Second Moment of Brownian Motion

Since $E[B_t]=0$, the second moment equals the variance.

Recall that $\text{Var}(B_t)=t$ for standard Brownian motion.

1. 1. Relate second moment to variance

   $E[B_t^2]=\text{Var}(B_t)+(E[B_t]{)}^2$, and $E[B_t]=0$.

2. 2. Substitute the variance

   Since $\text{Var}(B_t)=t$, the second moment equals $t$.

   $$E[B_t^2]=t$$

Answer: t