The fBm B H is a zero mean Gaussian process with the following covariance function:

This was derived in Kolmogorov [17] and further studied in Mandelbrot and van Ness [18], where a stochastic integral representation was derived in terms of the Brownian motion^[2]. ${\int }_0^t[{\left(t-s\right)}^{H-\frac{1}{2}}]dB(s))$ is called the Liouville form of a fBm and it holds many properties of the fBm except that it has non-stationary increments. The classical Itô theory cannot handle a stochastic integral in terms of fBm because B^H is not a semi-martingale if H ≠ ½. Two approaches have been used to evaluate stochastic integrals with respect to fBm. If H> 1/2, the Riemann-Stieltjes stochastic integral can be defined using Young’s integral [19]. Suppose we have a financial time series representing the price of a stock over a certain period of time, denoted by S(t). We want to calculate the Riemann-Stieltjes stochastic integral of the stock price with respect to a cumulative trading volume, denoted by V(t), which represents the total number of shares traded up to time t. Let's assume S(t) represents the closing price of a stock at time t. We use the following definitions. Integrator function - consider V(t) to be the cumulative trading volume up to time t, which we obtain from historical trading data. Partition - choose a partition of the time interval into subintervals. For simplicity, let's consider a uniform partition with n subintervals. Piecewise linear approximation - approximate V(t) on each subinterval using a piecewise linear function. This can be done by interpolating the cumulative trading volume between the available data points. Calculate the Riemann-Stieltjes integral using Young's integral approach. For each subinterval, compute the integral of S(t) with respect to the piecewise linear approximation of V(t). This can be done by multiplying the stock price at each subinterval by the change in the approximated cumulative trading volume. Refinement- repeat the above steps with finer and finer partitions, refining the piecewise linear approximation of V(t) and calculating the Riemann-Stieltjes stochastic integral using Young's integral. By repeating the process with increasingly finer partitions, the Riemann-Stieltjes stochastic integral will converge to a well-defined value, representing the integral of the stock price with respect to the cumulative trading volume over the given time period. This can provide insights into the relationship between trading activity and stock price movements. The main difference between the Riemann-Stieltjes and the Young integral is that the former one uses a fixed partition of the time interval, while the latter one allows for an arbitrary partition of the time interval. This makes the Young’s integral more flexible and easier to use in stochastics. The Riemann-Stieltjes integral cannot be used to define an integral of a function with respect to fBm, because the Stieltjes measure is sometimes not defined for stochastic processes with non-stationary increments like fBm. A Stieltjes measure is a generalisation of the Lebesgue measure to allow for integration over sets with irregular boundaries. A Stieltjes measure is defined on a measurable space (X, A), where X is a set and A is a σ-algebra of subsets of X. A Stieltjes measure is a function μ: A → [0, ∞] that satisfies the following properties:

The third property is called the continuity property. It ensures that the Stieltjes measure is not sensitive to small changes in the sets that are being measured. Stieltjes measures can be used to define integrals of functions with respect to sets with irregular boundaries. For example, the Stieltjes measure can be used to define the integral of a function with respect to a Cantor set. These are fractional sets that are constructed by repeatedly removing a middle third of a line segment. Cantor sets have no Lebesgue measure, but they do have a Stieltjes measure. This means that we can define the integral of a function with respect to a Cantor set. Here is an example of how to use a Stieltjes measure to define the integral of a function with respect to a set with an irregular boundary:

Let X = [0, 1] and let A be the σ-algebra of all Borel subsets of X. Let g(t) be the Cantor function. The Cantor function is a continuous function that is defined on the interval [0, 1] and has the following properties:

The Cantor function is a pathological function, which means it has some unusual features. The Cantor function has no Lebesgue measure, but it does have a Stieltjes measure. It is a set of points that are sparse but not empty. We can define the Stieltjes measure for the Cantor function as follows:

μ(A) = g(b) - g(a), where A is a Borel subset of X and [a, b] is the smallest interval of the form [a, b] that contains A. We can now use the Stieltjes measure to define the integral of a function f(t) with respect to the Cantor function$: \int f(t) dg(t) [\mathrm{0,1}]=\mu (\int f(t) dt) [\mathrm{0,1}].$ This integral is defined for all functions f(t) that are measurable with respect to the Lebesgue measure. The Cantor function is not measurable with respect to the Lebesgue measure, but we can still define the Stieltjes measure for the Cantor function with respect to the Lebesgue measure. This is because the Stieltjes measure is a more general measure than the Lebesgue measure.

The Lebesgue measure is a measure that is defined on the Borel sets of the real line. The Borel sets of the real line are the sets that can be formed from the open sets of the real line using the operations of countable union, countable intersection, and complementation. The Stieltjes measure is a measure that is defined on the Borel sets of the real line, but it is not restricted to the Borel sets. The Stieltjes measure can be defined on any set of real numbers, even sets that are not Borel sets.

When we define the Stieltjes measure for the Cantor function with respect to the Lebesgue measure, we are using the Lebesgue measure to define the sets that the Stieltjes measure is defined on. However, we are not using the Lebesgue measure to define the values of the Stieltjes measure. The values of the Stieltjes measure are defined by the Cantor function. The Stieltjes measure for the Cantor function can be used to define the integral of a function with respect to the Cantor function. The integral of a function with respect to the Cantor function is defined as follows: $\int f(t) dg(t) [\mathrm{0,1}]=\mu (\int f(t) dt) [\mathrm{0,1}]$,

where f(t) is the function that we want to integrate; g(t) is the Cantor function; μ is the Stieltjes measure for the Cantor function.

The integral of a function with respect to the Cantor function can be used to calculate the area under the graph of the Cantor function. The area under the graph of the Cantor function is equal to the Stieltjes measure of the Cantor set.

In summary, both Young's integral and Stieltjes measures are valuable mathematical tools, but their applicability depends on the characteristics of the stochastic process being studied and the mathematical framework in which they are used. Young's integral is tailored to the needs of fBm and related processes, while Stieltjes measures are more general and can be applied to a wider range of scenarios.

If H < ½, Malliavin calculus techniques are used to approximate B_H(t) by a semi-martingale process [20]. The scope for this paper is to derive results for MFFBM processes when h > ½. The theoretical framework we are relying on when h < ½ is from the Molchan-Golosov integral and adopted by [21]. It allows us to change the Hurst exponent using a specified transformation. This will be the main contribution in the future work. Returning to the proposition that if H> 1/2, then Riemann-Stieltjes stochastic integral can be defined using Young’s integral [19].

