So far we have calculated all the integrals with respect to time. For example- \[\begin{aligned} &\mathbb{R(t)} = \mathbb{\int_{0}^{t}\Theta(u).du} \\ &\mathbb{I(t)} = \mathbb{\int_{0}^{t}\Delta(u).dW(u)} \\ \end{aligned}\]
Let’s further generalize the process. Instead of just integrating with respect to time we will integrate with an Ito process.
First, we define an Ito Process. Let \(\mathbb{W(t), \ t \geq 0}\) be a Brownian motion, and let \(\mathbb{F(t), \ t \geq 0}\) be an associated filtration. An Ito process is a stochastic process of the form-
\[\mathbb{X(t)} = \mathbb{X(0)} + \ \mathbb{\int_0^{t} \Delta(u).dW(u) + \ \int_0^{t} \Theta(u).du}\]
where \(\mathbb{X(0)}\) is non-random and \(\mathbb{\Delta(u)}\) and \(\mathbb{\Theta(u)}\) are adapted stochastic processes.
In differential form, \[\mathbb{dX(t) = \Delta(t)\ dW(t) \ + \Theta(t) \ dt}\]
Let \(\mathbb{\Gamma(.)}\) be an adapted process. Then we define the integral of \(\mathbb{\Gamma(.)}\) with respect to Ito process \(\mathbb{X}\) is defined as-
\[\mathbb{\int_{0}^{t} \Gamma(u) \ dX(u)} = \mathbb{\int_{0}^{t} \Gamma(u) \ \Delta(u).dW(u) \ + \int_{0}^{t} \Gamma(u) \ \Theta(u).du}\]
To prove this, we are going to follow the same procedure as Ito- Doeblin formula for Brownian motion. We are going to use Ito process \(\mathbb{X(t)}\) in place of of the Brownian motion \(\mathbb{W(t)}\). Just open the proof the Ito Doeblin formula for Brownian motion (https://financewithbrownie.wixsite.com/home/post/stochastic-calculus-part-5) on the side and compare step by step for easier understanding.
Ito- Doeblin formula for Ito process:
Consider the partition of \(\mathbb{[0,T]}\) as- \(\mathbb{\Pi =\{0=t_0< t_1 < \cdots< t_n= T\}}\) \[\begin{aligned} \mathbb{f(T,X(T))- f(0,X(0))} &= \mathbb{\sum_{j=0}^{n-1} \ [f(t_{j+1}, X(t_{j+1})) -f(t_{j}, X(t_{j}))]} \qquad \text{[Now apply Taylor series expansion]} \\ &= \mathbb{\sum_{j=0}^{n-1}f_{t}(t_{j},X(t_{j}))\ (t_{j+1}-t_{j})} \ + \ \mathbb{\sum_{j=0}^{n-1}f_{x}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))} \\ & + \ \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{xx}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))^2} \\ & + \ \mathbb{\sum_{j=0}^{n-1}f_{tx}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))\ (t_{j+1}-t_{j})} \\ & + \ \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{tt}(t_{j},X(t_{j}))\ (t_{j+1}-t_{j})^2}\ +\ \text{higher order terms} \\ \end{aligned}\]
You see the difference?
So let’s see what happens to the right hand side, when \(||\Pi|| \rightarrow 0\).
