Properties of fuzzy Ito integrals with respect to fuzzy Brownian motion

Abstract

Integral of a stochastic process with respect to Brownian motion is called Ito integral. Here the stochastic process and Brownian motion are random as well as fuzzy. Hence the Ito integral is fuzzy Ito integral. This paper deals with the properties of fuzzy Ito integral for simple adapted process with respect to fuzzy Brownian motion. The quadratic variance and covariance of FII are discussed. The concept of fuzzy simple adapted process, fuzzy martingale, fuzzy functions are used to derive the properties of fuzzy Ito integrals.

Keywords

Fuzzy Ito integral(FII)fuzzy Brownian motion(FBM)fuzzy simple adapted process(FSAP)quadratic variance and covariance of fuzzy Ito integral fuzzy martingale(FM)

List of abbreviation:-

FBM

Fuzzy Brownian Motion

FII

Fuzzy Ito Integral

Stochastic Process

FRV

Fuzzy Random Variable

Gaussian Process

FSP

Fuzzy Stochastic Process

RAP

Regular Adapted Process

Fuzzy Number

Fuzzy Martingale

Brownian Motion

FSAP

Fuzzy Simple Adapted Process

FSV

Fuzzy Set Valued

SAP

Simple Adapted Process

FRAP

Fuzzy Regular Adapted Process

1 Introduction

Stochastic processes(SP) which depends on time is introduced by Joseph Doob. It takes a major role in many areas, such as physics, chemistry, biology, microeconomics, finance etc.

Brownian motion (BM) is an example of SP, which describe the position of a particle at a time t i.e. B(t), t∈ [0,T]. An Ito integral is defined as the integral value of SP with respect to BM which takes the value $\int_{0}^{T} X (t) dB (t)$ , t∈[0,T].

Fuzzy set theory was introduced by Zadeh [1]. He proposed a fuzzy set, whose membership grade ranges between zero and one. Fuzzy random variables(FRV) are random variables whose parameters are not exactly known, which can be estimated from the confidence interval. So these parameters can be treated as fuzzy numbers(FN). FN was introduced by Kwakernaak [2]. He discussed the expectation of FRV, characteristic functions of fuzzy events, conditional expectation, and some results related to independent FRVs.

Puri and Ralescu [3] developed the concept of FRV, which plays a tool for representing the relationship between the outcomes of a random experiment in exact data. Buckley [26] defined FRV in a different way which is more convenient for computational point of view. Ranarahu et al. [19, 20] discussed the multi objective bilinear programming problem using the fuzzy programming technique. Sahoo and Dash [21] discussed a single period inventory fuzzy probabilistic model by using the fuzzy chance constraints. Rakesh et al. [25] discussed the comparative investigation of modeling and optimizing creation cost for smart production by using α-cut technique and quadrilateral fuzzy number. After that many researchers have used FRV in different branches of mathematics. The concept of fuzzy Gaussian process and convergence theorem for fuzzy martingale(FM) have been introduced by Puri and Ralescu [4, 5]. Further Stojakovic [17] discussed FM for a simple form of fuzzy process. Fuzzy expectations and FM have been discussed by Li and Ogura [6 –9]. Li and Guan [10] discussed GP and BM in terms of Ito integral(II). Li and Zheng [24] proved the existence and uniqueness of stochastic integral equations.

Kim and Kim [11, 12] developed the fuzzy set valued SP with respect to BM. Further in their the discussion they have concluded that fuzzy set valued(FSV) diffusion process with respect to FBM contained in fuzzy stochastic integral. FSV Ito integral(FII) was developed by Li and Ren [13]. Malinoskis [15, 16] developed fuzzy stochastic integral equation and the existence and uniqueness of random fuzzy differential equation. Seya et al. [14] discussed FII is driven by FBM and defined metric between two fuzzy numbers. Bandyopadhyay and Kar [22] developed the Type-1 and Type-2 fuzzy stochastic and fuzzy partial stochastic differential the equations by using fuzzy parameters. Dash et al. [23] discussed on fuzzy stochastic option pricing problem.

In the present study, some properties of FII are discussed. Many researchers have discussed the BM as a function of crisp time but in general time may not be fixed or crisp rather some uncertainty will be there. Here uncertain times is considered as fuzzy time.

The fuzzy Ito integral(FII) with respect to fuzzy Brownian motion(FBM) is a natural tool in the study of theory of fuzzy stochastic differential equations driven by a fuzzy Brownian motion. FII of fuzzy simple function with respect to FBM is a fuzzy number.

The organized of the said paper is as follows: In section-2 contains prerequisites. The characteristic of FII of a simple adapted process with respect to FBM is discussed in section-3. In section-4 the quadratic variance and covariance of fuzzy Ito integral are discussed. Finally the conclusion and concluding remarks are made in section-5 and following supporting reference.

2 Basic Preliminaries

2.1 Fuzzy number(FN) [14]

It follows (14) “Let X= I R be a universal set, the set of fuzzy numbers denoted as F (I R) and defined as $F (I R) = {\tilde{u} : I R ⟼ [0, 1]}$ where $\tilde{u}$ satisfies

(i) $\tilde{u}$ is normal,

(ii) $\tilde{u}$ is convex,

(iii) $\tilde{u}$ is upper semi continuos.

and $\tilde{u} \in F (I R)$ is a FN.”

2.2 α- cut [14]

It follows (14) “Let $\tilde{u} \in F (I R)$ be a fuzzy number, the α-cut of $\tilde{u}$ , for every α ∈ [0, 1] , the set $[\tilde{u}]_{α} = {x \in I R : \tilde{u} (x) \geq α} = [{\tilde{u}}_{α}^{L}, {\tilde{u}}_{α}^{U}]$ $where [\tilde{u}]_{α}^{L} = inf_{x \in I R} {x \in [\tilde{u}]_{α}}$ $and [\tilde{u}]_{α}^{U} = sup_{x \in I R} {x \in [\tilde{u}]_{α}} .$ the support of $\tilde{u}$ is given by $[\tilde{u}]_{0} = Support (\tilde{u}) = {x \in I R : \tilde{u} (x) > 0} .^{″}$

2.3 Fuzzy martingale(FM) [14]

It follows (14) “The sequence ${{\tilde{X}}_{n}, {\tilde{F}}_{n}}_{n}$ of fuzzy random variables and σ- algebra is a fuzzy martingale if for each n ≥ 1,

(i) ${\tilde{X}}_{n} is {\tilde{F}}_{n}$ measurable and $E ∥ \sup {\tilde{X}}_{n} ∥ < \infty .$ (ii) $E ({\tilde{X}}_{n + 1} ∣ {\tilde{F}}_{n}) = {\tilde{X}}_{n}$ .

