# Fuzzy Fractional Brownian Motion: Review and Extension
## 4 Conclusions  
This paper reviews a new approach to option pricing that incorporates fuzziness or uncertainty in parameters like expected return rate, volatility, jump intensity, and jump magnitudes. Traditional option pricing models like Black-Scholes assume that these parameters are crisp values. The proposed model treats them as fuzzy numbers to allow participants to estimate option prices based on their risk preferences and beliefs about the uncertain parameters. In Section 2, we build on [15] to show an alternative proof to extend [PERSON]'s jump-diffusion model for underlying asset price dynamics to include fractional Brownian motion and fuzzy processes. The asset price follows a stochastic differential equation driven by Brownian motion, fractional Brownian motion, and a fuzzy jump process. Detailed mathematical derivations are provided to incorporate the fuzziness into the asset price dynamics using fuzzy arithmetic and fuzzy stochastic calculus. An expectation formula is derived for the fuzzy asset price process that depends on the fuzzy drift, volatility, jump parameters, and the expectation of the log jump sizes. The article builds on and extends prior work on fuzzy option pricing models. We start by considering the usual probability space (\(\Omega\), F, P) that captures the randomness via the probability measure P on the sample space \(\Omega\) and a filtration \(\{\mathrm{Ft}\}\). This represents the standard stochastic/Brownian motion components. To incorporate fuzziness, we propose to construct a product measure space by taking the product of the above probability space with another measure space that captures the fuzziness inherent in model parameters like expected return rate \(\mu\), volatility \(\sigma\), jump intensity \(\lambda\), etc. Essentially, we represent the fuzzy parameters using fuzzy numbers or fuzzy sets defined on appropriate measure spaces. By taking products of these measure spaces with the original probability space, we obtain a new joint \"product measure space\" that simultaneously captures both the randomness from the stochastic processes and the fuzziness from the fuzzy parameters. On this product space, we can then define stochastic processes that have fractional, jump-diffusion dynamics given by SDEs driven by Brownian motions, but where the coefficients (drift, volatility, etc.) are now fuzzy numbers instead of constants. The use of product measure spaces allows us to apply fuzzy arithmetic, fuzzy calculus, and fuzzy stochastic analysis on these mixed \"fractional fuzzy\" processes in a theoretically consistent manner within the standard measure-theoretic probability framework. In summary, the key idea is to start with classical probability spaces for the stochastic components, construct separate measure spaces for the fuzzy parameters, and then form product measure spaces that merge the randomness and fuzziness into a unified model for asset price dynamics. This enables extending stochastic models like fractional Brownian motion to incorporate fuzzy coefficients and parameters. The hybrid systems modelling approach presented in this paper is particularly suited for domains and subfields where systems exhibit long-range dependence and fuzziness. In the context of hybrid systems, fuzziness typically refers to situations where imprecision is inherent. This can be due to various factors, such as variability in data, lack of precise information, subjective judgement, or complex underlying processes. The areas where this framework can be applied include financial risk management, telecommunications and network management, insurance modelling, and healthcare. The theoretical extensions proposed in this manuscript require practical validation and testing in empirical studies. It is of course entirely plausible that other models are equally capable of modelling the uncertainty of the processes and we emphasise that the point of this manuscript is not to show a superior approach to others.