# Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the law of $S_t$ knowing the value at $0$ and at $T$, for $t \in [0,T]$.

Thank you very much.

• the solution is known for sigma constant but I didnt find anything about this case. – mawchne Sep 15 '18 at 18:24
• The variance between $t_1$ and $t_2$ is going to be the average $\sigma_t^2$ in this interval i.e. $V(t_1,t_2)=\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\sigma^2_t dt$. You could try to work from this, I haven't tried but I have a feeling things may get extremely complicated, especially for a completely general $\sigma_t$. Do we know anthing about how $\sigma_t$ varies with $t$? – Alex C Sep 15 '18 at 21:18
• Hi Alex, let's say it is constant wise, if that may simplify things – mawchne Sep 16 '18 at 7:36