Making a power of Brownian motion a martingale

HardStochastic Processes~10m

Let $W_t$ be a standard Brownian motion. The process $X_t = W_t^{10} - \displaystyle\int_0^t a_s\,ds$ is a martingale for a suitable adapted process $a_t = A\,W_t^{B}$ . Find $A \cdot B$ .

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Itô on $W_t^{10}$ : the drift is $\tfrac12 f''(W_t)$ with $f(w)=w^{10}$ .
$f''=90w^8$ , so drift $=45W_t^8$ ; matching $a_t=AW_t^B$ gives $A=45,B=8$ .

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Making a power of Brownian motion a martingale

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