1) ⊙ \int_{0}^{t} f (s, x_{n - 1} (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} (- 1) \int_{0}^{t} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉] . \end{matrix}$

The following functions κ₁, κ₂, κ₃ are known (cf. [48]) as examples of function κ appearing in (C2). Let L > 0 and let δ ∈ (0, 1) be sufficiently small. Define $\begin{matrix} κ_{1} (u) = Lu 1_{[0, \infty)} (u), \\ κ_{2} (u) = u log (u^{- 1}) 1_{[0, δ]} (u) \\ + [δ log (δ^{- 1}) + κ_{2}^{'} (δ -) (u - δ)] 1_{(δ, \infty)} (u), \\ κ_{3} (u) = u log (u^{- 1}) log log (u^{- 1}) 1_{[0, δ]} (u) \\ + [δ log (δ^{- 1}) log log (δ^{- 1}) \\ + κ_{3}^{'} (δ -) (u - δ)] 1_{(δ, \infty)} (u), \end{matrix}$ where $κ_{i}^{'} (δ -)$ (i = 2, 3) denotes left-sided derivative of κ_i at δ. Due to the form of κ₁ we see that the Lipschitz condition is a special case of the proposed non-Lipschitz condition (C2).

Before formulation of the main result of this paper we provide some useful assertions.

Lemma 3.6. (Bihari’s inequality, see e.g. Theorem 1.8.2 in it[35]). Let T > 0 and c > 0. Let $κ : ℝ_{+} \to ℝ_{+}$ be a continuous nondecreasing function such that κ (t) >0 for every t > 0. Let u (·) be a Borel measurable bounded nonnegative function on [0, T], and let v (·) be a nonnegative integrable function on [0, T]. If $u (t) ⩽ c + \int_{0}^{t} v (s) κ (u (s)) ds for every t \in [0, T],$ then $u (t) ⩽ J^{- 1} (J (c) + \int_{0}^{t} v (s) ds)$ holds for all such t ∈ [0, T] that $J (c) + \int_{0}^{t} v (s) ds \in Dom (J^{- 1})$ , where $J (r) = \int_{1}^{r} \frac{ds}{κ (s)}$ , r > 0, and J^-1 is the inverse function of J. Moreover, if c = 0 and $\int_{0 +} \frac{ds}{κ (s)} = \infty$ then u (t) =0 for every t ∈ [0, T].

Lemma 3.7. Let conditions (C0)-(C4) be satisfied. Then x_n’s described in (C4) are well defined d_∞-continuous fuzzy stochastic processes belonging to $L^{2} (\tilde{I} \times Ω, N; F (R^{d}))$ .

Proof. Since $x_{0} \in L^{2} (Ω, A_{0}, P; F (ℝ^{d}))$ and x₀ (t) $\overset{\tilde{I} P . 1}{=} x_{0}$ , we infer immediately that the initial process $x_{0} \in L^{2} (\tilde{I} \times Ω, N; F (ℝ^{d}))$ . Further assuming that $x_{n - 1} \in L^{2} (\tilde{I} \times Ω, N; F (ℝ^{d}))$ we obtain that: - due to (C1), the composition $f (\cdot, \cdot, x_{n - 1} (\cdot, \cdot)) : \tilde{I} \times Ω \to F (ℝ^{d})$ is a nonanticipating fuzzy stochastic process, - due to (C1), the composition $g^{k} (\cdot, \cdot, x_{n - 1} (\cdot, \cdot)) : \tilde{I} \times Ω \to ℝ^{d}$ is a nonanticipating $ℝ^{d}$ -valued stochastic process, k = 1, 2, …, m, - due to (C2) and (C3), $f (\cdot, \cdot, x_{n - 1} (\cdot, \cdot)) \in L^{2} (\tilde{I} \times Ω, N; F (ℝ^{d}))$ and g^k (· , · , $x_{n - 1} (\cdot, \cdot)) \in L^{2} (\tilde{I} \times Ω, N; ℝ^{d})$ , k = 1, 2, …, m, - due to Proposition 2.1(i), the fuzzy stochastic process $(t, ω) \mapsto \int_{0}^{t} f (s, ω, x_{n - 1} (s, ω)) ds$ belongs to $L^{2} (\tilde{I} \times Ω, N; F (R^{d}))$ and obviously the single-valued Itô process $(t, ω) \mapsto (\int_{0}^{t} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s)) (ω)$ belongs to $L^{2} (\tilde{I} \times Ω, N; ℝ^{d})$ , - due to Proposition 2.1(ii), the fuzzy stochastic process $(t, ω) \mapsto \int_{0}^{t} f (s, ω, x_{n - 1} (s, ω)) ds$ is d_∞-continuous, and obviously the stochastic processes $(t, ω) \mapsto (\int_{0}^{t} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s)) (ω)$ are continuous. Since $F (R^{d})$ is closed under operations ⊕ and ⊖, and the sum and Hukuhara difference of two fuzzy random variables are still fuzzy random variables, we conclude that x_n’s possess the property stated in assertion. □