$B_{H,\epsilon }\left(t\right)=a{\int }_0^t{\phi }^{\epsilon }\left(s\right)ds+{\epsilon }^{a}B(t)$, where ${\phi }^{\epsilon }\left(s\right)={\int }_0^t{\left(t-s+\epsilon \right)}^{a-1}]dB(s)$Proof using integration by parts technique: $\int uv=u\int v-\int u\text{'}\int v$. For an alternative proof of this theorem please see Thao (2006) Theorem 2.1 [22].

So, where ${\phi }^{\epsilon }\left(s\right)={\int }_0^t{\left(t-s+\epsilon \right)}^{a-1}]dB(s)$The process B_H,ϵ (t) converges to B_H(t) in L²(Ω) when ϵ tends to zero because it is square integrable [22].

This section provides some necessary background for fuzzy processes. The material mainly comes from the fuzzy set theory of Zadeh [23]. Here we will define a number of measures that are used in this paper. The Lebesgue measure is widely used to measure lengths, areas, volumes, and higher-dimensional extents of sets. It is defined on the Borel σ-algebra of subsets of Euclidean space. For example, the Lebesgue measure of an interval [a, b] in one dimension is simply the length of the interval, given by [b – a]. Counting measure - in combinatorics and discrete mathematics, the counting measure is a fundamental measure that assigns the number of elements in a set. It is defined on any set and is particularly useful for counting finite sets. For instance, the counting measure of a finite set A is equal to the cardinality (number of elements) of A. Probability measure - in probability theory, probability measures assign probabilities to events in a sample space. A probability measure is a measure that is defined on a σ-algebra of subsets of the sample space and satisfies the properties of non-negativity, assigning probability 0 to the empty set, and countable additivity. For example, the uniform distribution on the interval [0, 1] is a probability measure that assigns equal probability to all sub-intervals of [0, 1]. Hausdorff measure - in geometric measure theory, the Hausdorff measure is used to quantify the "size" or "extent" of fractional sets. It is defined on metric spaces and captures the concept of dimension. For instance, the Hausdorff measure of a fractional set like the Cantor set or the Sierpinski triangle measures the "length" or "area" of the set in a fractional sense.

Let K(R) be the family of all nonempty, compact, and convex subsets of R (field). The Hausdorff metric d _H is defined by the following:

A metric space is said to be separable if there exists a countable dense subset. If A, B, C ⊂ K(R) then:

Let Ω, A, P be a probability space. The mapping F: Ω-> K(R) is called A-measurable if it satisfies {w ϵ Ω : F(w) ⋂ C ≠ φ} ϵ A, for every closed set C ⊂ R. This means that for any element w that belongs to space Ω such that F(w) is disjoint with C, it is not equal to zero for every closed set C ⊂ R. A closed set means that all limiting and interior points belong to a set. A multifunction F ⊂ M is said to be L^p – integrably bounded if p^th power is integrable for p ≥ 1, iff (if and only if) there exists h ⊂ L^p(Ω, A, P, R₊) such that the norms metric (the norm ||.|| induces a metric on F, known as the metric induced by the norm or the norm metric) such that ||F|| ≤ h P-a.e (almost everywhere - event holds for almost all elements in a set, with the exception of a negligible subset of elements according to the probability measure P), R₊ = [0, ∞) and ||F|| = d_H(F, {0}) = sup |f|, f ϵ F. This means the norms in terms of metric is the greatest value of F. For example, imagine we have a function F that takes some inputs and gives us sets (groups of things) as outputs. Now, this function F is special in two ways:

The first point means that if we look at the sets produced by this function, they're not too wild. They don't stretch out forever or become enormous. The second point is about measuring. We can measure the size of these sets by giving them a number that says how big they are. Now, this function F can't be too unpredictable. It has to follow certain rules. If we square the sizes of its sets and add them all up, that should be a reasonable number (integrable). This is where the "L^p" comes in. "p" is just a number, and when it's 1 or bigger, it means things are behaving well.

To make sure these sets aren't too big, there's another function (let's call it "h") that keeps an eye on them. It's like a supervisor. This supervisor function, "h," is also measured in a special way, and it makes sure our function's sets don't get too big. So, in simple terms, an L^p-integrably bounded multifunction is like a machine that produces sets. These sets are controlled and supervised to make sure they're not too big, and we can measure their sizes using numbers.

The membership function u on an interval closed set: R -> [0,1] is defined for a fuzzy set u ⊂ R, where u(x) is the degree of membership of x in the fuzzy set u. Let us denote by F(R) the fuzzy set u: R -> [0,1] such that [u]^a ⊂ K(R) for every a ⊂ [0,1], where [u]^a ={x ϵ R: u(x) ≥ a}. Define domain d_∞: F(R) x F(R) -> [0,∞) codomain /space of images and F(R) with d_∞ is a complete metric space by d_∞(u, v) = sup d_H ([u]^a, [v]^a), where a ⊂ [0,1]. This means that we are defining a function d_∞ that takes two inputs from the set F(R) which represents the set of all real valued functions and returns a value in the range [0, ∞) – the non-negative real numbers. We will now show that d_∞ is a metric in F(R) and (F(R) d_∞) is a complete metric space. To accomplish this, we are considering F(R) equipped with the metric d_∞ to establish if it is a complete metric space. A metric space consists of a set equipped with a metric, which is a function that measures the distance between elements in the set. In this case, F(R) is the set, and d_∞ is the metric we are defining. The metric d_∞(u,v) is defined as the supremum or the least upper bound of the Hausdorff distance between the equivalence classes [u]^a and [v]^a, where a belongs to the interval [0,1]. Here [u]^a and [v]^a represent equivalence classes of functions u and v, respectively, under the relation defined by the parameter a. The Hausdorff distance measures the “closeness” between two sets by considering the furthest distance from each set to the other. It is the maximum distance of a set to the nearest point in the other set. In this case, the sets [u]^a and [v]^a are being compared, and their Hausdorff distance is computed. The supremum is then taken over all values of a in the interval [0,1]. By defining the metric d_∞ this way, we establish that F(R) is a complete metric space. Completeness means that every Cauchy sequence in F(R) converges to a limit within F(R). In other words, F(R) with the metric d_∞ satisfies the property that any sequence of functions that gets arbitrarily close to each other eventually has a limit within the same set. So, we have defined the metric d_∞ on the set of real valued functions F(R) and established F(R) equipped with d_∞ as a complete metric space. The metric measures the Hausdorff distance between equivalence classes of functions under a parameter a in the interval [0,1]. For every u, v, w, z ϵ F(R), λ ϵ R, we have the following properties:

Expanding the work of [16], particularly in 6 and 7 below, we will now define a fuzzy random variable as a function that assigns fuzzy sets to elements in probability space. We will introduce the concept of measurability with respect to different metrics such as d_∞ and d_s and describe the properties and relationships between these metrics and notion of a fuzzy random variable. We will break it down step by step:

In other words, a σ-algebra is a collection of subsets of Ω that includes the empty set, is closed under taking of complements, and is closed under countable unions.

