Say that $F(t,x)$ is the solution to a partial differential equation with stochastic representation $$F(t,x)=\mathbb{E}_x[\Phi(X_t)]$$ where $X_t$ is a process with dynamics $$dX_t=\alpha(X_t)dt+dW_t \qquad X_0=x$$ for continuous functions $\alpha$ and $\Phi$, and $W_t$ being a Wiener process. How do we use Girsanov formula to show that $F(t,x)$ can be expressed by $$F(t,x)=\mathbb{E}_x\Bigg[ \exp\Bigg\{\int_0^t\alpha(W_u)dW_u-\frac{1}{2}\int_0^t\alpha^2(W_u)du\Bigg\}\Phi(W_t)\Bigg]$$ My first thought was to use use an equivalent risk neutral measure $Q$, define the Girsanov kernel $\varphi_t=-\alpha(X_t)$ such that $$dX_t=dW^Q_t$$ under this new measure. Then we define the likelihood process $L_t=\frac{dQ}{dP}$ which satisfies $dL_t=\varphi_tL_tdW_t$ given by $$L_t=e^{-\int_0^t \alpha(X_s)dW_s+\frac{1}{2}\int_0^t \alpha^2(X_s)ds}$$Then we would have $$\begin{aligned}\mathbb{E}_x[\Phi(X_t)]&=\mathbb{E}^Q_x\bigg[\frac{1}{L_t}\Phi(X_t)\bigg]\\&=\mathbb{E}^Q_x\bigg[\exp\Bigg\{\int_0^t\alpha(X_u)dW_u-\frac{1}{2}\int_0^t\alpha^2(X_u)du\Bigg\}\Phi(X_t)\bigg]\end{aligned}$$ From here it looks like I'm almost done but I don't know how to proceed.


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