On a stochastic heat equation with first order fractional noises and applications to finance

Abstract

a b s t r a c tIn this paper, we propose a class of stochastic heat equations with first order fractional noises. We define a first order noise through the adjoint operator of the first order operator, where the operation of the stochastic integral can be avoided. In this framework, the existence and uniqueness of the solution of the equation will be established. Further, we give the regularity of the solution. Finally, we model the term structure of forward rate with the solutions and give the conditions under which the market is arbitrary-free.

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

“…A stochastic generalized Burgers equations driven by fractional noises has been studied in [24]. For more contributions about stochastic calculus with fractional noise, we refer the reader to [4,9,17,20,22,23,28,31] and the references therein.…”

Convergence of numerical solutions for a class of stochastic age-dependent capital system with fractional Brownian motion

“…In the last few years, some articles have been published choosing fractional Brownian motion (fBm) as an underlying diffusive process (e.g., Refs. [7][8][9][14][15][16][17] and references therein). For example, Jiang et al [9] proposed a class of stochastic heat equations with first order fractional noises and established the existence and uniqueness of the solution of the equation.…”

“…[7][8][9][14][15][16][17] and references therein). For example, Jiang et al [9] proposed a class of stochastic heat equations with first order fractional noises and established the existence and uniqueness of the solution of the equation. Wang et al [15] studied the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fBm as a model of asset prices.…”

Mixed Fractional Heat Equation Driven by Fractional Brownian Sheet and Lévy Process

“…The motivation comes from wide applications of fBm. We refer, among others, to Duncan et al [1], Hu [2], Jiang et al [3,4], Liu and Yan [5], Sobczyk [6], Tindel et al [7], Mishura et al [8], and the references therein. On the other hand, as is well known, SPDEs driven by Lévy noise constitute a very important research direction and many significant researches have been carried out.…”