The purpose of defining a σ-algebra is to identify a set of subsets of Ω that can be considered "measurable" in a certain sense. The elements of Σ are often referred to as measurable sets, and they form a structure that allows us to define measures (such as probability measures) on these sets.

By specifying a σ-algebra on Ω, we establish a framework for defining and working with measurable functions, integrating functions, and constructing measures. The σ-algebra provides a systematic way to determine which subsets of Ω are considered "measurable" within the given context, allowing us to analyse and reason about these subsets in a rigorous mathematical manner. In summary, a σ-algebra of subsets of Ω is a collection of subsets that satisfies specific properties, including containing the empty set, being closed under complementation, and being closed under countable unions. It provides a foundation for defining measurable sets and functions, allowing us to study measures and perform various mathematical operations on these sets.

d_s(u, v) := inf max {sup|λ(t) – t|, sup d_H(X_u(t), X_u(λ(t)))} where λ∈Φ, t ∈ [0,1] and t∈[0,1], Φ denotes the set of strictly increasing continuous functions λ: [0,1] → [0,1] satisfying λ(0) = 0 and λ(1) = 1. X_u and X_v are càdlàg (right-continuous with left limits) representations of the fuzzy sets u and v in F(R), respectively. d_H represents the Hausdorff distance between fuzzy sets.

Next, we expand the work of [16] in fuzzy random random variable and fuzzy stochastic process. We define a fuzzy random variable X: Ω → F(R) is said to be L^p -integrably bounded for p ≥ 1 if the equivalence class [X]^α belongs to the space L^p (Ω, A, P; K(R)) for every α ∈ [0,1]. Here, the random variables X, Y ∈ L^p (Ω, A, P; K(R)) are identical if P(d_∞(X, Y) = 0) = 1. This means that for X and Y to be considered identical, the probability that their distance d_∞ is equal to zero should be 1. d_∞ measures the “distance” or dissimilarity between two fuzzy variables X and Y. When the distance d_∞ between X and Y is zero, it indicates that X and Y are indistinguishable from each other in terms of their fuzzy characteristics. For a fuzzy random variable X: Ω → F(R) and p ≥ 1, the following conditions are equivalent: a) X ∈ L^p(Ω, A, P; F(R)). b) The equivalence class [X]⁰ belongs to L^p(Ω, A, P; F(R)). c) The norm of [X]⁰ denoted as ||[X]⁰|| belongs to L^p(Ω, A, P; R+), where R+ represents the non-negative real numbers. The statement then asserts that these three conditions (a, b, and c) are equivalent. In other words, if any one of these conditions is true, the other two conditions will also be true, and vice versa. This demonstrates the relationships between the integrability of a fuzzy random variable, its equivalence class, and the norm of that equivalence class in different spaces. The notation I := [0, T] signifies the interval from 0 to T, which means it is a closed interval from 0 to T inclusive and (Ω, A, P) represents a complete probability space equipped with a filtration {A_t}t∈I. A filtration is a sequence or family of sub-σ-algebras {A_t}t∈I defined on a probability space. In this case, the filtration is defined for each t in the interval I = [0, T]. A complete probability space refers to a probability space where every subset of a null set (a set with measure zero) is also measurable and has measure zero. A complete probability space means that not only is the sample space, the σ-algebra, and the probability measure defined, but it also has an additional property that every subset of a null set (a set with measure zero) is also measurable. In other words, if you have a set of outcomes that is so unlikely (e.g., its probability measure is zero), then any subset of that unlikely set is also considered measurable. Being measurable means that it's part of the σ-algebra, so you can assign probabilities to it. The requirement that these subsets of null sets are measurable ensures that there are no hidden or unmeasurable sets that could potentially lead to issues in probability theory. It essentially fills in any gaps or irregularities in the probability space, making it more robust.

The filtration is an increasing and right-continuous family of sub-σ-algebras of A, which contains all P-null sets. The filtration {A_t}t∈I is said to be increasing, which means that as t increases, the sub-σ-algebra A_t becomes larger or contains more sets. In other words, if s ≤ t, then A_s is a subset of A_t. This property ensures that the information or knowledge available at time s is also available at any later time t ≥ s. The filtration {A_t}t∈I is also right-continuous, which means that for any t in the interval I, the sub-σ-algebra A_t is equal to the intersection of all sub-σ-algebras A_s with s > t. In other words, the information available at time t is already available at any slightly earlier time t - ε, where ε is a small positive value. The filtration {A_t}t∈I is required to contain all P-null sets. A P-null set is a subset of the sample space Ω with measure zero. By including all P-null sets in each sub-σ-algebra A_t, the filtration captures all negligible information about the underlying probability space.

A fuzzy stochastic process X is said to be d_∞-continuous if almost all its trajectories, denoted as X(·, ω), are d_∞-continuous functions. Here, the trajectories refer to the mappings X(·, ω) that associate each time point t in the interval I with a fuzzy set X(t, ω) in F(R). The d_∞-continuity of these trajectories means that they exhibit continuity with respect to the d_∞ metric, which measures the dissimilarity between fuzzy sets. A fuzzy stochastic process X is considered measurable if the equivalence class [X]^α: I × Ω → K(R) is a measurable multifunction for all α in the interval [0, 1]. Here, [X]^α represents the equivalence class of X with respect to the parameter α, and K(R) denotes the set of fuzzy sets over the real numbers. A measurable multifunction [X]^α: I × Ω → K(R) maps a measurable set in the time domain I and the sample space Ω to a set of measurable fuzzy sets in the value domain R. For example, Let X(t) be a fuzzy stochastic process that represents the price of a stock at time t. The parameter α can represent the level of confidence that we have in the price prediction. For example, α = 0.95 could represent a 95% confidence level. We can define the equivalence class [X(t)]α as follows: [X(t)]^⍺ = {Y ∈ K(R) | Y(x) ≥ α for all x ∈ R}. The parameter α represents the level of confidence in the price prediction. It quantifies the minimum membership value required for a fuzzy set to be considered as part of the equivalence class [X(t)]^⍺. This equivalence class contains all fuzzy sets that are at least as precise as X(t) with respect to the confidence level α. The equivalence class [X(t)]^⍺ is defined as the set of fuzzy sets Y in K(R) that have a membership value greater than or equal to α for all x in the real numbers (R). In other words, [X(t)]^⍺ contains all fuzzy sets that are at least as precise as X(t) with respect to the confidence level α.We can then show that the multifunction [X(t)]^⍺: I × Ω → K(R) is measurable. This means that the fuzzy stochastic process X(t) is measurable. By stating that the multifunction [X(t)]^⍺: I × Ω → K(R) is measurable, we are asserting that the mapping from the Cartesian product of the index set I and the sample space Ω to the set of fuzzy sets K(R) is a measurable multifunction. This means that the pre-images of measurable sets in K(R) under [X(t)]^⍺ are measurable sets in I × Ω. The pre-images, in this context, refer to the subsets of the domain I × Ω that get mapped to a specific set in the target space K(R) under the mapping [X(t)]^⍺: I × Ω → K(R).To put it simply, if we have a set A in K(R), the pre-image of A under [X(t)]^⍺ is the set of all points (i, ω) in the domain I × Ω such that [X(t)]^⍺(i, ω) belongs to A.

