Fuzzy Fractal Brownian Motion: Extensions and Applications
Authors: Urumov Georgy , Chountas Panagiotis , Chaussalet Thierry 
Published: January 2025

Over the course of this research, we have developed a multifaceted framework for pricing derivatives and modelling asset dynamics that integrates fuzzy set theory, fractional Brownian motion (fBm), jump processes, and classical no-arbitrage principles under a risk-neutral measure. This approach does not merely extend one well-known model. It synthesises multiple advanced elements from mathematical finance and stochastic calculus into a single comprehensive structure.
We began by defining an SDE in which fuzzy concepts (upper/lower functions for membership levels) are combined with fractional Brownian motion having Hurst exponent H &gt; ½. Standard fractional finance work typically involves only a fractional Brownian model or a deterministic fuzzy model, but not both simultaneously. Our step of allowing the SDE’s drift and diffusion terms to depend on fuzzy processes introduces extra flexibility in capturing uncertainty. This alone is more advanced than most single-parameter fractional or fuzzy finance models.
Instead of using a purely classical PDE ∂_t u+Lu=0, we designed a custom PDE with embedded exponential and hyperbolic functions for negative drift. This is an extension of standard PDE–SDE bridging via Feynman–Kac which typically places all operators on the left-hand side, ensuring a known diffusion generator. Our method is a hybrid approach, creating a direct path to solving an enriched PDE that captures both fractional behaviour through exponents involving σt^2H and fuzzy membership constraints through special boundary or integral conditions.
We then invoke Girsanov’s theorem to shift from the “real-world” measure P to a “risk-neutral” measure Q. This step is typical in modern finance, but we apply it to a fuzzy fractional mixed jump environment. We demonstrate that the drift changes while the diffusion remains unaffected, combined with our no-arbitrage condition μ=r−λ(J−1), it ensures consistency with fundamental asset pricing principles. This bridging of fuzzy fractional assets to classical “risk-neutral” arguments is innovative because standard fuzzy finance seldom performs measure change in this manner.
Additionally, we draw on Cheridito’s mixed fractional Brownian framework, referencing the condition H&gt;¾ for certain regularity or H&gt; ½ for existence-uniqueness. We then incorporate Poisson i.i.d. jump sequences (Vi) to capture jump risk. This yields an SDE or PDE that unifies continuous fractional noise, discrete jumps, and fuzzy uncertainty. Such synergy between fractional noise, fuzzy membership, and jumps is rarely attempted, making our model significantly more unified than classical jump-diffusions or single-parameter fractional SDEs.
Our final expressions for option pricing iterated expectations with respect to σ(∑Yi), integrals involving Φ(d1) and Φ(d2), and summations over Poisson counts reflect an extension of the standard Black–Scholes–Merton logic. By combining fuzzy membership, fractional exponents, jump expansions under the risk-neutral measure, we arrived at a unique closed-form payoff integral. In other words, there is only one semimartingale (or fuzzy fractional path) that satisfies our SDE/PDE.