# Stochastic differential equation with quadratic drift and volatility

I am looking for an exact (closed-form) solution to the SDE: $$\begin{equation} dX_t = \alpha X_t(A_0 - X_t) dt + \beta X_t(A_0 + X_t) dW_t \end{equation}$$ for a Wiener process $$dW_t$$ with initial condition $$X(0) = X_0$$. I have tried different substitutions such as $$\begin{equation} d\left[\frac{1}{A_0}\ln\left(\frac{x}{x-A_0}\right)\right] \end{equation}$$ but after applying the Ito formula I always end up with an extra $$X_t dt$$ term on the right hand side: $$\begin{equation} d\left[\frac{1}{A_0}\ln\left(\frac{x}{x-A_0}\right)\right] = C_1 z + C_2 W_t + C_3 X_t dt \end{equation}$$ where $$C_j$$ are constants. If I then integrate both sides I end up with an integral equation that has no solution. Is there any other way to simplify the expression, at least to eliminate the need for an integral equation? Any approximations/asymptotic expansions would also be useful

• What makes you believe that a closed form exists...? – saz Nov 14 '18 at 6:46