Section 18.4

Ito Integrals and the Isometry

Means and variances of stochastic integrals and of time integrals of W, quadratic variation.

Toolkit

An Itô integral $I_t=\int_0^t f_s\,dW_s$ (with $f$ adapted, $\mathbb{E}[\int_0^t f_s^2\,ds]<\infty$ ) is a martingale with

\mathbb{E}[I_t]=0,\qquad \operatorname{Var}(I_t)=\mathbb{E}\Big[\int_0^t f_s^2\,ds\Big]=\int_0^t \mathbb{E}[f_s^2]\,ds,

the Itô isometry. More generally $\mathbb{E}[I_tJ_t]=\int_0^t\mathbb{E}[f_sg_s]\,ds$ for $J_t=\int_0^t g_s\,dW_s$ .

For deterministic time-integrals of $W$ , use $\operatorname{Cov}(W_u,W_v)=\min(u,v)$ and Fubini:

\operatorname{Var}\Big(\int_0^t W_s\,ds\Big)=\int_0^t\int_0^t \min(u,v)\,du\,dv,

or integrate by parts, $\int_0^t W_s\,ds = tW_t-\int_0^t s\,dW_s=\int_0^t(t-s)\,dW_s$ , then apply the isometry.

Quadratic variation: for a partition into $2^n$ pieces, $\sum (\Delta W)^2\to t$ in $L^2$ , since increments are independent with $\mathbb{E}[(\Delta W)^2]=\Delta t$ , $\operatorname{Var}((\Delta W)^2)=2(\Delta t)^2$ .

Problems