Let $W$ and $B$ be correlated one dimensional Brownian motions with constant correlation coefficient $r \in (-1, 1)$, that is, we have $d\langle W, B \rangle_t = r \, dt.$ We assume we have $B_0 = v$ for some arbitrary initial condition $v \in \mathbb R$.
Consider the SDE
$$dX_t = B_t X_t \, dW_t$$
with initial condition $X_0 = x_0$ a.s.
Question: Is there an explicit solution to the above SDE, modulo stochastic/deterministic integrals without explicit $X$ terms?
First, we move to independent BMs
$$W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t=c_{1}W_{t}^{1}+c_{2}Z_{t},$$
where $Z_t$ is a BM independent of $W_t^1$. So we condition on $Z_{t}$ to make it deterministic
$$dX_{t}=(c_{1}W_{t}+c_{2}Z_{t})X_{t}dW_{t}.$$
By applying Itô for $Y=\ln(X)$ we have
$$dY=\frac{1}{X}dX+\frac{-1}{2X^{2}}((c_{1}W_{t}+c_{2}Z_{t})X_{t})^{2}dt$$
$$=c_{1}W_{t}+c_{2}Z_{t}dW_{t}-\frac{1}{2}(c_{1}W_{t}+c_{2}Z_{t})^{2}dt$$
and so
$$X=\exp\left(\int c_{1}W_{t}+c_{2}Z_{t}dW_{t}-\frac{1}{2}\int (c_{1}W_{t}+c_{2}Z_{t})^{2}dt\right).$$