If properties (ii) is replaced by $({ii}^{'}) E ({\tilde{X}}_{n + 1} ∣ {\tilde{F}}_{n}) \geq {\tilde{X}}_{n},$ $(E ({\tilde{X}}_{n + 1} ∣ {\tilde{F}}_{n}) \leq {\tilde{X}}_{n}),$ then ${{\tilde{X}}_{n}, {\tilde{F}}_{n}}_{n}$ is called fuzzy sub-martingale (super-martingale) respectively”.

2.4 Bounded fuzzy function [14]

It follows (14) “A fuzzy function $\tilde{G} : I R \to F_{c} (I R)$ is bounded if and only if for every α lies between 0 and 1, for i ∈ {L, U}, ${\tilde{G}}_{α}^{i}$ is a bounded fuzzy function(FN), where F_c (I R) denote the family of all convex subset of I R″ .

2.5 Fuzzy stochastic process(FSP) [14]

It follows (14) “A fuzzy family ${\tilde{X} (t), t \in [0, T], 0 < T < \infty}$ is called a fuzzy stochastic process on probability space (Ω, A, P) if and only if ∀ α ∈ [0, 1] , the process ${\tilde{X}}_{α} (t, ω) = [{\tilde{X}}_{α}^{L} (t, ω), {\tilde{X}}_{α}^{U} (t, ω)]$ (1) is an interval valued stochastic process(SP) on (Ω, A, P) and ${\tilde{X}}_{α} (t, ω) = ⋃_{α \in [0, 1]} {\tilde{X}}_{α} (t, ω)^{″} .$

2.6 The interval valued BM [14]

It follows (14) “A family ${\bar{B} (t), t \in [0, T], 0 < T < \infty}$ is an interval valued Brownian motion on probability space (Ω, A, P) if and only if ${\bar{B}}_{α} (t, ω) = [{\bar{B}}_{α}^{L} (t, ω), {\bar{B}}_{α}^{U} (t, ω)]$ (2) where ${\bar{B}}_{α}^{L} (t, ω) and {\bar{B}}_{α}^{U} (t, ω)$ are Brownian motions such that $P [{\bar{B}}_{α}^{L} (t, ω) \leq {\bar{B}}_{α}^{U} (t, ω)] = 1^{″}$ (3)

2.7 Fuzzy Brownian Motion[FBM] [14]

It follows (14) “A Fuzzy stochastic process ${\tilde{B} (t) t \in [0, T], 0 < T < \infty}$ is called a fuzzy Brownian motion on probability space (Ω, A, P) if and only if ∀ α ∈ [0, 1], the process ${\tilde{B}}_{α} (t, ω) = [{\tilde{B}}_{α}^{L} (t, ω), {\tilde{B}}_{α}^{U} (t, ω)]$ (4) is an interval Brownian motion on (Ω, A, P) and ${\tilde{B}}_{α} (t, ω) = ⋃_{α \in [0, 1]} {\tilde{B}}_{α} (t, ω)^{″} .$

2.8 fuzzy Ito isometry [14]

It follows (14) “Let $\tilde{φ} (t, ω)$ be a bounded fuzzy simple function, ∀ α ∈ [0, 1] ,

$\begin{matrix} E (\int_{S}^{T} {\tilde{φ}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω))^{2} \\ = \int_{S}^{T} E ({\tilde{φ}}_{α}^{2} (t, ω)) {dt}^{″} \end{matrix}$ (5)

2.9 Simple adapted process(SAP) [18]

It follows (18) “A process X = {X (t) , 0 ≤ t ≤ T} is called a simple adapted process if there exist time 0 = t₀ < t₁ < . . . . < t_n = T and random variables ξ₀, ξ₁, . . . . ξ_n-1, such that ξ₀ is a constant, ξ_i is F_{t
_i} measurable which depends upon the values of B (t) for t≤t_i but not the values of B(t) for t > t_i, F_{t
_i} is called filtration. and $E (ξ_{i}^{2}) < \infty$ for i = 0, 1, . . . n - 1, such that $X (t) = ξ_{0} I_{0} (t) + \sum_{i = 0}^{n - 1} ξ_{i} I_{[t_{i}, t_{i + 1}]} (t),$ (6) where I_{[t_i,t_i+1]} (t) denotes the indicator function defined as

I_{[t_i,t_i+1]} (t) = ${\begin{matrix} 1 if t \in [t_{i}, t_{i + 1}] \\ 0 otherwise . \end{matrix}$

and I₀ = 1.

For simple adapted process, the Ito integral $\int_{0}^{T} X (t) dB (t)$ is defined as $\int_{0}^{T} X (t) dB (t) = \sum_{i = 0}^{n - 1} ξ_{i} (B (t_{i + 1}) - B (t_{i}))^{″} .$ (7)

2.10 Filtration [18]

It follows (18) “A stochastic process X (t) is called adapted to the filtrationF_t, if ∀ t=0,1,2,...,T, X (t) is F_t measurable which means the σ - field generated by Brownian motion up to time t”.

2.11 Fuzzy simple process(FSP)

It follows (14) “A fuzzy random process $\tilde{X} (t)$ is said to be fuzzy simple process if there exist times 0 = t₀ < t₁ < . . . . . . . . . < t_n = T and fuzzy random variables ${\tilde{ξ}}_{0}, {\tilde{ξ}}_{1}, . . . ., {\tilde{ξ}}_{n - 1}$ such that $\tilde{X} (t) = {\tilde{ξ}}_{0} I_{0} (t) \oplus \sum_{i = 0}^{n - 1} {\tilde{ξ}}_{i} (ω) \otimes I_{[t_{i}, t_{i + 1}]} (t)^{″}$ (8)

2.12 Fuzzy simple adapted process(FSAP)

It follows (14) “A process $\tilde{X} = {\tilde{X} (t), 0 \leq t \leq T}$ is called a fuzzy simple adapted process if there exist times 0 = t₀ < t₁ < . . . . < t_n = T and fuzzy random variables ${\tilde{ξ}}_{0}, {\tilde{ξ}}_{1}, . . . ., {\tilde{ξ}}_{n - 1} .$ ${\tilde{ξ}}_{i}$ is ${\tilde{F}}_{t_{i}}$ measurable which depends upon the value of fuzzy Brownian motion of ${\tilde{B}}_{t}$ for t≤t_i and $E ({\tilde{ξ}}_{i}^{2}) ≺ \infty$ for i = 0, 1, . . . n - 1, $\tilde{X} (t, ω) = {\tilde{ξ}}_{0} I_{0} (t) \oplus \sum_{i = 0}^{n - 1} \tilde{ξ} (ω) \otimes I_{[t_{i}, t_{i + 1}]} (t) .$ (9)I_{[t_i,t_i+1]} (t) denotes the indicator functions”.

