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Reciprocal SDE

A release-checked medium problem for training Stochastic Calculus.

Question

Let $B_t$ be a standard Brownian Motion. Suppose that $X_t$ is some process such that $\dfrac{1}{X_t}$ satisfies the SDE $d\left(\dfrac{1}{X_t}\right) = \dfrac{1}{X_t}\left(2dt - dB_t\right)$. The SDE that $X_t$ satisfies is in the form $dX_t = X_t(adt + bdW_t)$ for some integers $a$ and $b$. Find $ab$.

Practice focus

This Pure Math problem is tagged Stochastic Calculus. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.

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