In other words, the pre-image of a set A in K(R) consists of all the elements (i, ω) in I × Ω that get mapped to A under the mapping [X(t)]^⍺. By saying that the pre-images of measurable sets in K(R) under [X(t)]^⍺ are measurable sets in I × Ω, it means that if we consider a measurable set A in K(R), the pre-image of A under [X(t)]α is a measurable set in I × Ω. This implies that the behaviour of the fuzzy stochastic process X(t) with respect to such pre-images is well-behaved and can be characterised by measurable sets in the domain I × Ω. For example, we have a fuzzy stochastic process X(t) that represents the temperature at a particular location over time. The parameter α represents the confidence level in our temperature predictions. The domain of the process is the Cartesian product of the time index set I and the sample space Ω. Let's assume I = {1, 2, 3} represents three time points and Ω = {A, B} represents two possible outcomes (e.g., sunny, or cloudy days). The target space K(R) represents the set of fuzzy sets over the real numbers. Equivalence class- for a specific time point t, let's say X(t) generates a fuzzy set that represents the temperature with respect to the confidence level α. Measurable sets in K(R), let's consider a measurable set A that represents a specific range of temperatures. Now, let's define the pre-image of A under [X(t)]^⍺: the pre-image of A under [X(t)]^⍺ is the set of all (i, ω) in I × Ω such that [X(t)]^⍺(i, ω) belongs to A. In other words, it consists of all the time points and sample outcomes for which the fuzzy set generated by X(t) at that particular time and outcome falls within the range represented by A. For example, let's assume A represents temperatures between 20 and 30 degrees Celsius. If [X(t)]^⍺ (2, B) represents a fuzzy set with membership values indicating temperatures around 25 degrees Celsius, then (2, B) would be in the pre-image of A because the temperature falls within the range represented by A. In this case, the pre-image of the measurable set A under [X(t)]^⍺ would be a subset of I × Ω containing all the time points and sample outcomes where the fuzzy sets generated by X(t) fall within the temperature range represented by A.

We propose that the condition that [X]^⍺ be measurable for all α in the interval [0, 1] ensures that the fuzzy stochastic process X can be integrated with respect to time and the sample space.

B(I) is the Borel σ-algebra, which consists of all subsets of the interval I. To determine measurability, [X]^α: I × Ω → K(R) should be B(I) ⊗ A-measurable. Here, B(I) ⊗ A represents the product σ-algebra, which is generated by the Borel σ-algebra B(I) and the σ-algebra A. This requirement ensures that the equivalence class [X]^α is measurable with respect to the joint σ-algebra generated by B(I) and A. In this context, K(R) denotes the set of fuzzy sets over the real numbers. Fuzzy sets are sets that assign membership degrees or degrees of truth to elements, allowing for a more flexible representation of uncertainty or partial truth. B(I) refers to the Borel σ-algebra, which is the σ-algebra generated by the open sets in the interval I. The Borel σ-algebra consists of all subsets of I that can be formed by taking unions, intersections, and complements of open sets. B(I) ⊗ A represents the product σ-algebra, which is the smallest σ-algebra generated by the sets of the form B × A, where B is a Borel set from B(I) and A is a set from the σ-algebra A. The requirement of [X]^α being B(I) ⊗ A-measurable ensures that the equivalence class [X]^α is measurable with respect to the joint σ-algebra generated by B(I) and A. This joint σ-algebra allows us to analyse and reason about the measurability properties of the fuzzy stochastic process X across both the time domain (Borel sets in I) and the probability space (sets in A). Suppose we have two sets: B(I), which represents the Borel σ-algebra generated by the open sets in the interval I, and A, which represents a σ-algebra defined on a sample space Ω. We want to generate a σ-algebra that captures the joint measurability of sets from B(I) and A. The product σ-algebra B(I) ⊗ A is defined as the smallest σ-algebra generated by sets of the form B × A, where B is a Borel set from B(I) and A is a set from the σ-algebra A. Let's consider a specific example to illustrate this concept. Assume that I is the interval [0, 1] and A is the power set of Ω = {a, b}, which contains all possible subsets of Ω. B(I): B(I) is the Borel σ-algebra on I, generated by the open sets in I. It includes sets such as open intervals, closed intervals, half-open intervals, and countable unions or intersections of such intervals. A: A is the power set of Ω, which includes all subsets of Ω. In this case, A = {∅, {a}, {b}, {a, b}}. To construct the product σ-algebra B(I) ⊗ A, we consider all possible combinations of Borel sets from B(I) and sets from A. For example, let's take B = (0.2, 0.8) (an open interval in I) and A = {a} (a singleton set in A). The set B × A is the Cartesian product of B and A, which results in the set {(x, a) | 0.2 < x < 0.8}. Similarly, we can consider other combinations of Borel sets from B(I) and sets from A. The product σ-algebra B(I) ⊗ A is then defined as the smallest σ-algebra that includes all such sets B × A for all possible choices of B from B(I) and A from A. In this example, the product σ-algebra B(I) ⊗ A would consist of sets such as B × A, where B represents Borel sets from B(I) and A represents sets from A. It captures the joint measurability of sets from B(I) and A, allowing us to reason about sets that involve both the real number domain and the sample space. Imagine we want to analyse rainfall data for different cities over time. We have a set of cities, say {London, Los Angeles, Paris}, and we want to consider different time intervals for our analysis. Our set of cities is like the interval I in our previous discussion. The time intervals could represent events or outcomes (such as days with rainfall data) in our probability space, denoted by the σ-algebra A. Now, we want to understand the measurability of rainfall data across both the cities and time intervals. This is where the product σ-algebra B(I) ⊗ A comes into play. For each city, we can create Borel sets, which are sets formed from open intervals. In this context, these could represent different subsets of time periods for each city. For example, we might have the Borel set for London's rainy days, the Borel set for Los Angeles' rainy days, and so on. σ-Algebra A - this represents our probability space, which includes various outcomes or events in our analysis. In this case, each outcome or event is a different time interval (e.g., a specific date or range of dates). Now, the product σ-algebra B(I) ⊗ A is the smallest σ-algebra that contains all possible combinations of sets formed by pairing a Borel set from B(I) with a set from σ-algebra A.

