The SDE of the reciprocal

Let $B_t$ be a standard Brownian motion, and suppose $X_t$ is a process whose reciprocal satisfies $d\!\left(\tfrac{1}{X_t}\right) = \tfrac{1}{X_t}\big(2\,dt - dB_t\big)$. The process $X_t$ then satisfies an SDE of the form $dX_t = X_t\big(a\,dt + b\,dW_t\big)$ for integers $a, b$. Find $a\cdot b$.