2.13 Regular adapted process(RAP) [18]

It follows (18) “A stochastic process X(t) is called a regular adapted process if there exists time 0 = t₀ < t₁ < . . . . < t_n = T, $\int_{0}^{T} X^{2} (t) dt < \infty^{″} .$

2.14 Fuzzy regular adapted process(FRAP)

A FSP $\tilde{X} = {\tilde{X} (t), t \in [0, T]}$ is said to be FRAP if ∀ α ∈ [0, 1] , $\int_{0}^{T} {\tilde{X}}_{α}^{2} (t) dt ≺ \infty .$

2.15 Quadratic variation and Covariance of Ito integral [18]

It follows (18) “The Ito integral $Y (t) = \int_{0}^{T} X (s) dB (s), 0 \leq t \leq T,$ is a random function of t, which is continuous and adapted processes.

The quadratic variation of Y(t) is defined as $[Y] (t) = lim_{δ_{n} \to 0} \sum_{i = 0}^{n - 1} (Y (t_{i + 1}^{n}) - Y (t_{i}^{n}))^{2}$ (10) where for each n, ${t_{i}}_{i = 0}^{n}$ is a partition of [0,T] and the limit is taken over all partitions with $δ_{n} = \max_{1 \leq i \leq n} (t_{i + 1}^{n} - t_{i}^{n})$ .

The covariance of Y(t) for t, u ≥ 0 is $Cov (Y (t), Y (t + u)) = \int_{0}^{T} X^{2} (s) {ds}^{″}$ (11)

3 Properties of fuzzy Ito integral

Ito integral:- Stochastic integrals with respect to BM is called Ito integrals i.e $\int_{0}^{T} X (t) dB (t) .$

Fuzzy Ito integral(FII):- For each α ∈ [0, 1], the fuzzy stochastic integral of ${\tilde{X}}_{α} (t)$ with respect to a fuzzy Brownian motion ${\tilde{B}}_{α} (t, ω)$ , is called FII which is defined on [0,T] i.e $\int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω)$ . For each α ∈ [0, 1] , ${\tilde{X}}_{α} (t) = [{\tilde{X}}_{α}^{U} (t), {\tilde{X}}_{α}^{L} (t)]$ and BM ${\tilde{B}}_{α} (t) = [{\tilde{B}}_{α}^{U} (t), {\tilde{B}}_{α}^{L} (t)]$ are interval-valued FSP and FBM respectively.

Fuzzy Ito integral of the simple process:-

The fuzzy Ito integral of fuzzy simple process $\tilde{X} (t)$ with respect to a fuzzy Brownian motion, is defined on [0,T], ∀ α ∈ [0, 1] .

$\begin{matrix} \int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) \\ = \sum_{i = 0}^{n - 1} ({\tilde{ξ}}_{i})_{α} (ω) \otimes [{\tilde{B}}_{α} (t_{i + 1}, ω) ⊖ {\tilde{B}}_{α} (t_{i}, ω)] \end{matrix}$ (12)

Theorem 3.1. A fuzzy simple process $\tilde{X} (t)$ satisfy regular fuzzy adapted process, ∀ α ∈ [0, 1] , $\int_{0}^{T} {\tilde{X}}_{α}^{2} (t) dt ≺ \infty$ with almost sure. The Fuzzy Ito integral $\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t)$ satisfy the following properties.

(i)Linearity: If $\tilde{X} (t)$ and $\tilde{Y} (t)$ are FSP, ${\tilde{B}}_{t}$ be a FBM, A and B are some constants, ∀ α ∈ [0, 1] , $\int_{0}^{T} (A {\tilde{X}}_{α} (t, ω) \oplus B {\tilde{Y}}_{α} (t, ω)) d {\tilde{B}}_{α} (t, ω)$ $\begin{matrix} = A \int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) \\ \oplus B \int_{0}^{T} {\tilde{Y}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) \end{matrix}$ (ii) For an indicator function I(t) of an interval [a, b] , ∀ α ∈ [0, 1] , $\int_{0}^{T} I_{[a, b]} (t) d {\tilde{B}}_{α} (t) = {\tilde{B}}_{α} (b) ⊖ {\tilde{B}}_{α} (a) .$ $and \int_{0}^{T} I_{[a, b]} (t) {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω)$ $= \int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) .$ (iii) Fuzzy Ito integral satisfies zero mean property i.e ∀ α ∈ [0, 1] , $E (\int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t)) = \tilde{0}$ .

Proof. (i) Let $\tilde{X} (t, ω)$ and $\tilde{Y} (t, ω)$ be two fuzzy simple processes,

Let $△ X = {X (t, ω) | X (t, ω) \in {\tilde{X}}_{α} (t, ω)},$ $△ B = {B (t, ω) | B (t, ω) \in {\tilde{B}}_{α} (t, ω)},$ $△ Y = {Y (t, ω) | Y (t, ω) \in {\tilde{Y}}_{α} (t, ω)} .$ ∀ α ∈ [0, 1] , ${\tilde{X}}_{α} (t . ω)$ and ${\tilde{Y}}_{α} (t, ω)$ are two real intervals.

By definition of linearity,

$\begin{matrix} \int_{0}^{T} (A {\tilde{X}}_{α} (t, ω) \oplus B {\tilde{Y}}_{α} (t, ω)) d {\tilde{B}}_{α} (t, ω) \\ = {\int_{0}^{T} AX (t, ω) dB (t, ω) | △ X} \oplus \\ {\int_{0}^{T} BY (t, ω) dB (t, ω) | △ Y} \end{matrix}$ $= {A \int_{0}^{T} X (t, ω) dB (t, ω) | △ X}$ $\oplus {B \int_{0}^{T} Y (t, ω) dB (t, ω) | △ Y}$ . $\begin{matrix} \Rightarrow \int_{0}^{T} (A {\tilde{X}}_{α} (t, ω) \oplus B {\tilde{Y}}_{α} (t, ω)) d {\tilde{B}}_{α} (t, ω) = \\ A \int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) \oplus B \int_{0}^{T} {\tilde{Y}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) \end{matrix}$

where the α- cut of the fuzzy Ito integrals are, $\int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω)$ $= [\int_{0}^{T} {\tilde{X}}_{α}^{L} (t, ω) d {\tilde{B}}_{α}^{L} (t, ω), \int_{0}^{T} {\tilde{X}}_{α}^{U} (t, ω) d {\tilde{B}}_{α} (t, ω)] .$ $and \int_{0}^{T} {\tilde{Y}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) =$ $[\int_{0}^{T} {\tilde{Y}}_{α}^{L} (t, ω) d {\tilde{B}}_{α}^{L} (t, ω), \int_{0}^{T} {\tilde{Y}}_{α}^{U} (t, ω) d {\tilde{B}}_{α}^{U} (t, ω)] .$ (ii) By the definition of indicator functions,

