In the previous blog, we have seen some properties of Ito integral like- Ito isometry and Quadratic variation of Ito integral. We have extensively proved these properties. But we can use some tricks we learned in Brownian motion.
If you remember, for Brownian motion \(\mathbb{W(t), \ t>0}\) we did have- \[\begin{aligned} &\mathbb{dW(t).dW(t) = dt} \\ \end{aligned}\] We interpret this equation as the statement that Brownian motion accumulates quadratic variation at rate one per unit time. It is another way of writing- \[\begin{aligned} \mathbb{[W,W](t) = t} \qquad \text{for} \ \mathbb{t \geq 0} \end{aligned}\] Now, the Ito integral formula is- \[\begin{aligned} \mathbb{I(t)} = \mathbb{\int_{0}^{t}\Delta(u).dW(u)} \end{aligned}\] In differenial form, this can also be written as- \[\mathbb{dI(t)} = \mathbb{\Delta(t).dW(t)}\]
Hence, from the above form we can write- \[\begin{aligned} \mathbb{dI(t).dI(t)} &= \ \mathbb{\Delta^2(t).dW(t).dW(t)} \\ &= \ \mathbb{\Delta^2(t).dt} \\ \end{aligned}\]
This equation says that the Ito integral \(\mathbb{I(t)}\) accumulates quadratic variation at rate \(\mathbb{\Delta^2{(t)}}\) per unit time.
\[\begin{aligned} \mathbb{I(t)} = \mathbb{\int_{0}^{t}\Delta(u).dW(u)} \equiv \mathbb{dI(t)} = \mathbb{\Delta(t).dW(t)} \end{aligned}\]
Now, since the integral expression of \(\mathbb{I(t)}\) is coming from integrating both sides of the differential equation from time \(\mathbb{T=0}\) to \(\mathbb{T=t}\) the correct expression should be- \[\begin{aligned} &\int_{T=0}^{T=t}\mathbb{dI(u)} = \int_{T=0}^{T=t}\mathbb{\Delta(u).dW(u)} \\ \implies &\mathbb{I(t)-I(0)} = \int_{T=0}^{T=t}\mathbb{\Delta(u).dW(u)} \\ \implies &\mathbb{I(t)} = \mathbb{I(0)} + \int_{T=0}^{T=t}\mathbb{\Delta(u).dW(u)} \\ \end{aligned}\]
Generally we take \(\mathbb{I(0) = 0}\) and ignore that in the Ito integral expression.
Previously, in the Ito integral the integrand \(\mathbb{\Delta(t)}\) is considered to be simple process, i.e. it is constant over intervals. We still assume that \(\mathbb{\Delta(t)}, \ \mathbb{t \geq 0}\) is adapted to Filtration \(\mathbb{F(t), \ \mathbb{t \geq 0}}\). We also assume the square integrability condition, i.e.
\[\mathbb{E\int_{0}^{T}\Delta^2(t).dt} < \infty\]
Again, like previous case in order to define \(\mathbb{\int_0^{T}\Delta(t).dW(t)}\), we approximate \(\mathbb{\Delta(t)}\) by simple processes. Consider a partition- \[\mathbb{0 = t_0 < t_1 < t_2 < \cdots < t_m = T}\] The value of the simple process is constant over each sub-interval- \(\mathbb{[t_j, t_{j+1})}\). As the maximal step size of the partition, i.e. \(\mathbb{\max_{j}(t_{j+1}-t_j)}\) approaches zero, the approximating integrand will become a better and better approximation of the continuously varying one.
In general, it is possible to choose a sequence \(\mathbb{\{\Delta_n(t)\}}\) of simple processes such that as \(\mathbb{n \rightarrow \infty}\), these processes converge to the continuously varying \(\mathbb{\Delta(t)}\).
By converge we mean- \[\mathbb{E \int_{0}^{T}|\Delta_n(t)-\Delta(t)|.dt = 0}\]
Now, since we have taken \(\mathbb{\Delta_n(t)}\) to be simple process, the Ito integral- \[\mathbb{\int_{0}^{t}\Delta_n(u).dW(u)} \ \text{is defined for} \quad \mathbb{0 \leq t \leq T}\] This integral has all the properties Ito integrals of Ito process. We define the Ito integral for continuously varying \(\mathbb{\Delta(t)}\) by-
\[\mathbb{I(t)} = \mathbb{\int_{0}^{t}\Delta(u).dW(u)} = \mathbb{\underset{n \rightarrow \infty}{\lim} \mathbb{\int_{0}^{t}\Delta_n(u).dW(u)}} \qquad \ \mathbb{0 \leq t \leq T}\]
Let us see all the properties of Ito integral.
Theorem: Let \(\mathbb{T}\) be a positive constant and let \(\mathbb{\Delta(t), \ 0 \leq t \leq T}\) be a adapted stochastic process that satisfies- \[\mathbb{E\int_{0}^{T}\Delta^2(t).dt} < \infty\] Then the Ito integral- \[\mathbb{I(t)} = \mathbb{\int_{0}^{t}\Delta(u).dW(u)} = \mathbb{\underset{n \rightarrow \infty}{\lim} \mathbb{\int_{0}^{t}\Delta_n(u).dW(u)}} \qquad \ \mathbb{0 \leq t \leq T}\]
has the following properties-
In the next blog, we are going to use these properties to find some very important results. Happy reading.