Proposition 3.8. Under conditions (C0)-(C4) there exists a constant M₁ > 0 such that for every $n \in ℕ$ $𝔼 sup_{t \in \tilde{I}} d_{\infty}^{2} (x_{n} (t), 〈 0 〉) ⩽ M_{1} .$

Proof. Denote $e_{n} (t) = 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n} (u), 〈 0 〉)$ for $n \in ℕ$ and $t \in \tilde{I}$ . Then $\begin{matrix} e_{n} (t) = \\ 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{0} ⊖ ((- 1) ⊙ \int_{0}^{u} f (s, x_{n - 1} (s)) ds \oplus \\ 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{u} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉), 〈 0 〉) \\ ⩽ (m + 2) [𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) + \\ 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (\int_{0}^{u} f (s, x_{n - 1} (s)) ds, 〈 0 〉) \\ + \sum_{k = 1}^{m} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (〈 \int_{0}^{u} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉, 〈 0 〉)] . \end{matrix}$

Applying Proposition 2.1(iv) and the Doob inequality, we get $\begin{matrix} e_{n} (t) ⩽ (m + 2) [𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) \\ + 2 t 𝔼 \int_{0}^{t} (d_{\infty}^{2} (f (s, x_{n - 1} (s)), f (s, 〈 0 〉)) \\ + d_{\infty}^{2} (f (s, 〈 0 〉), 〈 0 〉)) ds \\ + 8 \sum_{k = 1}^{m} 𝔼 \int_{0}^{t} (d_{\infty}^{2} (〈 g^{k} (s, x_{n - 1} (s)) 〉, 〈 g^{k} (s, 〈 0 〉) 〉) \\ + d_{\infty}^{2} (〈 g^{k} (s, 〈 0 〉) 〉, 〈 0 〉)) ds] . \end{matrix}$

Due to assumptions (C2) and (C3), we have $\begin{matrix} e_{n} (t) ⩽ (m + 2) [𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) + 2 Ct (t + 4 m) \\ + 2 (t + 4 m) 𝔼 \int_{0}^{t} κ (d_{\infty}^{2} (x_{n - 1} (s), 〈 0 〉)) ds] . \end{matrix}$

Let us observe that for the function κ (·) described in (C2), we can find positive constants a, b such that κ (u) ⩽ a + bu for u ⩾ 0. Then $\begin{matrix} e_{n} (t) ⩽ (m + 2) [𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) + 2 t (C + a) (t + 4 m) \\ + 2 b (t + 4 m) 𝔼 \int_{0}^{t} d_{\infty}^{2} (x_{n - 1} (s), 〈 0 〉) ds] \\ ⩽ (m + 2) [𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) + 2 t (C + a) (t + 4 m) \\ + 2 b (t + 4 m) \int_{0}^{t} 𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n - 1} (u), 〈 0 〉) ds] . \end{matrix}$