I_[a,b] (t)= ${\begin{matrix} 1 if t \in [a, b] \\ 0 otherwise \end{matrix}$ for each α ∈ [0, 1] , the α cut of $\int_{0}^{T} I_{[a, b]} (t) d \tilde{B} (t)$ is ${\int_{0}^{T} I_{[a, b]} (t) dB (t) | △ B} = {\int_{a}^{b} dB (t) | △ B}$ $= {B (b) ⊖ B (a) | △ B}$ $\Rightarrow \int_{0}^{T} I_{[a, b]} (t) d {\tilde{B}}_{α} (t, ω) = {\tilde{B}}_{α} (b) ⊖ {\tilde{B}}_{α} (a)$ (13) Similarly the α - cut of $\int_{0}^{T} I_{[a, b]} (t) \tilde{X} (t, ω) d \tilde{B} (t, ω)$ is $= {\int_{0}^{T} X (t, ω) dB (t, ω) | △ X, △ B}$ $= \int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω) .$

□

(iii) $Let △ ξ = {ξ_{i} | ξ_{i} \in ({\tilde{ξ}}_{i})_{α}}$ $△ B_{1} = {B (t_{i + 1}) | B (t_{i + 1}) \in {\tilde{B}}_{α} (t_{i + 1})}$ and $△ B_{2} = {B (t_{i}) | B (t_{i}) \in {\tilde{B}}_{α} (t_{i})} .$

By fuzzy simple process, $\tilde{X} (t, ω) = \sum_{i = 0}^{n - 1} {\tilde{ξ}}_{i} (ω) \otimes I_{[t_{i}, t_{i + 1}]} (t)$ (14) ∀ α ∈ [0, 1] , the α-cut of $\int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω)$ is $= {\sum_{i = 0}^{n - 1} ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | △ ξ, △ B_{1}, △ B_{2}}$ Since $({\tilde{ξ}}_{i})_{α}$ are square integrable, $So {\int_{- \infty}^{\infty} | ξ_{i} |^{2} d ξ | △ ξ} ≺ \infty,$ on the interval [0,T]. $\begin{matrix} \Rightarrow {\int_{0}^{T} | ξ_{i} |^{2} d ξ | △ ξ} ≺ \infty \end{matrix}$ $\Rightarrow {\int_{- \infty}^{\infty} | ξ_{i} | d ξ | △ ξ} exist .$ $\Rightarrow | ({\tilde{ξ}}_{i})_{α} | ≺ \infty$ $\begin{matrix} \Rightarrow {| ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | | \\ △ ξ, △ B_{1}, △ B_{2}} ≺ \infty \end{matrix}$ $\begin{matrix} \Rightarrow {E | ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | \\ | △ ξ, △ B_{1}, △ B_{2}} ≺ \infty \end{matrix}$ (15) Then by Cauchy-Schwartz inequality in (15) $\begin{matrix} {E | ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | | △ ξ, △ B_{1}, △ B_{2}} ≼ \\ {\sqrt{E (ξ_{i})^{2} E (B (t_{i + 1}) ⊖ B (t_{i}))^{2}} | \\ △ ξ, △ B_{1}, △ B_{2}} ≺ \infty \end{matrix}$

$\begin{matrix} So {E | \sum_{i = 0}^{n - 1} ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | | △ ξ, △ B_{1}, △ B_{2}} ≼ \end{matrix}$

$\sum_{i = 0}^{n - 1} {E | ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | | △ ξ, △ B_{1}, △ B_{2}}$ $≺ \infty$ Then by using Fuzzy martingale property, the α-cut becomes ${E (ξ_{i} (B (t_{i + 1}) ⊖ B (t_{i})) | F_{ti}) | △ ξ, △ B_{1}, △ B_{2}}$ $= {ξ_{i} E ((B (t_{i + 1} ⊖ B (t_{i})) | F_{ti}) | △ ξ, △ B_{1}, △ B_{2}} .$ $= {ξ_{i} (B (t_{i}) ⊖ B (t_{i})) | △ ξ, △ B_{1}, △ B_{2}} = \tilde{0} .$ $\begin{matrix} \Rightarrow {E \sum_{i = 0}^{n - 1} ξ_{i} (t) (B (t_{i + 1}) ⊖ B (t_{i})) | △ ξ, \\ △ B_{1}, △ B_{2}} = \tilde{0} . \end{matrix}$ $\Rightarrow E (\int_{0}^{T} {\tilde{X}}_{α} (t, ω) d {\tilde{B}}_{α} (t, ω)) = \tilde{0} .$ (16)

□

Theorem 3.2. Let the sequence of fuzzy simple stochastic process ${\tilde{X}}^{n} (t)$ convergence to the fuzzy stochastic process $\tilde{X} (t)$ , by fuzzy simple adapted process, ∀ α ∈ [0, 1], the sequence of simple fuzzy stochastic integrals $\int_{0}^{T} {\tilde{X}}_{α}^{n} (t) d {\tilde{B}}_{α} (t)$ converges to the limit $\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t)$ .

Proof. Let $△ X_{1} = {X (t) | X (t) \in {\tilde{X}}_{α} (t)} △ X_{2} = {X (t_{i}^{n}) | X (t_{i}^{n}) \in {\tilde{X}}_{α} (t_{i}^{n})}, △ X_{3} = {X (t_{i}^{m}) | X (t_{i}^{m}) \in {\tilde{X}}_{α} (t_{i}^{m})}, △ B = {B (t) | B (t) \in {\tilde{B}}_{α} (t)}, △ B_{1} = {B (t_{i}^{m}) | B (t_{i}^{m}) \in {\tilde{B}}_{α} (t_{i}^{m})}, △ B_{2} = {B (t_{i + 1}^{m}) | B (t_{i + 1}^{m}) \in {\tilde{B}}_{α} (t_{i + 1}^{m})}, △ B_{3} = {B (t_{i}^{n}) | B (t_{i}^{n}) \in {\tilde{B}}_{α} (t_{i}^{n})}, △ B_{4} = {B (t_{i + 1}^{n}) | B (t_{i + 1}^{n}) \in {\tilde{B}}_{α} (t_{i + 1}^{n})}$ .

