STOCHASTIC CALCULUS 04
The Itô Integral
The Itô integral defines integration with respect to Brownian motion and gives rigorous meaning to the stochastic term in an SDE.
The main idea
\[\boxed{\int_0^T H_t\,dW_t}\]
It accumulates contributions against random Brownian increments.
1. Construction
Divide time as
\[0=t_0\lt t_1\lt\cdots\lt t_n=T.\]Define \(\Delta W_i=W(t_{i+1})-W(t_i)\). The Itô sums are
\[\boxed{\sum_{i=0}^{n-1}H_{t_i}\Delta W_i}\]
The coefficient is evaluated at the left endpoint, so it uses information available before the next increment.
2. Definition idea
\[\boxed{\int_0^T H_t\,dW_t=\lim\sum_{i=0}^{n-1}H_{t_i}\Delta W_i}\]
The limit is taken in a stochastic sense.
3. Basic example
\[\int_0^T dW_t=W_T.\]4. Mean and Itô isometry
\[E\left[\int_0^T H_t\,dW_t\right]=0,\]\[\boxed{E\left[\left(\int_0^T H_t\,dW_t\right)^2\right]=E\left[\int_0^T H_t^2\,dt\right]}\]
5. SDE integral form
\[\boxed{X_t=X_0+\int_0^t a(X_s,s)\,ds+\int_0^t b(X_s,s)\,dW_s}\]
6. Constant coefficients
For \(dX_t=\mu dt+\sigma dW_t\),
\[X_t=x_0+\mu t+\sigma W_t.\]Hence \(E[X_t]=x_0+\mu t\) and \(\operatorname{Var}(X_t)=\sigma^2t\).