Solving root stochastic differential equation

Question

I'm concerned with the following SDE: $$d Y_t= v \,dt + \sqrt {|Y_t|} \,d W_t$$ with $Y_0=-a$, $v>0$ being a constant, $a>0$ and $W_t$ as standard Brownian Motion. Do you have hints how to solve the SDE? Furthermore, I am interested in results (like density or moments) about the first hitting time of $a$ by the corresponding stochastic process.

Thanks a lot!

Answer 1

Question asks for hints but given the 6+ years since asking I'm posting my attempt at a solution.

We first assume the existence of some function $F(W_t)=Y_t$ such that \begin{equation} dY_t = \frac{\partial F}{\partial W_t}dW_T + \frac{\partial^2 F}{\partial W_t^2}dt. \end{equation} From comparison with the given equation of the SDE, we find that

1. $\frac{\partial F}{\partial W_t}=\sqrt{|F(W_t)|}$
2. $\frac{\partial^2 F}{\partial W_t^2}=v$
3. $F(t=0)=-a$

Integrating 2. w.r.t. $W_t$, we find that $(\star)$ $\frac{\partial F}{\partial W_t}=vW_t + c_0$ where $c_0$ is some constant of integration. Integrating again and using $d(W_t^2)=2W_t dW_t + dt$, we find that \begin{equation} F(W_t)=\frac{v}{2}(W_t^2-t)+c_0 W_t + c_1, \end{equation} where $c_1$ is again a constant of integration.

Using the above with 3., we find $F(t=0)=c_1=-a$. Then, using 1. and $(\star)$ at $t=0$, we find that $c_0=\sqrt{|-a|}=\sqrt{a}$ since $a>0$.

Therefore, $Y_t$ is given by \begin{equation} Y_t=\frac{v}{2}(W_t^2-t)+\sqrt{a}W_t-a. \end{equation}