brownian motion
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2022 ◽  
Vol 2022 ◽  
pp. 1-8
Longjin Lv ◽  
Changjuan Zheng ◽  
Luna Wang

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed-form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

2022 ◽  
Vol 9 ◽  
Han Gao ◽  
Rui Guo ◽  
Yang Jin ◽  
Litan Yan

Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

Yuk Leung

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

Aleksander A Stanislavsky ◽  
Aleksander Weron

Abstract Stochastic resetting with home returns is widely found in various manifestations in life and nature. Using the solution to the home return problem in terms of the solution to the corresponding problem without home returns [Pal et al. Phys. Rev. Research 2, 043174 (2020)], we develop a theoretical framework for search with home returns in the case of subdiffusion. This makes a realistic description of restart by accounting for random walks with random stops. The model considers stochastic processes, arising from Brownian motion subordinated by an inverse infinitely divisible process (subordinator).

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