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 positive 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".

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".

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

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".

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 positive 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".