To prove $\int_{0}^{T} {\tilde{X}}_{α}^{n} (t) d {\tilde{B}}_{α} (t)$ converges to a $lim_{n \to \infty} \int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t),$ i.e is to show $\begin{matrix} {lim_{n \to \infty} \int_{0}^{T} E (X^{n} (t) ⊖ X (t))^{2} dt | \\ △ X_{1}, △ X_{2}} = \tilde{0} \end{matrix}$ (17) Fuzzy simple process can be expressed as ${\tilde{X}}^{n} (t) = {\tilde{X}}_{0} I_{0} (t) \oplus \sum_{i = 0}^{n - 1} \tilde{X} (t_{i}^{n}) I_{[t_{i}^{n}, t_{i + 1}^{n + 1}]} (t),$ (18) where $δ_{n} = \max_{i} (t_{i + 1}^{n} - t_{i}^{n}) \to 0$ as n→ ∞.

On a partion [0,T], the sequence of fuzzy stochastic process ${\tilde{X}}^{n} (t)$ equals to fuzzy stochastic process $\tilde{X} (t)$ but for small interval $(t_{i}^{n}, t_{i + 1}^{n})$ may be different.

As $\tilde{X} (t)$ is a fuzzy simple adapted process, for each α ∈ [0, 1] $\int_{0}^{T} E ({\tilde{X}}_{α}^{2} (t)) dt ≺ \infty .$ $\Rightarrow {\int_{0}^{T} E (X^{2} (t)) dt | △ X_{1}} ≺ \infty .$ The α - cut for fuzzy Ito integral is $\int_{0}^{T} {\tilde{X}}_{α}^{n} (t) d {\tilde{B}}_{α} (t)$ $\begin{matrix} = {\sum_{i = 0}^{n - 1} [X (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))] | \\ △ X_{2}, △ B_{3}, △ B_{4}} \end{matrix}$ Given ${\tilde{X}}^{n} (t)$ converges to $\tilde{X} (t),$ so for each α ∈ [0, 1] $\begin{matrix} {\int_{0}^{T} X^{n} (t) dB (t) | △ B, △ X_{1}} \\ = {\sum_{i = 0}^{n - 1} [X (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))] | \\ △ X_{2}, △ B_{3}, △ B_{4}} \\ ⟶ {\int_{0}^{T} X (t) dB (t) | △ B △ X_{1}} \end{matrix}$ (19) Squareing equation (19) both sides

$\begin{matrix} {(\int_{0}^{T} X^{n} (t) dB (t))^{2} | △ B, △ X_{1}} = \\ {\sum_{i = 0}^{n - 1} X^{2} (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))^{2} | △ X_{2}, △ B_{3}, △ B_{4}} \\ \to {(\int_{0}^{T} X (t) dB (t))^{2} | △ B, △ X_{1}} \end{matrix}$ $\begin{matrix} \Rightarrow {\int_{0}^{T} (X^{n})^{2} (t) dt | △ X_{1}} = \\ {\sum_{i = 0}^{n - 1} X^{2} (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))^{2} | △ X_{2}, △ B_{3}, △ B_{4}} \\ \to {\int_{0}^{T} X^{2} (t) dt | △ X_{1}} \end{matrix}$ $Now {\int_{0}^{T} E (X^{n} (t))^{2} dt | △ X_{1}} =$ $\begin{matrix} {\sum_{i = 0}^{n - 1} E (X^{2} (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))^{2}) \\ | △ X_{2}, △ B_{3}, △ B_{4}} \\ \to {\int_{0}^{T} {EX}^{2} (t) dt | △ X_{1}} \end{matrix}$ $\begin{matrix} = {\sum_{i = 0}^{n - 1} E (X^{2} (t_{i}^{n})) E (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))^{2} \\ | △ X_{2}, △ B_{3}, △ B_{4}} \end{matrix}$ $\begin{matrix} = {\sum_{i = 0}^{n - 1} {EX}^{2} (t_{i}^{n}) (t_{i + 1}^{n} ⊖ t_{i}^{n}) | \\ △ X_{2}, △ B_{3}, △ B_{4}} \to {\int_{0}^{T} E (X^{2} (t)) dt | △ X_{1}} \end{matrix}$

∀ α ∈[0, 1] , the α - cut of fuzzy Ito integral ${\tilde{J}}_{n} (t)$ be ${\tilde{J}}_{n} = \int_{0}^{T} {\tilde{X}}_{α}^{n} (t) d {\tilde{B}}_{α} (t)$ $\begin{matrix} = {\sum_{i = 0}^{n - 1} X (t_{i}^{n}) (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n})) \\ | △ X_{2}, △ B_{3}, △ B_{4}} . \end{matrix}$ By zero mean property fuzzy Ito integral ${\tilde{J}}_{n} (t)$ $E ({\tilde{J}}_{n} (t)) = \tilde{0} .$ similarly ∀ α ∈ [0, 1] the α - cut of fuzzy Ito integral of ${\tilde{J}}_{m} (t)$ be ${\tilde{J}}_{m} (t) = \int_{0}^{T} {\tilde{X}}_{α}^{m} (t) d {\tilde{B}}_{α} (t) .$ $\begin{matrix} = {\sum_{i = 0}^{n - 1} X (t_{i}^{m}) (B (t_{i + 1}^{m}) ⊖ B (t_{i}^{m})) | \\ △ X_{3}, △ B_{1}, △ B_{2}} . \end{matrix}$ By zero mean property fuzzy Ito integral ${\tilde{J}}_{m} (t),$ $E ({\tilde{J}}_{m} (t)) = \tilde{0}$ . Using fuzzy Ito isometry property,