In our example, it means we're looking at measurable sets that are related to both specific cities (Borel sets) and specific time intervals (σ-algebra A). It allows us to answer questions like: "How does rainfall in London relate to specific time intervals, and how does it compare to rainfall in Los Angeles during those times?" So, B(I) ⊗ A helps us explore the measurability of events that occur at the intersection of these two domains: cities and time intervals.

A process X is said to be nonanticipating if, for every α ∈ [0, 1], the multifunction [X]^α is measurable with respect to the σ-algebra N. The σ-algebra N is defined as follows: N := {A ∈ B(I) ⊗ A : A^t ∈ A for every t ∈ I}, where A^t represents the set of outcomes ω such that the pair (t, ω) belongs to the set A. In other words, N consists of subsets of I × Ω for which the corresponding time t belongs to the set A.

Before proceeding further, we need to state a few theorems.

The Fubini's theorem [24], also known as the Fubini-Tonelli theorem, is a fundamental result in measure theory and integration theory. It provides a condition under which the iterated integrals of a measurable function over a product space can be computed by performing separate integrations. Formally, let (Ω₁, A₁, μ₁) and (Ω₂, A₂, μ₂) be two measure spaces, where Ω₁ and Ω₂ are sets, A₁ and A₂ are σ-algebras of subsets of Ω₁ and Ω₂, respectively, and μ₁ and μ₂ are measures defined on A₁ and A₂, respectively. Consider the product measure space (Ω, A, μ) defined as the product of the two measure spaces, where Ω = Ω₁ × Ω₂, A = A₁ ⊗ A₂ (the product σ-algebra), and μ is the product measure. Fubini's theorem states that if f: Ω → [0, +∞] is a non-negative measurable function, then the iterated integrals ∫Ω₁∫Ω₂ f(ω₁, ω₂) dμ₁(ω₁) dμ₂(ω₂) and ∫Ω₂∫Ω₁ f(ω₁, ω₂) dμ₂(ω₂) dμ₁(ω₁) are both well-defined and equal, provided at least one of them is finite. In other words, under appropriate conditions, the order of integration can be interchanged without affecting the result.

The Aumann integral, also known as the integral with respect to a fuzzy measure, is a generalization of traditional integration theory that applies to fuzzy sets. It was introduced by Robert J. Aumann, a mathematician and economist, to extend the concept of integration to accommodate uncertainty and partial truth [25]. In traditional integration theory, the integral of a function measures the "area under the curve" or the accumulation of a quantity over a given range. However, when dealing with fuzzy sets, where membership degrees represent degrees of truth or uncertainty, the concept of area or accumulation needs to be generalised. The Aumann integral addresses this by considering the integration of a fuzzy set with respect to a fuzzy measure. A fuzzy measure assigns a degree of "importance" or "size" to subsets of a set, similar to how a traditional measure assigns a value to subsets. The Aumann integral captures the concept of "accumulation" of fuzzy sets over a range of values. Formally, let X be a fuzzy set defined on a set Ω, and let μ be a fuzzy measure defined on the power set of Ω. The Aumann integral of X with respect to μ, denoted as ∫ X dμ, is a generalized notion of integration that takes into account the degrees of truth or membership of X. The Aumann integral is typically defined in terms of a level-wise integral. For each α in the interval [0, 1], the Aumann integral ∫ X dμ is equal to the integral of the α-cut of X (i.e., the subset of elements with membership degree at least α) with respect to the α-cut of the fuzzy measure μ. The Aumann integral allows for the incorporation of fuzzy sets and fuzzy measures into integration theory, enabling the quantification of accumulation and aggregation of fuzzy information. For example, let's consider a product space where X is the set of real numbers ℝ, and L is the lattice of closed intervals [a, b] on the real line, ordered by inclusion. Suppose we have a function f: ℝ × [0, 1] → ℝ defined as f(x, [a, b]) = x²;. Here, x represents a point in the real line, and [a, b] represents a closed interval. To compute the Aumann integral of f, we divide the lattice of closed intervals into levels based on the length of the intervals, such as intervals of length 1, intervals of length 1/2, intervals of length 1/4, and so on. We then integrate the function f(x, [a, b]) = x²; over each level, considering the measure on the real line. Finally, we combine these integrals according to lattice operations to obtain the Aumann integral. The Aumann integral of the function f(x,[a,b]) = x² over the product space X x L, where X is the set of real numbers and L is the lattice of the closed intervals on the real line is given by: where μ is the Lebesgue measure on the real line and λ is the counting measure on the lattice L. Since the counting measure gives a weight of 1 to each interval, the integral over L is simply the sum of the values of f(x,[a,b]) over all intervals [a,b] ϵ L. Therefore, the Aumann integral becomes:

The sum over all intervals [a,b] can be replaced by an integral over the real line, since the intervals are contiguous and non-overlapping.

Evaluating the inner integral, we get:

Evaluating the outer integral, we get:

The reason the Aumann integral is infinite lies in the nature of the counting measure and the unbounded function. For any given x, there are infinitely many intervals [a, b] in L that contain x. Since the function f(x, [a, b]) = x² depends only on x and is unbounded for large values of x, summing its value over infinitely many intervals will inevitably lead to an infinite result. Therefore, the Aumann integral of the function f(x, [a,b]) = x² over the product space X x L is infinite. To obtain a finite Aumann integral, we could restrict the lattice L to a subset of intervals with finite measure or specific properties.

