Markovian Lifts and Multifactor Approximations for Weighted Sub-fractional Brownian Motion
Abstract
We construct a Hilbert-space Markovian lift of weighted sub-fractional Brownian motion for 0<b<2, including the logarithmic case b=1. A Laplace factorization represents the process as the boundary value of a forward curve. In the weighted Sobolev space with derivative weight rγ/(1+r)2, 0<γ<1, right translation is strongly continuous and the noise operator is Hilbert--Schmidt if and only if b+γ>1. The state norm bounds the total variation of initial curves on compact intervals. For locally bounded deterministic temporal weights, this bound allows the driven scalar equation to be restarted from every lifted state. Under spatial Lipschitz assumptions, with additional time and derivative regularity in the Young case, the solution and forward curve form a time-inhomogeneous Markov process with Cb-Feller transitions; both coefficients may have linear growth. Discretization of the Laplace variable gives independent Ornstein--Uhlenbeck factors coupled to the original process through the same Gaussian noise. The nonlinear approximations converge uniformly in time in every finite Lp. For uniform spectral partitions with balanced cutoffs, the solution rate is M−(b−1)/3 for 1<b<2, while the uniform driver rate is M−b/4√(log(2+M)) for 0<b≤1. Under stronger coefficient assumptions, we derive error bounds for Young solutions and for the fully discrete scheme, retaining the dependence on the spectral cutoff and specifying joint mesh conditions.
Related
- Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation
- Convergence in Hölder norms for Markovian approximations of stochastic Volterra equations
- Strong Measurability and Adapted Approximations with Applications to Non-Markovian Control
- Limit theorems for supercritical branching processes in Markovian environment with linear fractional offspring distributions
- Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion
BibTeX
@misc{ramirezgonzalez2026markovian,
title = {{Markovian Lifts and Multifactor Approximations for Weighted Sub-fractional Brownian Motion}},
author = {J. H. Ramirez-Gonzalez and Edison Cuba},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03926},
eprint = {2611.03926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03926},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Markovian Lifts and Multifactor Approximations for Weighted Sub-fractional Brownian Motion
AU - J. H. Ramirez-Gonzalez
AU - Edison Cuba
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03926
UR - https://arxiv.org/abs/2611.03926
AB - We construct a Hilbert-space Markovian lift of weighted sub-fractional Brownian motion for 0<b<2, including the logarithmic case b=1. A Laplace factorization represents the process as the boundary value of a forward curve. In the weighted Sobolev space with derivative weight r^γ/(1+r)^2, 0<γ<1, right translation is strongly continuous and the noise operator is Hilbert--Schmidt if and only if b+γ>1. The state norm bounds the total variation of initial curves on compact intervals. For locally bounded deterministic temporal weights, this bound allows the driven scalar equation to be restarted from every lifted state. Under spatial Lipschitz assumptions, with additional time and derivative regularity in the Young case, the solution and forward curve form a time-inhomogeneous Markov process with C_b-Feller transitions; both coefficients may have linear growth. Discretization of the Laplace variable gives independent Ornstein--Uhlenbeck factors coupled to the original process through the same Gaussian noise. The nonlinear approximations converge uniformly in time in every finite L^p. For uniform spectral partitions with balanced cutoffs, the solution rate is M^−(b−1)/3 for 1<b<2, while the uniform driver rate is M^−b/4√(log(2+M)) for 0<b≤1. Under stronger coefficient assumptions, we derive error bounds for Young solutions and for the fully discrete scheme, retaining the dependence on the spectral cutoff and specifying joint mesh conditions.
ER -