Let \(a\) and \(\sigma\) be continuous deterministic functions, \(B\) a BM and \(X\) the solution of \(d X_{t}=a(t) X_{t} d t+\sigma(t) d B_{t}, X_{0}=x\).

Let \(T_{0}=\inf \left\{t \geq 0, X_{t} \leq 0\right\}\). Prove that, for \(x>0, y>0\),

\[\mathbb{P}\left(X_{t} \geq y, T_{0} \leq t\right)=\mathbb{P}\left(X_{t} \leq-y\right)\]

Use the fact that \(X_{t} e^{-A_{t}}=W_{\alpha(t)}^{(x)}\) where \(A_{t}=\int_{0}^{t} a(s) d s\) and \(W^{(x)}\) is a Brownian motion starting from \(x\). Here \(\alpha\) denotes the increasing function \(\alpha(t)=\int_{0}^{t} e^{-2 A(s)} \sigma^{2}(s) d s\). Then, use the reflection principle to obtain \(\mathbb{P}\left(W_{u}^{(x)} \geq z, T_{0} \leq u\right)=\mathbb{P}\left(W_{u}^{(x)} \leq-z\right)\).