$\begin{matrix} ({\tilde{J}}_{n} ⊖ {\tilde{J}}_{m}) = {(\int_{0}^{T} X^{n} (t) dB (t) ⊖ \\ \int_{0}^{T} X^{m} (t) dB (t)) | △ X_{2}, △ X_{3}, △ B} . \end{matrix}$ $\begin{matrix} So E ({\tilde{J}}_{n} ⊖ {\tilde{J}}_{m})^{2} = {E (\int_{0}^{T} X^{n} (t) dB (t) ⊖ \\ \int_{0}^{T} X^{m} (t) dB (t))^{2} | △ X_{2}, △ X_{3}, △ B} \end{matrix}$ $\begin{matrix} = {E ((\int_{0}^{T} X^{n} (t) ⊖ \int_{0}^{T} X^{m} (t)) dB (t))^{2} | \\ △ X_{2}, △ X_{3}, △ B} \end{matrix}$ $= {E \int_{0}^{T} (X^{n} (t) ⊖ X^{m} (t))^{2} dt | △ X_{2}, △ X_{3}} .$ $\begin{matrix} = {E \int_{0}^{T} (X^{n} (t) ⊖ X (t) \oplus X (t) ⊖ X^{m} (t))^{2} dt | \\ △ X_{1}, △ X_{2}, △ X_{3}} \end{matrix}$ $\begin{matrix} = {E [\int_{0}^{T} (X^{n} (t) ⊖ X (t))^{2} dt \oplus E \int_{0}^{T} (X^{m} (t) ⊖ \\ X (t))^{2} dt \oplus 2 \int_{0}^{T} (X^{n} (t) ⊖ X (t)) (X (t) - X^{m} (t)) \\ dt] | △ X_{1}, △ X_{2}, △ X_{3}} \end{matrix}$

$≼ {2 E \int_{0}^{T} (X^{n} (t) ⊖ X (t))^{2} dt \oplus$ $2 E \int_{0}^{T} (X^{m} (t) ⊖ X (t))^{2} dt | △ X, △ X_{2}, △ X_{3}}$ $\Rightarrow lim_{n \to \infty} E (({\tilde{J}}_{n})_{α} ⊖ ({\tilde{J}}_{m})_{α})^{2}$ $\begin{matrix} ≼ {lim_{n \to \infty} \int_{0}^{T} 2 E (X^{n} (t) ⊖ X (t))^{2} dt \oplus \\ 2 E lim_{n \to \infty} \int_{0}^{T} (X^{m} (t) ⊖ X (t))^{2} dt | \\ △ X_{1}, △ X_{2}, △ X_{3}} \end{matrix}$ By mean square converges property, $\begin{matrix} {lim_{n \to \infty} \int_{0}^{T} E (X^{n} (t) ⊖ X (t))^{2} dt | \\ △ X_{1}, △ X_{2}} \\ = \tilde{0} \\ and {lim_{n \to \infty} \int_{0}^{T} E (X^{m} (t) ⊖ X (t))^{2} dt \\ | △ X_{1}, △ X_{3}} = \tilde{0} \end{matrix}$ $So lim_{n \to \infty} E (({\tilde{J}}_{n} (t)) - ({\tilde{J}}_{m} (t)))^{2} = \tilde{0}$ $\begin{matrix} \Rightarrow \int_{0}^{t} {\tilde{X}}_{α}^{n} (t) d {\tilde{B}}_{α} (t) converges \\ \int_{0}^{t} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t) . \end{matrix}$ □

Theorem 3.3. If a FSP $\tilde{X} (t)$ is continuous and satisfies simple fuzzy adapted process then ∀ α ∈ [0, 1] , the fuzzy Ito integrals $\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t)$ and $\int_{0}^{T} f ({\tilde{B}}_{α} (t)) d {\tilde{B}}_{α} (t)$ exist, the function f is continuous on I R.

Proof. Let $△ X = {X (t) | X (t) \in {\tilde{X}}_{α} (t)}, △ B = {B (t) | B (t) \in {\tilde{B}}_{α} (t)} .$

Given $\tilde{X} (t)$ be a continuous simple fuzzy adapted process.

So $\forall α \in [0, 1], {\int_{0}^{T} X^{2} (t) dt | △ X} ≺ \infty$ $\Rightarrow {\int_{0}^{T} X^{2} (t) dt | △ X}$ exist. $\Rightarrow {\int_{0}^{t} X (t) dB (t) | △ X, △ B}$ exist.

Given the function f is continuous on I R.

Hence $f (\tilde{B} (t))$ is continuous on I R as $\tilde{B} (t)$ is continuous.

$\Rightarrow \int_{0}^{T} f (\tilde{B} (t) d \tilde{B} (t)$ is continuos on I R.

So ∀ α ∈ [0, 1] , ${\int_{0}^{T} f (\tilde{B} (t)) d \tilde{B} (t) | △ B}$ is continuous.

So $\int_{0}^{T} f ({\tilde{B}}_{α} (t)) d {\tilde{B}}_{α} (t)$ exist.

Theorem 3.4. The two fuzzy Ito integrals, ∀ α ∈ [0, 1] $\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t)$ and $\int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α} (t)$ satisfy regular fuzzy adapted process, then $\begin{matrix} E (\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t) \otimes \int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α} (t)) = \\ \int_{0}^{T} E ({\tilde{X}}_{α} (t) \otimes {\tilde{Y}}_{α} (t)) dt . \end{matrix}$

