fractional brownian motion
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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.

2022 ◽  
Vol 0 (0) ◽  
Sadibou Aidara ◽  
Ibrahima Sane

Abstract This paper deals with a class of deplay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {\frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout this paper is the divergence-type integral.

2022 ◽  
Vol 4 (1) ◽  
pp. 15-30
T. Moussa ◽  
Ba Demba Bocar ◽  
D. Bou

In this paper, we study some models without jumps of stochastic differential equations directed by a fractional Brownian motion.

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Panhong Cheng ◽  
Zhihong Xu

A new framework for pricing European vulnerable options is developed in the case where the underlying stock price and firm value follow the mixed fractional Brownian motion with jumps, respectively. This research uses the actuarial approach to study the pricing problem of European vulnerable options. An analytic closed-form pricing formula for vulnerable options with jumps is obtained. For the purpose of understanding the pricing model, some properties of this pricing model are discussed in the paper. Finally, we compare and analyze the pricing results of different pricing models and discuss the influences of basic parameters on the pricing results of our proposed model by using numerical simulations, and the corresponding economic analyses about these influences are given.

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Weijun Ma ◽  
Wei Liu ◽  
Quanxin Zhu ◽  
Kaibo Shi

This paper examines the dynamics of the exponential population growth system with mixed fractional Brownian motion. First, we establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mixed fractional Brownian motion. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mixed fractional Brownian motion. Second, we propose two sufficient and necessary conditions for the almost sure exponential stability and the k th moment exponential stability of the solution of the constant coefficient exponential population growth system with mixed fractional Brownian motion. Furthermore, we conduct some large deviation analysis of this mixed fractional population growth system. To the best of the authors’ knowledge, this is the first paper to investigate how the Hurst index affects the exponential stability and large deviations in the biological population system. It is interesting that the phenomenon of large deviations always occurs for addressed system when 1 / 2 < H < 1 . Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.

2021 ◽  
Vol 5 (1) ◽  
pp. 371-379
Nguyen Thu Hang ◽  
Pham Thi Phuong Thuy ◽  

The aim of this paper is to study the tail distribution of the CEV model driven by Brownian motion and fractional Brownian motion. Based on the techniques of Malliavin calculus and a result established recently in [<a href="#1">1</a>], we obtain an explicit estimate for tail distributions.

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