Hence $e_{n} (t) ⩽ C_{1} + C_{2} \int_{0}^{t} e_{n - 1} (s) ds$ , and consequently $max_{1 ⩽ n ⩽ k} e_{n} (t) ⩽ C_{1} + C_{2} \int_{0}^{t} max_{1 ⩽ n ⩽ k} e_{n - 1} (s) ds$ for $k \in ℕ$ , where $C_{1} = 2 \tilde{T} (C + a) (\tilde{T} + 4 m)$ and $C_{2} = 2 b (\tilde{T} + 4 m)$ . Since $max_{1 ⩽ n ⩽ k} e_{n - 1} (t) ⩽ 𝔼 d_{\infty}^{2} (x_{0},$ $〈 0 〉) + max_{1 ⩽ n ⩽ k} e_{n} (t)$ , we have $\begin{matrix} max_{1 ⩽ n ⩽ k} e_{n} (t) & ⩽ & C_{1} + C_{2} \tilde{T} 𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉) \\ + C_{2} \int_{0}^{t} max_{1 ⩽ n ⩽ k} e_{n} (s) ds . \end{matrix}$

Then, by Gronwall’s inequality, $max_{1 ⩽ n ⩽ k} e_{n} (t) ⩽ (C_{1} + C_{2} \tilde{T} 𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉)) exp {C_{2} t}$ . Hence we infer that for every $n \in ℕ$ it holds $e_{n} (\tilde{T}) ⩽ (C_{1} + C_{2} \tilde{T} 𝔼 d_{\infty}^{2} (x_{0}, 〈 0 〉)) exp {C_{2} \tilde{T}}, t \in \tilde{I},$ which ends the proof. □

Proposition 3.9. Under conditions (C0)-(C4) there exists a constant M₂ > 0 such that for every $n, ℓ \in ℕ$ and every $t \in \tilde{I}$ it holds $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) ⩽ \\ M_{2} \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n + ℓ - 1} (u), x_{n - 1} (u))) ds . \end{matrix}$

Proof. Let us fix $n, ℓ \in ℕ$ and $t \in \tilde{I}$ . Then, using Proposition 2.1(iv), Doob’s inequality and (C2) we obtain $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ = 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{0} ⊖ ((- 1) ⊙ \int_{0}^{u} f (s, x_{n + ℓ - 1} (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{u} g^{k} (s, x_{n + ℓ - 1} (s)) {dB}^{k} (s) 〉), \\ x_{0} ⊖ ((- 1) ⊙ \int_{0}^{u} f (s, x_{n - 1} (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{u} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉)) . \end{matrix}$

Hence $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ ⩽ & (m + 1) [t 𝔼 \int_{0}^{t} d_{\infty}^{2} (f (s, x_{n + ℓ - 1} (s)), \\ f (s, x_{n - 1} (s)) ds \\ + 4 \sum_{k = 1}^{m} 𝔼 \int_{0}^{t} ∥ g^{k} (s, x_{n + ℓ - 1} (s)) \\ - g^{k} (s, x_{n - 1} (s)) ∥^{2} ds] \\ ⩽ & (m + 1) (t + 4 m) \int_{0}^{t} 𝔼 κ (d_{\infty}^{2} (x_{n + ℓ - 1} (s), \\ x_{n - 1} (s))) ds . \end{matrix}$

By Jensen’s inequality, we have further $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ ⩽ (m + 1) (t + 4 m) \int_{0}^{t} κ (𝔼 d_{\infty}^{2} (x_{n + ℓ - 1} (s), \\ x_{n - 1} (s))) ds \\ ⩽ (m + 1) (\tilde{T} + 4 m) \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n + ℓ - 1} (u), \\ x_{n - 1} (u))) ds \end{matrix}$ for every $t \in \tilde{I}$ . □

Proposition 3.10. Under conditions (C0)-(C4) there exists a constant M₃ > 0 such that for every $n, ℓ \in ℕ$ and every $t \in \tilde{I}$ it holds $𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) ⩽ M_{3} t .$

