medium Pure Math
A release-checked medium problem for training Stochastic Calculus.
Let $B_t$ be a standard Brownian Motion. Fix $t > 0$, and define $\Delta_{m,n} = B_{tm2^{-n}} - B_{t(m-1)2^{-n}}$. Evaluate $\mathbb{E}\left[\left(\displaystyle \sum_{m=1}^{2^n}\Delta_{m,n}^2 - t\right)^2\right]$ as a function of $n$ and $t$. Evaluate this function with $t = 1$ and $n = 5$.
This Pure Math problem is tagged Stochastic Calculus. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
Pure Math · IMC, WorldQuant
Pure Math · Akuna, WorldQuant
Pure Math
Pure Math
Pure Math · Goldman Sachs
Pure Math