Stochastic integration with respect to the sub-fractional Brownian motion with

2012 ◽  
Vol 82 (2) ◽  
pp. 240-251 ◽  
Author(s):  
Guangjun Shen ◽  
Chao Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Integration

On Delayed Logistic Equation Driven by Fractional Brownian Motion

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Nguyen Tien Dung
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Approximate Method ◽  
Explicit Expression ◽  
Stochastic Integral ◽  
Logistic Equation ◽  
The Other ◽  
Stochastic Integration

In this paper we use the fractional stochastic integral given by Carmona et al. (2003, “Stochastic Integration With Respect to Fractional Brownian Motion,” Ann. I.H.P. Probab. Stat., 39(1), pp. 27–68) to study a delayed logistic equation driven by fractional Brownian motion which is a generalization of the classical delayed logistic equation. We introduce an approximate method to find the explicit expression for the solution. Our proposed method can also be applied to the other models and to illustrate this, two models in physiology are discussed.

STOCHASTIC INTEGRATION FOR FRACTIONAL BROWNIAN MOTION IN A HILBERT SPACE

2006 ◽  
Vol 06 (01) ◽  
pp. 53-75 ◽  
Author(s):  
T. E. DUNCAN ◽  
J. JAKUBOWSKI ◽  
B. PASIK-DUNCAN
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Malliavin Calculus ◽  
Stochastic Integration ◽  
Smooth Functions ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Stochastic Operator ◽  
Definition Of

A Hilbert space-valued stochastic integration is defined for an integrator that is a cylindrical fractional Brownian motion in a Hilbert space and an operator-valued integrand. Since the integrator is not a semimartingale for the fractional Brownian motions that are considered, a different definition of integration is required. Both deterministic and stochastic operator-valued integrands are used. The approach uses some ideas from Malliavin calculus. In addition to the definition of stochastic integration, an Itô formula is given for smooth functions of some processes that are obtained by the stochastic integration.

scholarly journals An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yong Xu ◽  
Bin Pei ◽  
Yongge Li
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Path Integrals ◽  
Delay Equations ◽  
Averaging Principle ◽  
Stochastic Integration ◽  
Differential Delay ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Convergence In Mean

An averaging principle for a class of stochastic differential delay equations (SDDEs) driven by fractional Brownian motion (fBm) with Hurst parameter in(1/2,1)is considered, where stochastic integration is convolved as the path integrals. The solutions to the original SDDEs can be approximated by solutions to the corresponding averaged SDDEs in the sense of both convergence in mean square and in probability, respectively. Two examples are carried out to illustrate the proposed averaging principle.

scholarly journals FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE

2003 ◽  
Vol 06 (01) ◽  
pp. 1-32 ◽  
Author(s):  
YAOZHONG HU ◽  
BERNT ØKSENDAL
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Standard Brownian Motion ◽  
Stochastic Integration ◽  
European Option ◽  
No Arbitrage ◽  
Black Scholes ◽  
White Noise Calculus ◽  
Replicating Portfolio

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Itô type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Itô type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Itô fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Itô fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

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