Following works of [16], we will now use Fubini’s theorem and Aumann integral to build the fuzzy integral and discuss its properties. A fuzzy stochastic process X is called L^p -integrably bounded (p ≥ 1) if there exists a real-valued stochastic process h ∈ L^p(I × Ω, N; R+), where I is the interval and Ω is the probability space, such that the fuzzy set ||[X(t, ω)]⁰|| is bounded by h(t, ω) for almost all (t, ω) in I × Ω. The notation ∣∣[X(t, ω)]⁰∣∣ represents the magnitude or absolute value of the equivalence class [X(t, ω)]⁰. L^p(I × Ω, N; F(R)) denotes the set of nonanticipating and L^p-integrably bounded fuzzy stochastic processes. Nonanticipating means that the process X does not rely on future information, but only on the current and past information. By employing the Fubini theorem, the fuzzy integral of a fuzzy stochastic process X, denoted as ${\int }_0^T\mathrm{X}(\mathrm{s},\mathsf{\omega })\mathrm{d}\mathrm{s}$ is defined. Here, T is a fixed time point, and the integral is taken over the interval from 0 to T. The integral is defined for ω in the set Ω minus N_X, where N_X is a subset of Ω belonging to the σ-algebra A with P(N_X) = 0, meaning it has measure zero. The fuzzy integral ${\int }_0^T\mathrm{X}\left(\mathrm{s},\mathsf{\omega }\right)\mathrm{d}\mathrm{s}$can be defined level-wise. For each α in the interval [0, 1] and for each ω in the set Ω minus N_X, the Aumann integral ${\int }_0^T[\mathrm{X}(\mathrm{s},\mathsf{\omega }){]}^a\mathrm{d}\mathrm{s}$ belongs to the set K(R), which represents fuzzy sets over the real numbers. Therefore, the fuzzy random variable ${\int }_0^T\mathrm{X}(\mathrm{s},\mathsf{\omega })\mathrm{d}\mathrm{s}$ belongs to the set F(R) for every ω in the set Ω\N_X.

Now we define the fuzzy stochastic Lebesgue-Aumann integral of X ϵ L¹(I x Ω, N; F(R)).

We will now discuss properties of this integral L_X. First, we introduce some more terminology. The material comes from Kolmogorov [26] and Zadeh [23]. The d∞ metric, also known as the supremum or uniform metric, is a mathematical measure of dissimilarity or distance between fuzzy sets. It quantifies the maximum discrepancy between the membership degrees of corresponding elements in two fuzzy sets. Formally, let A and B be two fuzzy sets defined on a common universe of discourse. The d∞ metric between A and B, denoted as d∞(A, B), is defined as the supremum or maximum of the absolute differences in membership degrees:$d\infty (A, B)=sup |\mu A(x)-\mu B(x)|,$ where μA(x) and μB(x) represent the membership degrees of the element x in the fuzzy sets A and B, respectively. The supremum is taken over all elements x in the universe of discourse. The d∞ metric provides a measure of the largest difference in membership degrees between corresponding elements of two fuzzy sets. It captures the maximum discrepancy or dissimilarity between the two sets, indicating how much they differ in terms of their membership degrees. Now we will define L1 norm. L1 refers to a mathematical space or norm in the context of functions and integrals. Specifically, L1 represents a function space of integrable functions with a finite integral norm. Formally, L1 is a function space defined over a given measure space (Ω, A, μ). It consists of all measurable functions f: Ω → ℝ (or f: Ω → ℂ) such that the integral of the absolute value of f, denoted as ∫|f| dμ, is finite: $L1(\Omega , A, \mu )=\{f: \Omega \to \mathbb{R}(or\mathbb{C})|\int \left|f\right| d\mu <\infty \}$.

In simpler terms, L1 consists of functions whose absolute value is integrable over the measure space, meaning that the area under the curve of the absolute value function is finite. The norm associated with the L1 space, denoted as ‖f‖₁, is defined as the integral of the absolute value of f: ‖f‖₁ = ∫|f| dμ. The L1 norm measures the size or magnitude of the function f in terms of its integral, capturing how spread out or concentrated the function is. L1 represents a function space of integrable functions, where the absolute value of the function is integrable. The associated norm, ‖f‖₁, measures the size of the function based on its integral. In the context of stochastic processes, the L¹ space refers to a function space that consists of processes that are integrable with respect to a given measure. Formally, let (Ω, A, P) be a probability space and let X be a stochastic process defined on that space. The L¹ space, denoted as L¹(Ω, A, P), consists of all stochastic processes X that satisfy the condition of integrability. This means that the expected value or integral of the absolute value of X is finite: L¹(Ω, A, P) = {X: Ω → ℝ (or ℂ) | E[|X|] < ∞}. Here, E[·] denotes the expected value or the integral with respect to the probability measure P. The absolute value |X| ensures that both positive and negative values contribute to the integrability condition. In simpler terms, a stochastic process X belongs to the L¹ space if the expected value of its absolute value is finite. This indicates that the process is well-behaved in terms of its average behaviour and does not exhibit extreme fluctuations or infinite values. Now we define the L^P space. The L^P space, also known as a p-norm space, is a function space that consists of functions with finite p-norms. Formally, the L^P space, denoted as L^p(Ω, A, μ), is defined over a given measure space (Ω, A, μ). It consists of all measurable functions f: Ω → ℝ (or ℂ) such that the p-th power of the absolute value of f, denoted as |f|^p, is integrable with respect to the measure μ: L^p(Ω, A, μ) = {f: Ω → ℝ (or ℂ) | ∫|f|^p dμ < ∞}. Here, p is a positive real number greater than or equal to 1. The p-norm of a function f, denoted as ‖f‖^p, is defined as the p-th root of the integral of |f|^p: ‖f‖^p = (∫|f|^p dμ)^(1/p). The L^P space is a function space that captures functions whose p-th power is integrable, indicating that the function is well-behaved in terms of its magnitude and spread. The norm associated with the L^P space, the p-norm, measures the size or magnitude of the function f with respect to its integral, emphasizing the contribution of higher values due to the exponent p. The L^P spaces have various properties, including completeness, which means that every Cauchy sequence of functions converges to a limit within the space. Different values of p lead to different L^P spaces. For instance, when p = 1, we have the L¹ space, and when p = 2, we have the L² space. In summary, the L^P space consists of functions with finite p-norms, where the p-th power of the absolute value is integrable. It provides a framework for studying well-behaved functions in terms of their magnitudes and spread, with the p-norm quantifying their sizes. In the context of stochastic processes, the L^P space refers to a function space that consists of processes that are p-integrable with respect to a given measure. Formally, let (Ω, A, P) be a probability space and let X be a stochastic process defined on that space. The L^P space, denoted as L^p(I × Ω, A × N ; F(R)), consists of all stochastic processes X that satisfy the condition of p-integrability. This means that the p-th power of the absolute value of X, denoted as |X|^p, is integrable with respect to the product measure A × N: L^p(I × Ω, A × N ; F(R)) = {X: I × Ω → F(R) | ∫∫|X(t, ω)|^p dP(t, ω) < ∞}. Here, p is a positive real number greater than or equal to 1. The integral is taken over the product measure P, which is the joint measure defined on the σ-algebra generated by the Cartesian product of the intervals I and the probability space (Ω, A, P). The p-integrability condition ensures that the p-th power of the absolute value of the process X is well-behaved and has a finite integral.

