Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, with $\Sigma(x)$ always a symmetric positive-definite matrix. Fix a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,\infty)},\mathbb{P})$ (satisfying the usual conditions) and supporting an $n$-dimensional Brownian motion $(W_t)_{t\in [0,\infty)}$. Define the $(\mathcal{F}_t)_t$-adapted process $(X_t)_{t\in [0,\infty)}$ as being the unique strong solution to: $$ X_t = x + \int_0^t [M_s^1+M_s^2X_s] ds + \int_0^t \Sigma_s dW_s $$

It is easy to show that (by discrizing, using the fact that the affine transformations of Gaussians is gaussian, and that Gaussianity is preserved under the relevant limits... well summarized in this post) under these conditions the marginals: $$ \mu_t(B):=\mathbb{P}\left(X_t\in B \right) \qquad B\in \mathcal{F}, $$ are non-degenerate and Gaussian.

Now, let $m:[0,\infty)\to \mathbb{R}^n$ and $\sigma:[0,\infty)\rightarrow \mathbb{R}^{n\times n}$ be the functions which map a time $t$ to the respective mean and covariance of $\mu_t$. My question is, is it true that (under mild conditions) $m$ and $\sigma$ are at-least $C^k$-functions?

[If so, does anyone know a reference to this (likely fact)?]


1 Answer 1


As (assuming $\Sigma\in L^2([0,\infty))$ $$ m_t=\mathbb{E}[X_t] = x + \int_0^t [M_s^1+M_s^2\mathbb{E}[X_s]] ds $$ by the fundamental theorem of calculus, the right hand side is differentiable in $t$, such that $$ \frac{d}{dt}m_t= M_t^1+M_t^2\mathbb{E}[X_t] = M_t^1+M_t^2m_t. $$ By the assumed differentiability on $M^1,M^2$ we have $m\in\mathcal{C}^{k+1}$. The same holds true for the variance.


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