A note on decompositions of the stochastic convolution driven by a white-fractional Gaussian noise

2019/12/08 by Ran Wang, Shiling Zhang, Wang, Ran +1
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1912.03684

openalex publication_date 2019/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let u = \u(t, x); (t,x)∈ \mathbb R₊× \mathbb R\ be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameter H∈(0, 1). For any given x∈ \mathbb R (resp. t∈ \mathbb R₊), we show a decomposition of the stochastic process t↦ u(t,x) (resp. x↦ u(t,x)) as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp. H) and a stochastic process with C^∞-continuous trajectories. Some applications of those decompositions are discussed.

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