Tail distribution of the integrated Jacobi diffusion process

Nguyen Tien Dung,Trinh Nhu Quynh

doi:10.19139/soic-2310-5070-760

Abstract

In this paper, we study the distribution of the integrated Jacobi diffusion processes with Brownian noise and fractional Brownian noise. Based on techniques of Malliavin calculus, we develop a unified method to obtain explicit estimates for the tail distribution of these integrated diffusions.

Highlights

IntroductionWhere the initial conditions X0 ∈ (0, 1), a, b, c are positive constants and Wt is a standard Brownian motion
We consider Jacobi diffusion process that is defined as the solution of the scalar stochastic differential equation ∫t ∫t √Xt = X0 + (a − bXs)ds + c Xs(1 − Xs)dWs, (1)where the initial conditions X0 ∈ (0, 1), a, b, c are positive constants and Wt is a standard Brownian motion
The Jacobi process was first used by De Jong et al [8] to model the exchange rates in a target zone and by Delbaen and Shirakawa [3] to model interest rates

Summary

Introduction

Where the initial conditions X0 ∈ (0, 1), a, b, c are positive constants and Wt is a standard Brownian motion. Recall that fractional Brownian motion (fBm) with Hurst index H ∈ (0, 1) is a centered Gaussian process W H = (WtH )t∈[0,T ] with covariance function: R(t, s) = 1 (|t|2H + |s|2H − |t − s|2H ). We will focus on providing the estimates for the tail distribution of Yt and YtH. The main tools of the present paper are the techniques of Malliavin calculus (stochastic calculus of variations) which have been successfully used to investigate many financial models, see e.g. Chapter 6 in [12]. By using the flexible transforms, we are able to bound Malliavin derivatives of Yt and YtH , and we obtain explicit estimates for the tail distributions. In Theorem 3.2, we obtain two explicit estimates (19) and (20) for the tail distribution of YtH.

Preliminaries

Jacobi process with Brownian noise

Jacobi process with fractional Brownian noise

Findings

Conclusion