Variance of Brownian motion at jump points

Question

How can one evaluate $Var(B(T_1))$, where $\{B(t), t \geq 0\}$ is a Standard Brownian motion and $\{T_n\}_{n \geq 1}$ denote the jump points of a Poisson($\lambda $) process?

Answer 1

From comments to give an answer:

You can use the law of total variance

So you said $$Var(B(T_1)) = E(Var[B(T_1)\mid T_1]) + Var(E[B(T_1) \mid T_1]) = E[T_1] + 0 = 1/\lambda$$

which looks correct to me

As an empirical check using R:

set.seed(2020)
lambda <- 0.2
cases <- 10^6
T1 <- rexp(cases, rate=lambda)
B1 <- rnorm(cases, mean=0, sd=sqrt(T1))
var(B1)
# 5.000312