Is there a specific term for this SDE?

Question

I know that $dX_t = aX_tdt + bX_tdW_t$ is the equation for the dynamics of the price of stock in the Black scholes options according to wikipedia.

Is there this equation something common?

$$dX_t = aX_tdt + bdW_t$$ where the solution is $X_t$ given that $X_0 = x$?

I am very new to studying SDEs and having a hard time understanding a lot of things. I am currently trying to solve this to prepare for a future class.

Answer 1

This is a special case of an Ornstein-Uhlenbeck process with $\mu=0$: $$dX_t = \theta(\mu - X_t)dt + \sigma dW_t\,\,. $$ Also using your notation $\theta=-a$ and $\sigma=b$.

Note: this only works (according to the Wikipedia definition) if $a<0$, since to be an Ornstein-Uhlenbeck process one requires that $\theta > 0$.

However, other definitions (see for example here) allow for that parameter to be non-positive (the only requirement for the parameter $\gamma=\theta$ given in that paper is that $\gamma \in \mathbb{R}$).

Likewise, Wikipedia says $\sigma >0$ while Maller et al says that $\sigma \ge 0$. So without restrictions on $b = \sigma$, technically your process is a generalization of the Ornstein-Uhlenbeck process.