The SDE of the reciprocal

MediumStochastic Processes~8m

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 $a\cdot b$ .

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Let $Y=1/X$ ; then $X=1/Y$ . Apply Itô to $g(y)=y^{-1}$ , keeping $\tfrac12 g''(dY)^2$ .
$(dY)^2=Y^2dt$ ; you get $dX=X(-dt+dB)$ , so $a=-1,b=1$ .

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The SDE of the reciprocal

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