Proof. $\tilde{X} (t) and \tilde{Y} (t)$ are fuzzy Ito integral, ∀ α ∈ [0, 1] , $\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t) and \int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α} (t) exists .$ $Let △ X = {X (t) | X (t) \in {\tilde{X}}_{α} (t)}$ $△ Y = {Y (t) | Y (t) \in {\tilde{Y}}_{α} (t)},$ $△ B = {B (t) | B (t) \in {\tilde{B}}_{α} (t)}$ $\begin{matrix} Define \tilde{A} = \int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t) = \\ {\int_{0}^{T} X (t) dB (t) | △ X, △ B}, \\ And \tilde{B} = \int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α} (t) = \\ {\int_{0}^{T} Y (t) dB (t) | △ Y, △ B} . \end{matrix}$ Given $\tilde{X} (t) and \tilde{Y} (t)$ satisfies FRAP. $\begin{matrix} Hence {E \int_{0}^{T} (X (t))^{2} dt | △ X} ≺ \infty and \\ {E \int_{0}^{T} (Y (t))^{2} dt | △ Y} ≺ \infty . \end{matrix}$ (20) Using $\tilde{A} \otimes \tilde{B}$ = $\frac{(\tilde{A} \otimes \tilde{B})^{2}}{2}$ $⊖ \frac{{\tilde{A}}^{2}}{2}$ $⊖ \frac{{\tilde{B}}^{2}}{2}$ $So \int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} (t) \otimes \int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α} (t)$ $\begin{matrix} = {\frac{(\int_{0}^{T} X (t) dB (t) \oplus \int_{0}^{T} Y (t) dB (t))^{2}}{2} ⊖ \\ \frac{(\int_{0}^{T} X (t) dB (t))^{2}}{2} ⊖ \frac{(\int_{0}^{T} Y (t) dB (t))^{2}}{2} | \\ △ X, △ Y, △ B} \end{matrix}$ $\begin{matrix} = {\frac{\int_{0}^{T} (X (t) \oplus Y (t))^{2} dt}{2} ⊖ \frac{\int_{0}^{T} X^{2} (t) dt}{2} ⊖ \\ \frac{\int_{0}^{T} Y^{2} (t) dt}{2} | △ X, △ Y} \end{matrix}$ $\begin{matrix} = {\frac{\int_{0}^{T} (X^{2} (t) \oplus Y^{2} (t) \oplus 2 X \otimes Y) dt}{2} ⊖ \\ \frac{\int_{0}^{T} X^{2} (t) dt}{2} ⊖ \frac{\int_{0}^{T} Y^{2} (t) dt}{2} | △ X, △ Y} \end{matrix}$ $\begin{matrix} = {\frac{\int_{0}^{T} X^{2} (t) dt}{2} \oplus \frac{\int_{0}^{T} Y^{2} (t) dt}{2} \oplus \\ \int_{0}^{T} (X (t) \otimes Y (t)) dt ⊖ \frac{\int_{0}^{T} X^{2} (t) dt}{2} \\ ⊖ \frac{\int_{0}^{T} Y^{2} (t) dt}{2} | △ X, △ Y} \end{matrix}$ $= {\int_{0}^{t} (X (t) \otimes Y (t)) dt | △ X, △ Y}$ $\begin{matrix} \Rightarrow E (\int_{0}^{T} {\tilde{X}}_{α} (t) d {\tilde{B}}_{α} \otimes \int_{0}^{T} {\tilde{Y}}_{α} (t) d {\tilde{B}}_{α}) = \\ {E (\int_{0}^{T} X (t) \otimes Y (t)) dt | △ X, △ Y} \end{matrix}$ (21) $= \int_{0}^{T} E ({\tilde{X}}_{α} (t) {\tilde{Y}}_{α} (t)) dt .$

□

Theorem 3.5. If $\tilde{X} (t)$ be a FAP such that for each α ∈ [0, 1] $\int_{0}^{T} E ({\tilde{X}}_{α} (s)^{2}) ds ≺ \infty,$ then ${\tilde{Y}}_{α} (t) = \int_{0}^{T} {\tilde{X}}_{α} d {\tilde{B}}_{α} (s)$ , 0 ≤ t ≤ T is a square integrable FM.

Proof. $△ B = {B (t) | B (t) \in {\tilde{B}}_{α} (t)}$ $△ X = {X (s) | X (s) \in {\tilde{X}}_{α} (s)}$ , ∀ α ∈ [0, 1] , $\tilde{X}$ be a adapted process, $So {\int_{0}^{T} E (X^{2} (s) ds) | △ X} ≺ \infty .$ (22) By fuzzy Ito isometry property, ${E (\int_{0}^{T} X^{2} (s) ds) | △ X} ≺ \infty .$ $\Rightarrow {| E (\int_{0}^{T} {\tilde{X}}^{2} (s) ds) | △ X} ≼ k .$ $\Rightarrow {| E (X^{2} (s) ds) | △ X} ≼ k .$ The variance of fuzzy Ito integrals are bounded i.e. second moment of fuzzy Ito integrals are bounded.

So $\tilde{X} (s)$ is bounded.

Hence $\int_{0}^{T} {\tilde{X}}_{α} (s) d {\tilde{B}}_{α} (s)$ is an square integrable FM. □

4 Quadratic variation and covariance of fuzzy Ito integral

Theorem 4.1. The quadratic variation of FII, ∀ α ∈ [0, 1] , $\int_{0}^{T} {\tilde{X}}_{α} (s) d {\tilde{B}}_{α} (s)$ is $\begin{matrix} [\int_{0}^{T} {\tilde{X}}_{α} (s) d {\tilde{B}}_{α} (s), \int_{0}^{T} {\tilde{X}}_{α} (s) d {\tilde{B}}_{α} (s)] \\ = \int_{0}^{T} {\tilde{X}}_{α}^{2} (s) ds . \end{matrix}$

Proof. $Let △ X = {X (s) | X (s) \in {\tilde{X}}_{α} (s)}, △ B = {B (s) | B (s) \in {\tilde{B}}_{α} (s)}, △ B_{1} = {B (t_{i}^{n}) | B (t_{i}^{n}) \in {\tilde{B}}_{α} (t_{i}^{n})}, △ B_{2} = {B (t_{i + 1}^{n}) | B (t_{i + 1}^{n}) \in {\tilde{B}}_{α} (t_{i + 1}^{n})}$ To prove this we consider a numerical illustration.

Let $\tilde{X} (t)$ take only two different values on [0,1]

I.e. on [0, 1/2] takes the value $({\tilde{ξ}}_{0})$ and on [1/2,1, 1/2,1] takes the value $({\tilde{ξ}}_{1})$ So $\tilde{X} (t) = {\tilde{ξ}}_{0} I_{[0, 1 / 2]} (t) \oplus {\tilde{ξ}}_{1} I_{[1 / 2, 1]} (t) .$ ∀ α ∈ [0, 1] , ${\tilde{Y}}_{α} (t) = {\int_{0}^{T} X (s) dB (s) | △ X, △ B}$ (23)

= ${\begin{matrix} {{\tilde{ξ}}_{0} B (t) | △ B} if t \leq \frac{1}{2} \\ {{\tilde{ξ}}_{0} B (1 / 2) \oplus {\tilde{ξ}}_{1} (B (t) ⊖ B (1 / 2)) | △ B} if t > \frac{1}{2} \end{matrix}$

Any partition of [0,T] and ∀ α ∈ [0, 1] ,

${\tilde{Y}}_{α} (t_{i + 1}^{n}) ⊖ {\tilde{Y}}_{α} (t_{i}^{n})$ = ${\begin{matrix} {{\tilde{ξ}}_{0} (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n})) | △ B}, if t_{i}^{n} < t_{i + 1}^{n} \leq 1 / 2 \\ {{\tilde{ξ}}_{1} (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n})) | △ B}, if 1 / 2 \leq t_{i}^{n} < t_{i + 1}^{n} \end{matrix}$

