See Exercise 1.7.1.8 for the notation. Prove that \(B\) defined by

\[d B_{t}=d W_{t}-\frac{\int_{-\infty}^{\infty} d y h^{\prime}(y) e^{-\left(y-W_{t}\right)^{2} /(2(T-t))}}{\int_{-\infty}^{\infty} d y h(y) e^{-\left(y-W_{t}\right)^{2} /(2(T-t))}} d t\]

is a \(\mathbb{Q}\)-Brownian motion. 

Exercise 1.7.1.8:

Give conditions on the function \(h\) so that the measure \(\mathbb{Q}\) defined on \(\mathcal{F}_{T}\) as \(\mathbb{Q}=h\left(W_{T}\right) \mathbb{P}\) is a probability equivalent to \(\mathbb{P}\). Prove that, for \(t<\left.T \mathbb{Q}\right|_{\mathcal{F}_{t}}=\left.L_{t} \mathbb{P}\right|_{\mathcal{F}_{t}}\) where

\[L_{t}=\int_{-\infty}^{\infty} d y h(y) \frac{e^{-\left(y-W_{t}\right)^{2} /(2(T-t))}}{\sqrt{2 \pi(T-t)}}\]

Prove that

\[L_{t}=1+\int_{0}^{t} d W_{s} \int_{-\infty}^{\infty} d y \frac{h(y) e^{-\left(y-W_{s}\right)^{2} /(2(T-s))}}{\sqrt{2 \pi(T-s)}} \frac{y-W_{s}}{T-s}\]

For \(h \in C^{1}\) with compact support, prove that

\[L_{t}=1+\int_{0}^{t} d W_{s} \int_{-\infty}^{\infty} d y \frac{h^{\prime}(y) e^{-\left(y-W_{s}\right)^{2} /(2(T-s))}}{\sqrt{2 \pi(T-s)}}\]