Proof. Let us fix $n, ℓ \in ℕ$ . Applying Proposition 3.8, assumption that κ is nondecreasing and Proposition 3.9 we obtain $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ ⩽ & M_{2} \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n + ℓ - 1} (u), x_{n - 1} (u))) ds \end{matrix}$ and further $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ ⩽ & M_{2} \int_{0}^{t} κ (𝔼 sup_{u \in I} d_{\infty}^{2} (x_{n + ℓ - 1} (u), x_{n - 1} (u))) ds \end{matrix}$

Thus for $t \in \tilde{I}$ $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n + ℓ} (u), x_{n} (u)) \\ ⩽ & M_{2} κ (2 𝔼 sup_{u \in I} d_{\infty}^{2} (x_{n + ℓ - 1} (u), 〈 0 〉) \\ + 2 𝔼 sup_{u \in I} d_{\infty}^{2} (x_{n - 1} (u), 〈 0 〉)) t ⩽ M_{2} κ (4 M_{1}) t . \end{matrix}$ □

Now we are in a position to prove our main result.

Theorem 3.11. Assume that conditions (C0)-(C4) are satisfied. Then equation (3.1) has a unique, possibly local solution.

Proof. To prove existence of the solution to (3.1) we shall apply the successive approximations defined in condition (C4). Due to Lemma 3.7 the nth approximation x_n is a well-defined d_∞-continuous fuzzy stochastic process for every $n \in ℕ$ .

Due to Proposition 3.9 we have: there exists a constant M₂ > 0 such that for every $n, ℓ \in ℕ$ and every $t \in \tilde{I}$ it holds $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n} (u), x_{ℓ} (u)) \\ ⩽ & M_{2} \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n - 1} (u), x_{ℓ - 1} (u))) ds . \end{matrix}$

Integrating both sides and applying Jensen’s inequality, we obtain (for every $τ \in \tilde{I}$ ) $\begin{matrix} \int_{0}^{τ} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n} (u), x_{ℓ} (u)) dt \\ ⩽ M_{2} \int_{0}^{τ} \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n - 1} (u), x_{ℓ - 1} (u))) dsdt \\ ⩽ M_{2} τ \int_{0}^{τ} κ (\frac{1}{t} \int_{0}^{t} 𝔼 sup_{u \in [0, s]} d_{\infty}^{2} (x_{n - 1} (u), x_{ℓ - 1} (u)) ds) dt . \end{matrix}$

Hence $i_{n, ℓ} (τ) ⩽ M_{2} \int_{0}^{τ} κ (i_{n - 1, ℓ - 1} (t)) dt$ , where $i_{n, ℓ} (τ) = \frac{1}{τ} \int_{0}^{τ} 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{n} (u), x_{ℓ} (u)) dt$ . Note that by Proposition 3.8 it holds $sup_{τ \in \tilde{I}} sup_{n, ℓ \in ℕ} i_{n, ℓ} (τ) < \infty$ . Thus, denoting $i (τ) = \underset{n, ℓ \to \infty}{lim sup} i_{n, ℓ} (τ)$ and applying Fatou’s lemma, we get $i (τ) ⩽ M_{2} \int_{0}^{τ} κ (i (t)) dt$ for $τ \in \tilde{I}$ , which implies that i (τ) =0 for every $τ \in \tilde{I}$ . Further it allows us to claim that $𝔼 d_{\infty}^{2} (x_{n} (t), x_{ℓ} (t)) \to 0$ as n, ℓ → ∞ for every $t \in \tilde{I}$ . Due to completeness of the metric space $(L^{2} (Ω, A_{t}, P; F (ℝ^{d})), δ_{2})$ , we infer that for every $t \in \tilde{I}$ there exists $x_{t} \in L^{2} (Ω, A_{t}, P; F (ℝ^{d}))$ such that $𝔼 d_{\infty}^{2} (x_{n} (t), x_{t}) ⟶ 0 as n \to \infty$ . Hence, by putting x (t, ω) = x_t (ω) we can define a fuzzy stochastic process $x : \tilde{I} \times Ω \to K (ℝ^{d})$ which is ${A_{t}}$ -adapted.