Case- 1 For t ≤ 1/2 $[{\tilde{Y}}_{α}] = lim_{δ_{n} \to 0} \sum_{i = 0}^{n - 1} ({\tilde{Y}}_{α} (t_{i + 1}^{n}) - {\tilde{Y}}_{α} (t_{i}^{n}))^{2}$ (24) $= {{\tilde{ξ}}_{0}^{2} lim_{δ_{n} \to 0} \sum_{i = 0}^{n - 1} (B (t_{i + 1}^{n}) ⊖ B (t_{i}^{n}))^{2} | △ B_{1}, △ B_{2}}$ $= {\tilde{ξ}}_{0}^{2} [{\tilde{B}}_{α}, {\tilde{B}}_{α}] (t)$ $= ({\tilde{ξ}}_{0})^{2} t$ $= {\int_{0}^{T} X^{2} (s) ds | △ X} .$ $= \int_{0}^{t} {\tilde{X}}_{α}^{2} (s) ds$ Case-2 for t > 1/2 $[{\tilde{Y}}_{α}] = lim_{δ_{n} \to 0} \sum_{i = 0}^{n - 1} ({\tilde{Y}}_{α} (t_{i + 1}^{n}) - {\tilde{Y}}_{α} (t_{i}^{n}))^{2}$ $\begin{matrix} = {\tilde{ξ}}_{0}^{2} lim_{δ_{n} \to 0} \sum_{t_{i} < 1 / 2} ({\tilde{B}}_{α} (t_{i + 1}^{n}) ⊖ {\tilde{B}}_{α} (t_{i}^{n}))^{2} \oplus {\tilde{ξ}}_{1}^{2} \\ lim_{δ_{n} \to 0} \sum_{t_{i} < 1 / 2} ({\tilde{B}}_{α} (t_{i + 1}^{n}) ⊖ {\tilde{B}}_{α} (t_{i}^{n}))^{2} \end{matrix}$ $= {\tilde{ξ}}_{0}^{2} [{\tilde{B}}_{α}, {\tilde{B}}_{α}] (1 / 2) \oplus {\tilde{ξ}}_{1}^{2} [{\tilde{B}}_{α}, \tilde{B} α] (1 / 2, t],$ for each α ∈ [0, 1] , $= {X^{2} (s) ds | △ X}$ $= \int_{0}^{T} {\tilde{X}}_{α}^{2} (s) ds$ $\Rightarrow \int_{0}^{T} {\tilde{X}}_{α} (s) ds =$ $\begin{matrix} [\int_{0}^{T} {\tilde{X}}_{α}^{L} (s) ds, \int_{0}^{T} {\tilde{X}}_{α}^{U} (s) ds] . \end{matrix}$

□

Theorem 4.2. Let $\tilde{X} (t, s)$ be a regular fuzzy adapted process, ${\tilde{Y}}_{α} (t)$ be a FII, then ∀ α ∈ [0, 1] , the covariance of ${\tilde{Y}}_{α} (t)$ is $\begin{matrix} Cov ({\tilde{Y}}_{α} (t), {\tilde{Y}}_{α} (t + u)) = \\ \int_{0}^{t} {\tilde{X}}_{α} (t, s) {\tilde{X}}_{α} (t + u, s) ds, t, u \geq 0 \end{matrix}$

Proof. $Let △ X_{1} = {X (t, s) | X (t, s) \in {\tilde{X}}_{α} (t, s)}, △ X_{2} = {X (t + u, s) | X (t + u, s) \in {\tilde{X}}_{α} (t + u, s)} △ B = {B (s) | B (s) \in {\tilde{B}}_{α} (s)},$ ∀ α ∈ [0, 1] , ${\tilde{Y}}_{α} (t) = {\int_{0}^{t} X (t, s) dB (t, s) | △ X_{1}, △ B}$ be a fuzzy Ito integral.

Given $\tilde{X} (t, s)$ a fuzzy regular adapted process ${\int_{0}^{t} X^{2} (t, s) ds | △ X_{1}} ≺ \infty .$ By zero mean property of FII ${E (\int_{0}^{t} X (t, s) dB (s)) | △ X_{1}, △ B} = \tilde{0}$ Simillarly ${E (\int_{0}^{t} X (t + u, s) dB (s))$ $| △ X_{2}, △ B} = \tilde{0} .$ So ∀ α ∈ [0, 1] , $\begin{matrix} {\tilde{Y}}_{α} (t + u) = {\int_{0}^{t + u} X (t + u, s) dB (s) \\ | △ X_{2}, △ B} \end{matrix}$ be a FII. $\begin{matrix} = {\int_{0}^{t} X (t + u, s) dB (s) | △ X_{2}, △ B} \\ \oplus {\int_{t}^{t + u} X (t + u, s) dB (s) | △ X_{2}, △ B} . \end{matrix}$ Since the FII ${\int_{t}^{t + u} X (t + u, s) dB (s) | △ X_{2}, △ B}$ is independent of ${\tilde{F}}_{t},$ $\begin{matrix} Cov ({\tilde{Y}}_{α} (t), {\tilde{Y}}_{α} (t + u)) \\ = {Cov (\int_{0}^{t} X (t, s) dB (s), \\ \int_{0}^{t + u} X (t + u, s) dB (s)) | △ X_{1}, △ X_{2}, △ B} . \end{matrix}$ $\begin{matrix} = {Cov (\int_{0}^{t} X (t, s) X (t + u, s) ds) | \\ △ X_{1}, △ X_{2}} \end{matrix}$ $= E {\int_{0}^{t} X (t, s) X (t + u, s) ds | △ X_{1}, △ X_{2}}$ (By zero mean property of FII). $= {\int_{0}^{t} X (t, s) X (t + u, s) ds | △ X_{1}, △ X_{2}}$ $= \int_{0}^{t} {\tilde{X}}_{α} (t, s) {\tilde{X}}_{α} (t + u, s) ds .$

□

5 Conclusion

In this paper, the properties of FII are discussed using the FBM. An Ito integral is said to be FII if the stochastic process and Brownian motion are random as well as fuzzy. Fuzzy Ito integral with respect to FBM is a suitable tool to find the solution of stochastic Differential equations. The Ito integral is continuous but no where differentiable function like Brownian motion. It has been studied the properties of fuzzy Ito integrals of the simple adapted process with respect to FBM. Fuzzy Ito integral arises when fuzzy random noise and white noise is introduced into an ordinary differential equations. Here fuzzy simple process, fuzzy simple adapted process, and fuzzy Ito integral of simple process are defined. The integrals of converging fuzzy simple process converge to the limit of fuzzy Ito integral is derived. Finally, the quadratic variance and covariance of fuzzy Ito integrals have been discussed. Throughout the paper we have used Buckley’s [26] concept of fuzzy probability. However it can be extended in different approach, which is the future scope of present work.

Footnotes

Acknowledgment

We would like to express our special thanks of gratitude to the editor in chef, Reza Langari and enormous reviewers who gave their valuable suggestions for improving our paper.

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