Having defined integrability conditions, we can now expand the work of [16] with exploring the Lebesgue-Aumann integral. If X ∈ L^p(I × Ω, N ; F(R)) for p ≥ 1, then L_X(·, ·) ∈ L^P(I × Ω, N ; F(R)): This property states that if the fuzzy stochastic process X belongs to the L^P space, which consists of processes that are p-integrable, then the fuzzy stochastic Lebesgue-Aumann integral L_x, defined over the same probability space, also belongs to the L^P space. In other words, the integrability of X carries over to the integrability of L_x. Let X ∈ L¹(I × Ω, N ; F(R)), then {L_X(t)}_t∈I is d_∞-continuous. This property states that if the fuzzy stochastic process X belongs to the L¹ space, which consists of processes that are integrable, then the trajectories of the fuzzy stochastic Lebesgue-Aumann integral {L_x(t)}_t∈I are d_∞-continuous. In other words, the integral L_x(t) is a d_∞-continuous function of time t for each fixed outcome ω. This implies that the fuzzy integral exhibits continuity with respect to the d_∞ metric, capturing the dependence and smoothness of the integral over time. We will now define an inequality that provides a bound on the maximum difference between fuzzy sets L_t,x(u) and L_{t, y}(u) over the interval [0, t] in terms of the cumulative difference between fuzzy sets X(s) and Y(s) over the sample interval. The inequality captures the relationship between the supremum and integral of the d_∞ metric almost everywhere (a.e.). This inequality states that the supremum of the d_∞ metric between L_t,x(u) and L_{t, y}(u) for u in in the interval [0, t] is bounded by the scaled integral of the d_∞ metric between X(s) and Y(s) for s in the interval [0, t] with the scaling factor depending on the value of p.

The L.H.S. represents the supremum (maximum) of the d_∞ metric between L_t,x(u) and L_t,y(u), where t is a fixed value and u ranges from 0 to t. Here, d^p_∞(a, b) denotes the d_∞ metric between two fuzzy sets or elements a and b. d^p_∞(L_t,x(u), L_t,y(u)) is the d_∞ metric between the fuzzy sets L_t,x(u) and L_t,y(u) at each u. It measures the maximum difference in membership degrees between corresponding elements of the fuzzy sets at a given u. The R.H.S. represents an integral involving the d_∞ metric between the fuzzy sets X(s) and Y(s), where s ranges from 0 to t. The integral is scaled by t^p–1, where p is a positive real number. It is the integral of the d_∞ metric between X(s) and Y(s) at each s. It measures the cumulative difference in membership degrees between corresponding elements of the fuzzy sets over the interval [0, t]. t^p–1 scales the integral by a factor that depends on the value of p. The value of p determines how the integral contributes to the overall inequality. The inequality holds almost everywhere (a.e), meaning it holds for all values of t except possibly for a set of measure zero.

We will now define the fuzzy stochastic Itô integral by using fuzzy random variable ${\int }_0^{T}X\left(s\right)dW(s)$, where W is a Wiener process. Let X ∈ L2(I × Ω, N; R), then {$〈{\int }_0^{T}X\left(s\right)dW(s)〉$}, where t ∈ I. This represents a fuzzy stochastic process defined over the interval I. For each fixed t, $〈{\int }_0^{T}X\left(s\right)dW(s)〉$ is a fuzzy set. The process captures the evolution of these fuzzy sets as t varies. Here, dW(s) represents the differential of a Wiener process or Brownian motion. The next part indicates that the fuzzy stochastic process $〈{\int }_0^{T}X\left(s\right)dW(s)〉$ belongs to the L² space over the product space I × Ω. The fuzzy sets generated by the integral have finite second moments or variances and take values in the space of fuzzy sets over the real numbers, denoted as F(R). The last proposition states that if X belongs to a family of square integrables then the next integration will become fuzzy stochastic differential equation and then that integration also belongs to the family of square integrables. Formally, let X ∈ L²(I × Ω, N ; R), then X belongs to the L² space, which consists of processes that have finite second moments or variances. The process X takes real-valued values and $〈{\int }_0^{T}X\left(s\right)dW(s)〉,t\in I$ is d_∞-continuous. In other words, as t varies within the interval I, the fuzzy sets $〈{\int }_0^{T}X\left(s\right)dW(s)〉$ exhibit continuity with respect to the d_∞ metric. As discussed earlier, the d_∞ metric, also known as the supremum or uniform metric, measures the maximum difference in membership degrees between corresponding elements of fuzzy sets. Thus, the statement asserts that the fuzzy sets $〈{\int }_0^{T}X\left(s\right)dW(s)〉$, obtained by integrating X with respect to the Wiener process, show continuity with respect to the d_∞ metric as t varies. This continuity property is significant because it indicates that the fuzzy stochastic process $〈{\int }_0^{T}X\left(s\right)dW(s)〉,t\in I$ has a smooth and well-behaved evolution with respect to the d_∞ metric. It suggests that the variations in the fuzzy sets, induced by the integral, are gradual and predictable as t changes within the interval I. In summary, the given statement states that if the stochastic process X belongs to the L² space, then the fuzzy stochastic process $〈{\int }_0^{T}X\left(s\right)dW(s)〉,t\in I$, obtained by integrating X with respect to a Wiener process, exhibits continuity with respect to the d^∞ metric. This property implies a smooth and predictable evolution of the fuzzy sets as t varies.

We will now look at the special case of H> ½ and use Liouville form fBm [22] [27]. This is a type of fractional Gaussian process that is defined by a covariance function that exhibits a power-law scaling. Specifically, the covariance function is given by: $C(t, s)=|t-s{|}^{(2H-2)}$, where 0 < H < 1 is the Hurst exponent. Now, in this case, H > ½ for the process to have finite moments of order greater than or equal to two. If H < ½, the process has infinitely many moments and the covariance function becomes singular. This means that the covariance function doesn't satisfy the conditions of positive definiteness that are typically expected for covariance functions. When H> ½, Liouville fBM does not have mean-reversion. When H< ½, the fBm process does not possess the self-similarity and stationarity properties required for the Liouville form. A fractional Brownian motion with H < ½ is often referred to as "rough" or "pathologically irregular." It exhibits strong long-range dependence, which means that distant points in the process are highly correlated. In the Liouville form, fBm is expressed as a stochastic integral involving a deterministic function and a standard Brownian motion. It has the form [22] : where the FSDE is given by equation (10), X(t) is the solution of the equation at time t, X₀ is the initial condition of X at time 0, f(s, X(s)) is a function that describes the drift of X, and g(s, X(s)) is a function that describes the diffusion of X, and B_H is the Liouville fBm that drives the equation. The integral term in equation represents the stochastic integral of g(s, X(s)) with respect to B_H(s), which is a type of integral that is defined in terms of the Itô integral and the covariance structure of the fBm. The angle brackets around the integral indicate that it is taken in the sense of the Stratonovich integral. The corresponding approximation equation to (10) is given by X^ε(t), where X^ε(t) is an approximation to X(t) that is obtained by discretising the integral term using a numerical scheme. The integral term in the approximation equation is approximated using a discrete version of the stochastic integral, which is defined in terms of the Riemann sum and the covariance structure of the discretised fBm, denoted by BH^ε(s) [27].

