Variance of an Itô integral

HardStochastic Processes~8m

Let $W_t$ be a standard Brownian motion. The variance $\operatorname{Var}\!\left(\int_0^t W_s\,dW_s\right)$ has the form $k\,t^2$ . Find the constant $k$ .

Show hints (2)+

$\int_0^t W_s\,dW_s = \tfrac12(W_t^2 - t)$ ; the constant doesn't affect variance.
Use $\operatorname{Var}(W_t^2)=\mathbb{E}[W_t^4]-t^2=3t^2-t^2=2t^2$ . (Or apply Itô isometry.)

Answer

Reveal answer →

0.5

Create a free account to see the full worked solution and track your progress.

Create a free account

Interview format: Market Maker, Systematic & Data-Driven Trading

Variance of an Itô integral

Answer

Related questions