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Consider an n-dimensional Ito process $$ X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dB^H(s), $$ where $1/3<H<1$ is the Hurst parameter for an $n$-dimensional fractional Brownian motion $(B^H(t))_{t\ge 0}$, and $\alpha,\beta$ are Lipschitz maps taking values in $\mathbb{R}^n$ and $\mathbb{S}(n)$. Here, $\mathbb{S}(n)$ denotes the set of $n\times n$ symmetric positive definite matrices and $x\in \mathbb{R}^n$.

Consider its solution operator $x\mapsto X_t^x$. Is this map (locally) Lipschitz in $x$ (for any fixed $t>0$)? Also, is it clear that $X_t^x$ is Gaussian?

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    $\begingroup$ The solution is Lipschitz continuous with respect to the initial condition with unit Lipschitz constant, because the noise and drift are both additive. Since $\alpha$ and $\beta$ are deterministic, $X_t^x$ is Gaussian, because the (Riemann-Stieltjes) integral of a deterministic function with respect to a fBM is Gaussian. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Jan 29, 2023 at 21:03
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    $\begingroup$ @NawafBou-Rabee Do you have a reference for the fact that the integral against an fBM of a deterministic function is Gaussian; and more importantly, what is the distribution of $X_t^x$. If $H=1/2$ I know how to compute this, but I have no idea in general. $\endgroup$
    – PhD_InStochastics
    Commented Jan 29, 2023 at 23:53
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    $\begingroup$ Here is how I think about. For simplicity, suppose $n=1$ and the integrand $\beta$ is a deterministic $C^1$-function. Let $I_t:=\int_0^t \beta(s) dB^H(s)$ and $R_{t,s} := E \{ B_t^H B_s^H \}$. Then any reasonable definition of the fractional stochastic integral would satisfy $I_t = \beta(t) B_t^H - \int_0^t \beta’(s) B_s^H ds$, and since $B^H$ is continuous (hence Riemann integrable) and furthermore starts at zero, $I_t$ is Gaussian with mean zero and variance $\beta(t)^2 R_{t,t} - 2 \int_0^t R_{s,t} \beta(t) ds + \int_0^t \int_0^t \beta’(s) \beta’(r) R_{s,r} ds dr$. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Jan 30, 2023 at 15:23

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As explained in the comments the Lip-continuity just follows from the linearity.

Similarly, for Gaussian, we can use that $\beta\in C^{1}$ and we can define the Young integral which is an approximation of sums of Gaussian eg. see "Stochastic differential equations driven by fractional Brownian motions".

The things below are for worse regularity for $b$ and even of the form $b(t,X_{t})$.

For the range $(1/2,1]$ as explained in the comments Gaussian follows by the linearity and definition of Young integral. "Stochastic differential equations driven by fractional Brownian motions"

enter image description here

See here for more general setting "Continuous dependence of solutions of stochastic differential equations driven by standard and fractional Brownian motion on a parameter"

For $H=1/2$, you can similarly use the Ito integral formulation and its approximation by sums of Gaussians see here.

For the $(1/3,1,2)$ or $(1/4,1/3)$, however one needs to use rough path lifts, enter image description here

which might not be Gaussian anymore eg.

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    $\begingroup$ If the integrand is deterministic and $C^1$, is the stochastic integral for $H<1/2$ really undefined? One can actually take as definition: $I_t := \beta(t) B_t^H - \int_0^t \beta’_s B_s^H ds$, which is a centered Gaussian random variable. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Jan 31, 2023 at 14:54
  • $\begingroup$ thank you, I updated. $\endgroup$
    – Thomas Kojar
    Commented Jan 31, 2023 at 19:20

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