This section is discussing a class of stochastic differential equations that are driven by a Liouville fBm, which is a type of fractional Gaussian process with a power-law covariance structure. Building on [16], we will now make a couple of assumptions to prove that (11) has a strong unique solution. First assumption(A1) is that for every u, v $\in$ F(R) and every t $\in$ I, there exists L > 0 such that max{d_∞(f(t, w,u), f(t, w, v)), |g(t, w, u) – g(t, w, v)|} ≤ Ld_∞(u,v). This assumption helps estimate upper bound using constant L. The second assumption (A2) is the Cauchy-Schwarz inequality. Suppose there is C > 0 such that max{d_∞(f(t, w,u⟨0⟩), |g(t, w, u)| } ≤ C(1+d_∞(u,⟨0⟩). Last assumption (A3) is that the mapping f: I x Ω x F(R) -> F(R) and g: I x Ω x F(R) -> F(R) are measurable with respect (w.r.t) to Standard Brownian motion. Formally, N $\otimes$ B_ds|B_ds - measurable and N $\otimes$ B_ds|B(R) – measurable, respectively. The product sigma-algebra is a way to combine two sigma-algebras (sets of events) from different sources. In this work, N represents the sigma-algebra associated with the underlying probability space, which captures the randomness or uncertainty inherent in the system. B_ds|B_ds denotes the sigma-algebra generated by the Brownian motion process B_ds. It contains all the events or information that can be described solely in terms of the Brownian motion process. The symbol ⊗ represents the product operation between sigma-algebras, which creates a larger sigma-algebra that combines the information from both sources. The product sigma-algebra N ⊗ Bds|Bds captures the combined information from the underlying probability space and the Brownian motion process. It allows us to consider events or random variables that depend on both the inherent randomness of the system (represented by N) and the randomness introduced by the Brownian motion process (represented by Bds|Bds).

This construction is important in the theory of stochastic processes and stochastic differential equations because it provides a way to define and work with stochastic integrals, which are integrals with respect to Brownian motion or other stochastic processes. The measurability condition (A3) in the context provided ensures that the drift and diffusion functions are well-defined and can be integrated with respect to the Brownian motion process, taking into account both the underlying probability space and the Brownian motion itself.

Next, we will consider the bound on the distance between two stochastic integrals involving Wiener process W(s). Let X(s) and Y(s) be two stochastic processes and let d²_∞ (·, ·) denote L2 distance between two processes. Then by Burkholder and Gundy inequality [28]:

Here, 〈·〉 denotes the quadratic variation of a stochastic process, which is a measure of the total variation of the process. The integral ${\int }_0^{u}X\left(s\right)dW\left(s\right)$represents the stochastic integral of X(s) with respect to the Wiener process up to time u. The inequality states that the distance between the stochastic integrals of X(s) and Y(s) with respect to W(s) is bounded by a multiple of the integral of their L₂ distance over time. The constant 4 in the inequality is a universal constant that arises from the properties of the Wiener process. (12) is a special case of the Burkholder-Davis-Gundy (BDG) inequality which we will explore further down the line and on which we will rely in other proofs. We will be using (12) and the assumptions above to show that (11) has a strong unique solution. We need to establish two points: 1. Existence of a strong solution. There exists a process X(t) that solves the FSDE a.s. for all t ≥ 0. 2. Uniqueness of the strong solution. The solution X(t) is unique a.s. Using equation (5) and integration by parts technique, we can re-write (11): where ${\phi }^{\epsilon }\left(s\right)={\int }_0^t{\left(t-s+\epsilon \right)}^{a-1}]dB(s)$, $\epsilon >0,a=H-1/2$.

The proof begins by introducing the Picard iterations [24], denoted as $X_n^{\epsilon }(t)$, which are obtained by iteratively applying the equation $X_n^{\epsilon }\left(t\right)=X_0+{\int }_0^{t}f\left(s, X_{n-1}^{\epsilon }\left(s\right)\right)ds+〈{\int }_0^t(a{\phi }^{\epsilon }(s)g\left(s, X_{n-1}^{\epsilon }\left(s\right)\right)ds+{\int }_0^t{\epsilon }^{a}g(s, X_{n-1}^{\epsilon }\left(s\right)dW(s)〉$, almost everywhere (a.e.,). These iterations aim to find an approximation for the solution of the equation. To approximate X(t), the Picard iteration scheme defines a sequence of iterates $X_n^{\epsilon }\left(t\right)$ that should converge to X(t) as n goes to infinity. Starting with $X_0^{\epsilon }\left(t\right)=X_0$, we calculate $X_1^{\epsilon }\left(t\right)$ by plugging X₀ into the right-hand side. Then $X_2^{\epsilon }\left(t\right)$ is calculate using $X_1^{\epsilon }\left(t\right)$ and so on. The key idea is that as n increases, the sequence $X_n^{\epsilon }\left(t\right)$ should get closer and closer to the true solution X(t), provided some condition on f, g and the initial data are satisfied to ensure convergence. The parameters ε and a are introduced to control the convergence behaviour, while φ is some approximating kernel. The Picard iterations provide a recursive way to get successively better approximations to the original equation’s solution by iteratively updating an integral equation approximation. Define the sequence j_n(t) = $E\underset{u\in [0,t]}{\sup }{d}_{\infty }^2(X_n^{\epsilon }\left(u\right),X_{n-1}^{\epsilon }\left(u\right))$. The sequence measures the distance between two consecutive iterations. When we take n=1, and by the assumption that there is C>0 such that $max\{d\infty (f(t, w,u⟨0⟩), |g(t, w, u)| \} \le C(1+d\infty (u,⟨0⟩$) and from Theorem 4.1 in Llendeto (2007), we know that:

Therefore, we have a constant 3 in: