# Try to extend the averaging theory to stochastic case

Consider a time-varying linear system \begin{align*} \dot x(t)=A(t)x(t) \end{align*} and its average system (the existence is assumed) \begin{align*} \dot x(t)=\bar A x(t), \quad \bar A=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}A(s)ds}{t-t_0}. \end{align*} By the classic averaging theory, if the average system is asymptotically stable, then there exists an $\varepsilon^* >0$ such that for all $\varepsilon \in (0, \varepsilon^*)$, the following fast time-varying system \begin{align*} \dot x(t)=A(t/\varepsilon)x(t) \end{align*} is asymptotically stable.

Can this result be extended to stochastic case as follows? For the stochastic linear system \begin{align*} d x(t)=A(t)x(t)dt+B(t)x(t)dB(t), \end{align*} suppose its time-average system \begin{align*} d x(t)=\bar A x(t)dt+\bar Bx(t)dB(t), \quad \bar A=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}A(s)ds}{t-t_0}, \bar B=\lim_{t\rightarrow \infty}\frac{ \int_{t_0}^{t}B(s)ds}{t-t_0} \end{align*} exists and it is asymptotically stable in a certain sense. Then can we conclude that the fast time-varying stochastic system \begin{align*} d x(t)=A(t/\varepsilon)x(t)dt+B(t/\varepsilon)x(t)dB(t), \end{align*} for all $\varepsilon \in (0, \varepsilon^*)$ is asymptotically stable in the same sense as above?

Any pointer will be helpful and be appreciated.

In fact, the exact question I am considering is motivated by the following fact which has been reported in for example [Dirk Aeyels and Joan Peuteman, On exponential stability of nonlinear time-varying differential equations, Automatica 35 (1999) 1091-1100]. There exists an $\varepsilon^*>0$ such that for any $\varepsilon \in (0, \varepsilon^*)$ the system $\dot x(t)=A(t/\varepsilon)x(t)$ has an exponentially stable null solution if:

1. there exists an increasing sequence of times $t_k, k\in \mathbb{Z}$ with $t_k\rightarrow \infty$ as $k\rightarrow \infty$ and $t_k\rightarrow -\infty$ as $k\rightarrow -\infty$;

2. there exists a finite $T>0$ such that $t_{k+1}-t_k \leq T, \forall k\in \mathbb{Z}$;

3. the sample average value of $A(t)$, $\bar A_k=\frac{\int_{t_k}^{t_{k+1}}A(s)ds}{t_{k+1}-t_k}$ is Hurwitz for all $k\in \mathbb{Z}$, i.e., there exist a positive definite matrix $P$ and a constant $v>0$ such that $P\bar A_k+\bar A_k^T P <-vI$.

Putting this issue in a stochastic case, I am now wondering whether the stability of the sample average stochastic systems $d x(t)=\bar A_k x(t)dt+\bar B_k x(t)dB(t)$ implies the stability of the fast time-varying stochastic system $d x(t)=\bar A(t/\varepsilon) x(t)dt+\bar B(t/\varepsilon) x(t)dB(t)$ for all $\varepsilon \in (0, \varepsilon^*)$?

I am posting it here hoping to get some answers or references. Thanks a lot.

The correct way of averaging the coefficient $B$ is $$\bar B^2 = \lim_{t\to\infty}{1\over t} \int_0^t B^2(s)\,ds\;.$$ In your particular case, this is quite easy to see "by hand" by looking at the explicit solution to your equation and using Lévy's characterisation of Brownian motion.