Exponential of Sums of different times of a Brownian Motion

Question

Let $\{B_s\}_{s\in[0,1]}$ be a Brownian motion, let $t_1 < \dots < t_n \in [0,1]$, I am interested in finding good upper and lower bounds for

$$ \mathbb{E}[\exp(B_{t_1}+ \dots + B_{t_n})]. $$

If it is not possible in general, are there a class of $t_1, \dots, t_n$ such that I can find such bounds?

Answer 1

The conditional distribution of $B_{t_{k}}$ given $B_{t_{k-1}}$ is normal with mean $B_{t_{k-1}}$ and variance $t_{k} - t_{k-1}$. Thus $$\mathbb E[\exp(s B_{t_k})| B_{t_{k-1}}] = \exp(s B_{t_{k-1}} + s^2 (t_k - t_{k-1})^2/2)$$ This lets you express everything in closed form. If I'm not mistaken, the end result is $$\exp\left( \sum_{k=1}^n \frac{(n+1-k)^2 (t_k - t_{k-1})^2)}{2}\right) $$ (where we take $t_0 = 0$)