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 $r^γ/(1+r)^2$, $0<γ<1$, 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 $M^{-b/4}\sqrt{\log(2+M)}$ for $0<b\le1$. 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.