Stability results for general linear stochastic ODE

Question

I am interested in the following time-invariant multivariate SDE: \begin{equation} dx_i = \sum_{j} a_{ij} x_j\,dt + \sum_{j,k} b_{ijk} x_k \, dW_j \end{equation} Despite its simplicity the general solution to SDEs of this form seems elusive (see e.g. Solution of multivariate Geometric Brownian Motion? and General solution to system of stochastic linear differential equations). I have a more limited question: what are the criteria under which solutions of this SDE converge almost surely to zero?

Answer 1

Writing $y = x/|x|$ and $r = \log |x|$ it is not difficult to see that $y$ solves an autonomous SDE and there exists a function $F$ such that $r(t) = r(0) + \int_0^t F(y(s))\ ds$. Since $y$ takes values in the sphere it admits an invariant measure $\mu$ which, under rather weak non-degeneracy assumptions on $A$ and $B$, is unique. The sharp criterion then is whether $\int F(y)\mu(dy)$ is strictly negative or not.

In general there is no closed form expression for $\mu$, but one can of course play around with Lyapunov functions to get sufficient conditions.

Answer 2

Since interested in convergence to zero or $x_{*}$, one approach is phrasing it as a gradient descent SDE An SDE perspective on stochastic convex optimization i.e. writing it as a convex function $f$ with

$$\partial_{i}f(x)=-\sum_{j}a_{ij}x_{j}$$

and so if we let $$f(x)=-\frac{1}{2}\sum_{i}\sum_{j}a_{ij}x_{j}x_{i}=-\frac{1}{2}x^{T}Ax,$$ we need the matrix $A=(a_{ij})$ to be negative definite and symmetric. For the volatility coefficient they needed bounded and Lipschitz.

Here are some other tools to help you study the growth of the process. The most comprehensive tool is Feller's test (Shreve-Karatzas theorem 5.29). Secondly, one can also use Lyapunov functions as done in "Stochastic Stability of Differential Equations". Also see "Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with Applications to Financial Markets∗".

Thirdly, if one is looking for invariant measures, you can use the stationary Fokker-Plank eg. https://math.stackexchange.com/questions/683775/invariant-measures-for-stochastic-processes and "Long-time dynamics of stochastic differential equations".