# Fuzzy Fractional Brownian Motion: Review and Extension
## 1 Introduction  
Traditional option pricing models rely on precise, fixed values for variables like expected return and volatility. However, these models fail to account for the inherent uncertainty surrounding these parameters. To address this limitation, recent approaches have incorporated fuzziness into the calculations, allowing investors to estimate option prices based on their individual risk tolerance and beliefs about the potential range of values for these variables. The aim of this paper is two-fold: first, to review the state-of-the-art methods in fuzzy fractional Brownian motion and prove the theoretical correctness of this model, and, second, to extend many novel proofs and theoretical considerations presented in the recent works of the other authors. In future work, we will present examples and applications and introduce a novel proof dealing with the constraint of h << %. This will pave the way to defining fuzzy fractional Brownian motion for h << %. In this paper, we employ a narrative review methodology to synthesise existing research and to identify gaps where our proposed model can offer improvements. This allows us to provide a comprehensive overview of the literature and enables expert interpretation and synthesis, which are essential for extending existing theoretical frameworks and offering novel proofs in this area. This paper delves into a unified framework that merges fractional Brownian motion with fuzzy processes, creating a combined measure space. We begin by constructing a product measure space that encompasses both the randomness inherent in fractionalBrownian motion and fuzziness of the associated parameters. The underlying asset price \(S_{t}\) is assumed to follow the following crisp dynamics:  
\[dS/S=\mu dt+\sigma dB(t)+\sigma dB_{t}^{H}+d\left(\sum_{1}^{N}(V_{i}-1)\right).\]  
In this equation, the terms represent various components: (i) the drift parameter \(\mu\) represents the expected return rate on the value of the underlying asset; (ii) the volatility parameter \(\sigma\) represents the standard deviation of the return rate on the value of the underlying asset; (iii) the term dB(t) represents a standard Brownian motion; (iv) the term \(\mathrm{dB}^{H}\)(t) represents an independent standard fractional Brownian motion with respect to a fractional Brownian motion parameterised by the Hurst exponent \(H\); (v) the term \(\mathrm{d}(\Sigma(V_{i}-1))\) represents a sum of jumps, where \(\{V_{i},\,i\geq 1\}\) is a sequence of independent and identically distributed non-negative random variables. The logarithm of \(V_{i}\), denoted as \(\gamma=\log(V)\), follows a normal distribution with a probability density function [1]:  
\[f_{y}(x)=\frac{1}{\sqrt{2\pi}\sigma_{t}}e^{-\frac{(x-y_{i})^{2}}{2\sigma_{t}^{ 2}}}\]  
The expected return rate \(\mu\), volatility \(\sigma\), jump intensity \(\lambda\), and jump magnitudes sequence \(\mathrm{Vi}\) in the underlying asset price dynamics cannot always be treated as constant real numbers. Therefore, a new model is proposed to incorporate both randomness and fuzziness in describing the dynamics of the process. When dealing with processes that involve non-standard or non-numeric variables, such as fuzzy numbers or other types of uncertain quantities, the traditional tools of real analysis may not be directly applicable. In such cases, alternative mathematical frameworks or extensions of real analysis are often employed to handle the specific characteristics and uncertainties present in the financial domain. In the next sections, we prove the no-arbitrage condition in the presence of uncertain or non-standard variables. This involves extending the mathematical framework to accommodate the specific characteristics of the variables under consideration. Ref. [2] conducted a literature review and categorised more than fifty papers on option pricing in a fuzzy setting, providing insights into different approaches and best practices for utilising fuzziness in option pricing. Refs.