In the sequel, we shall show that process x is a desired solution to (3.1). The fact $lim_{n, ℓ \to \infty} i_{n, ℓ} (τ) = 0$ for $τ \in \tilde{I}$ implies $𝔼 sup_{u \in \tilde{I}} d_{\infty}^{2} (x_{n} (u), x_{ℓ} (u)) ⟶ 0$ as n, ℓ → ∞. Now using Chebyshev’s inequality we obtain: for every ɛ > 0 $P (sup_{u \in \tilde{I}} d_{\infty}^{2} (x_{n} (u), x_{ℓ} (u)) > ɛ) ⟶ 0$ as n, ℓ → ∞. Thus we infer that there exists a subsequence {x_{n
_k} (· , ·)} of the sequence {x_n (· , ·)} such that $sup_{u \in \tilde{I}} d_{\infty} (x_{n_{k}} (u), x (u)) \overset{P . 1}{⟶} 0$ as k→ ∞. Therefore process x is d_∞-continuous and consequently $B (\tilde{I}) \otimes A$ -measurable. Since x is also ${A_{t}}$ -adapted, we obtain that x is $N$ -measurable. Now, having that $x (t) \in L^{2} (Ω, A, P; F (ℝ^{d}))$ for every $t \in \tilde{I}$ , we can write $𝔼 \int_{\tilde{I}} d_{\infty}^{2} (x (t), 〈 0 〉) dt ⩽ \tilde{T} sup_{t \in \tilde{I}} 𝔼 d_{\infty}^{2} (x (t), 〈 0 〉) < \infty$ which implies that $x \in L^{2} (\tilde{I} \times Ω, N; F (ℝ^{d}))$ . What is left is to prove that x is a solution, i.e., $\begin{matrix} d_{\infty} (x (t), x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉)) \overset{\tilde{I} P . 1}{=} 0 . \end{matrix}$

To this end let us note that for every $t \in \tilde{I}$ $\begin{matrix} 𝔼 d_{\infty}^{2} (x (t), x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉)) \\ ⩽ 2 𝔼 d_{\infty}^{2} (x_{n} (t), x (t)) \\ + 2 𝔼 d_{\infty}^{2} (\int_{0}^{t} f (s, x_{n - 1} (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉, \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉) \end{matrix}$

Note that the first summand above converges to zero. It remains to show that the second term converges to zero, too. Observe that $\begin{matrix} 𝔼 d_{\infty}^{2} (\int_{0}^{t} f (s, x_{n - 1} (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) 〉, \\ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉) \\ ⩽ (m + 1) [𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (\int_{0}^{u} f (s, x_{n - 1} (s)) ds, \\ \int_{0}^{u} f (s, x (s)) ds) \\ + \sum_{k = 1}^{m} 𝔼 sup_{u \in [0, t]} ∥ \int_{0}^{u} g^{k} (s, x_{n - 1} (s)) {dB}^{k} (s) \\ - \int_{0}^{u} g^{k} (s, x (s)) {dB}^{k} (s) ∥^{2}] \\ ⩽ (m + 1) [t 𝔼 \int_{0}^{t} d_{\infty}^{2} (f (s, x_{n - 1} (s)), f (s, x (s))) ds \\ + 4 \sum_{k = 1}^{m} 𝔼 \int_{0}^{t} ∥ g^{k} (s, x_{n - 1} (s)) - g^{k} (s, x (s)) ∥^{2} ds] \\ ⩽ (m + 1) (t + 4 m) 𝔼 \int_{0}^{t} κ (d_{\infty}^{2} (x_{n - 1} (s), x (s))) ds \\ ⩽ (m + 1) (t + 4 m) \int_{0}^{t} κ (𝔼 d_{\infty}^{2} (x_{n - 1} (s), x (s))) ds \\ ⩽ (m + 1) (\tilde{T} + 4 m) \tilde{T} sup_{s \in \tilde{I}} κ (𝔼 d_{\infty}^{2} (x_{n - 1} (s), x (s))) . \end{matrix}$

Since $sup_{s \in \tilde{I}} κ (𝔼 d_{\infty}^{2} (x_{n - 1} (s), x (s))) ⟶ 0$ as n→ ∞, we infer that for every $t \in \tilde{I}$ $\begin{matrix} 𝔼 d_{\infty}^{2} (x (t), x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉)) = 0 \end{matrix}$ and consequently for every $t \in \tilde{I}$ $\begin{matrix} d_{\infty} (x (t), x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉)) \overset{P . 1}{=} 0 . \end{matrix}$

Remembering that x is d_∞-continuous we obtain

$\begin{matrix} d_{\infty} (x (t), x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉)) \overset{\tilde{I} P . 1}{=} 0 . \end{matrix}$

Finally, we shall show the uniqueness of the local solution x. Let x, y denote two local solutions to (3.1) existing on the common interval $\tilde{I}$ . Define $j : \tilde{I} \to [0, \infty)$ as $j (t) : = 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x (u), y (u))$ for $t \in \tilde{I}$ . Then

$\begin{matrix} j (t) = 𝔼 sup_{u \in [0, t]} d_{\infty}^{2} (x_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, x (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, x (s)) {dB}^{k} (s) 〉), \\ y_{0} ⊖ ((- 1) ⊙ \int_{0}^{t} f (s, y (s)) ds \\ \oplus 〈 (- 1) \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, y (s)) {dB}^{k} (s) 〉)) \\ ⩽ (m + 1) [t 𝔼 \int_{0}^{t} d_{\infty}^{2} (f (s, x (s)), f (s, y (s))) ds \\ + 4 \sum_{k = 1}^{m} 𝔼 \int_{0}^{t} ∥ g^{k} (s, x (s)) - g^{k} (s, y (s)) ∥^{2} ds] \\ ⩽ (m + 1) (t + 4 m) \int_{0}^{t} 𝔼 κ (d_{\infty}^{2} (x (s), y (s))) ds \\ ⩽ (m + 1) (t + 4 m) \int_{0}^{t} κ (𝔼 d_{\infty}^{2} (x (s), y (s))) ds \\ ⩽ (m + 1) (\tilde{T} + 4 m) \int_{0}^{t} κ (j (s)) ds . \end{matrix}$

Hence, using Lemma 3.6, we obtain j (t) =0 for $t \in \tilde{I}$ . Therefore $𝔼 sup_{u \in \tilde{I}} d_{\infty}^{2} (x (u), y (u)) = 0$ which implies that $sup_{u \in \tilde{I}} d_{\infty} (x (u), y (u)) \overset{P . 1}{=} 0$ . Thus uniq-ueness of solution follows. □

It is worth mentioning that using non-Lipschitz condition (C2) it is also possible to obtain existence of a unique global solution to equation (3.4). In this case the proof relies on application of the following succesive approximations sequence $\begin{matrix} y_{0} (t) & \overset{I P . 1}{=} & x_{0}, and for n = 1, 2, \dots \\ y_{n} (t) & \overset{I P . 1}{=} & x_{0} \oplus \int_{0}^{t} f (s, y_{n - 1} (s)) ds \\ \oplus 〈 \sum_{k = 1}^{m} \int_{0}^{t} g^{k} (s, y_{n - 1} (s)) {dB}^{k} (s) 〉, \end{matrix}$ which should converge to the exact solution. As it is easily seen, now the condition (C4) is no more applied and a global solution is reachable, since no Hukuhara difference is involved in y_n’s. In fact, proceeding similarly as above we can arrive to the following assertion.

Proposition 3.12. Let the conditions (C0)-(C3) be satisfied. Then there exists a unique global solution to equation (3.4).

4 Application to set-valued stochastic differential equations of decreasing diameter

In this part of the paper we present some considerations of set-valued stochastic differential equations driven by m-dimensional Brownian motion and with random and non-Lipschitz coefficients. They are parallel to those established in Section 3 for fuzzy stochastic differential equations. Also all the derivations are similar to those contained in preceding section. Therefore we do not include the proofs.

A symbolic form of set-valued stochastic differential equations of decreasing diameter reads:

$dX (t) + [(- 1) F (t, X (t)) dt$ (4.1) $+ {(- 1) G (t, X (t)) d B (t)} \overset{I P . 1}{=} {0}, X (0) \overset{P . 1}{=} X_{0},$ where $F : I \times Ω \times K (ℝ^{d}) \to K (ℝ^{d})$ , G:I× Ω × $K (ℝ^{d}) \to ℝ^{d} \times ℝ^{m}$ and $X_{0} : Ω \to K (ℝ^{d})$ is a set-valued random variable. Since G = (G¹, G², …, G^m) where $G^{k} : I \times Ω \times K (ℝ^{d}) \to ℝ^{d}$ (k = 1, 2, …, m), we can rewrite (4.1) as follows $dX (t) + [(- 1) F (t, X (t)) dt$ (4.2) $\begin{matrix} + {\sum_{k = 1}^{m} (- 1) G^{k} (t, X (t)) {dB}^{k} (t)}] \overset{I P . 1}{=} {0}, \\ X (0) \overset{P . 1}{=} X_{0} . \end{matrix}$

Definition 4.1.By a local solution (existing on interval $\tilde{I} = [0, \tilde{T}] \subset I$ ) to equation (4.1) we mean a set-valued stochastic process $X : \tilde{I} \times Ω \to K (ℝ^{d})$ such that

$X \in L^{2} (\tilde{I} \times Ω, N; K (ℝ^{d}))$ ,

X is d_H-continuous,

it holds $X (t) \overset{\tilde{I} P . 1}{=} X_{0} ⊖ [(- 1) \int_{0}^{t} F (s, X (s)) ds$ (4.3) $+ {\sum_{k = 1}^{m} (- 1) \int_{0}^{t} G^{k} (s, X (s)) {dB}^{k} (s)}] .$

The right-hand side of (refreprezentacjaf) is understood in the meaning described earlier, i.e. the second term is the set-valued stochastic Lebesgue integral, while the third one is a sum of the

ℝ^{d}

-valued stochastic Itô integrals.

Definition 4.2. A local solution $X : \tilde{I} \times Ω \to K (ℝ^{d})$ to the equation (4.1) is said to be unique, if it holds $d_{H} (X (t), Y (t)) \overset{\tilde{I} P . 1}{=} 0$ , where $Y : \tilde{I} \times Ω \to K (ℝ^{d})$ is any other local solution to (4.1).

As a consequence of Remark 3.4 we can write the following observation.

Remark 4.3. If X is a local (or global) solution to (4.1) then for a.a. ω the function t ↦ diam (X (t, ω)) is nonincreasing.

In the sequel we shall write down the conditions imposed on the coefficients of the equation (4.1). They are as follows:

$X_{0} \in L^{2} (Ω, A_{0}, P; K (ℝ^{d}))$ ,

the mapping $F : (I \times Ω) \times K (ℝ^{d}) \to K (ℝ^{d})$ is $N \otimes B_{d_{H}} | B_{d_{H}}$ -measurable and G^k: (I× $Ω) \times K (ℝ^{d}) \to ℝ^{d}$ is $N \otimes B_{d_{H}} | B (ℝ^{d})$ -meas-urable,

P-a.e. it holds: $\begin{matrix} d_{H}^{2} (F (t, ω, A), F (t, ω, B)) ⩽ κ (d_{H}^{2} (A, B)) \\ for every t \in I and for any A, B \in K (ℝ^{d}), \\ ∥ G^{k} (t, ω, A) - G^{k} (t, ω, B) ∥^{2} ⩽ κ (d_{H}^{2} (A, B)) \\ for every k \in {1, 2, \dots, m} for every t \in I \\ and for any A, B \in K (ℝ^{d}), \end{matrix}$ where $κ : ℝ_{+} \to ℝ_{+}$ is like in the condition (C2) described in preceding section,

there exists a constant C > 0 such that P-a.e.: $\begin{matrix} d_{H}^{2} (F (t, ω, {0}), 〈 0 〉) ⩽ C for every t \in I, \\ ∥ G^{k} (t, ω, {0}) ∥^{2} ⩽ C for every k \in \\ {1, 2, \dots, m} and for every t \in I, \end{matrix}$

there exists $\tilde{T} \in (0, T]$ such that the sequ-ence of set-valued mappings $X_{n} : \tilde{I} \times Ω \to K (ℝ^{d})$ , $\tilde{I} = [0, \tilde{T}]$ , is well defined (the Hukuhara differences do exist), where $\begin{matrix} X_{0} (t) \overset{\tilde{I} P . 1}{=} X_{0}, and for n = 1, 2, \dots \\ X_{n} (t) \overset{\tilde{I} P . 1}{=} X_{0} ⊖ [(- 1) \int_{0}^{t} F (s, X_{n - 1} (s)) ds \\ + {\sum_{k = 1}^{m} (- 1) \int_{0}^{t} G^{k} (s, X_{n - 1} (s)) {dB}^{k} (s)}] . \end{matrix}$

Under conditions (c0)-(c4), proceeding similarly like in Section 3 we obtain the following properties for {X_n}.

X_n’s are d_H-continuous set-valued stocha-breakstic processes belonging to $L^{2} (\tilde{I} \times Ω, N; K (R^{d}))$ ,

there exists a constant M₁ > 0 such that $𝔼 sup_{t \in \tilde{I}} d_{H}^{2} (X_{n} (t), {0}) ⩽ M_{1}$ for every n,

there exists a constant M₂ > 0 such that for every $n, ℓ \in ℕ$ and every $t \in \tilde{I}$ it holds $\begin{matrix} 𝔼 sup_{u \in [0, t]} d_{H}^{2} (X_{n + ℓ} (u), X_{n} (u)) ⩽ M_{2} \\ \times \int_{0}^{t} κ (𝔼 sup_{u \in [0, s]} d_{H}^{2} (X_{n + ℓ - 1} (u), X_{n - 1} (u))) ds, \end{matrix}$

there exists a constant M₃ > 0 such that for every $n, ℓ \in ℕ$ and every $t \in \tilde{I}$ it holds $𝔼 sup_{u \in [0, t]} d_{H}^{2} (X_{n + ℓ} (u), X_{n} (u)) ⩽ M_{3} t .$

These properties allows us to proceed similarly as in the proof of Theorem 3.11 to obtain the existence and uniqueness of local solution to non-Lipschitz set-valued stochastic differential equations of decreasing diameter. In fact we obtain the following assertion.

Corollary 4.4. Let $X_{0} : Ω \to K (ℝ^{d})$ satisfy (c0) and let $F : I \times Ω \times K (ℝ^{d}) \to K (ℝ^{d})$ , $G^{k} : I \times Ω \times K (ℝ^{d}) \to ℝ^{d}$ , k = 1, 2, …, m, satisfy (c1)-(c4). Then equation (4.1) has a unique, possibly local solution.

Finally, we state an existence and uniqueness theorem for non-Lipschitz set-valued stochastic differential equations of increasing diameter $dX (t) \overset{I P . 1}{=} F (t, X (t)) dt$ (4.4) $+ {G (t, X (t)) d B (t)}, X (0) \overset{P . 1}{=} X_{0} .$

Corollary 4.5. Let $X_{0} : Ω \to K (ℝ^{d})$ satisfy (c0) and let $F : I \times Ω \times K (ℝ^{d}) \to K (ℝ^{d})$ , $G^{k} : I \times Ω \times K (ℝ^{d}) \to ℝ^{d}$ , k = 1, 2, …, m, satisfy (c1)-(c3). Then equation (4.1) possesses a unique global solution.

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