First term, \[\begin{aligned} \mathbb{\underset{||\Pi|| \rightarrow 0}{\lim}}\ \mathbb{\sum_{j=0}^{n-1}f_{t}(t_{j},X(t_{j}))\ (t_{j+1}-t_{j})} =\ \mathbb{\int_{0}^{T}f_{t}(t,X(t))\ dt} \end{aligned}\]
Second term,
\[\begin{aligned} \mathbb{\underset{||\Pi|| \rightarrow 0}{\lim}}\ \mathbb{\sum_{j=0}^{n-1}f_{x}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))} &=\ \mathbb{\int_{0}^{T}f_{x}(t,X(t))\ dX(t)} \\ &=\ \mathbb{\int_{0}^{T}f_{x}(t,X(t))\ [\Delta(t)\ dW(t) \ + \Theta(t) \ dt]} \\ &=\ \mathbb{\int_{0}^{T}f_{x}(t,X(t))\ \Delta(t)\ dW(t) \ +\ \int_{0}^{T}f_{x}(t,X(t))\ \Theta(t) \ dt} \\ \end{aligned}\]
Third term,
\[\begin{aligned} \mathbb{\underset{||\Pi|| \rightarrow 0}{\lim}}\ \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{xx}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))^2} =\ &\mathbb{\underset{||\Pi|| \rightarrow 0}{\lim}}\ \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{xx}(t_{j},X(t_{j}))\ [X,X](t_{j},t_{j+1})} \\ =\ &\frac{1}{2}\ \mathbb{\int_{0}^{T}f_{xx}(t,X(t))\ d[X,X](t)} \end{aligned}\]
Now we have seen that- \(\mathbb{[X,X](0,t)} = \mathbb{\int_{0}^{t}\Delta^2(u)\ du}\). In differential form it is- \(\mathbb{d[X,X](t)} = \mathbb{\Delta^2(t)\ dt}\). So, ultimately it becomes-
\[\begin{aligned} \mathbb{\underset{||\Pi|| \rightarrow 0}{\lim}}\ \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{xx}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))^2} =\ &\frac{1}{2}\ \mathbb{\int_{0}^{T}f_{xx}(t,X(t))\ d[X,X](t)} \\ =\ &\frac{1}{2}\ \mathbb{\int_{0}^{T}f_{xx}(t,X(t))\ \Delta^2(t)\ dt} \\ \end{aligned}\]
Similar to the third term we are going to write the fourth & fifth term.
Fourth term, \[\begin{aligned} \mathbb{\sum_{j=0}^{n-1}f_{tx}(t_{j},X(t_{j}))\ (X(t_{j+1})-X(t_{j}))\ (t_{j+1}-t_{j})} = &\ \mathbb{\int_{0}^{T}f_{tx}(t,X(t))\ [dX(t).\ dt]} \\ = &\ \mathbb{\int_{0}^{T}f_{tx}(t,X(t))\ [(\Delta(t)\ dW(t) \ + \Theta(t) \ dt).\ dt]} \\ = &\ \mathbb{\int_{0}^{T}f_{tx}(t,X(t))\ [\Delta(t)\ dW(t).\ dt + \Theta(t) \ dt.\ dt]} \\ = &\ 0 \end{aligned}\]
Fifth term,
\[\begin{aligned} \frac{1}{2}\ \mathbb{\sum_{j=0}^{n-1}f_{tt}(t_{j},X(t_{j}))\ (t_{j+1}-t_{j})^2}\ &=\ \frac{1}{2}\ \mathbb{\int_{0}^{T}f_{tt}(t,X(t))\ dt\ dt} \\ &=\ 0 \end{aligned}\]
Theorem: Ito- Doeblin formula for Ito process
Let \(\mathbb{X(t)}\ \mathbb{t \geq 0}\) be an Ito process as previously defined. Let \(\mathbb{f(t,x)}\) be a function for which the partial derivatives-
\[\begin{aligned} \mathbb{\frac{\delta}{\delta x}f(t,x)} = \mathbb{f_x(t,x)}; \quad \mathbb{\frac{\delta}{\delta t}f(t,x)} = \mathbb{f_t(t,x)}; \quad \mathbb{\frac{\delta^2}{\delta^2 x}f(t,x)} = \mathbb{f_{xx}(t,x)} \\ \end{aligned}\] are defined and continuous; Then for every \(\mathbb{T \geq 0}\), we have-
\[\begin{aligned} &\mathbb{f(T,X(T))} \\ =\ &\mathbb{f(0,X(0))}\ +\ \mathbb{\int_{0}^{T}f_{t}(t,X(t))\ dt}\ +\ \mathbb{\int_{0}^{T}f_{x}(t,X(t))\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{\int_{0}^{T}f_{xx}(t,X(t))\ d[X,X](t)} \\ =\ &\mathbb{f(0,X(0))}\ +\ \mathbb{\int_{0}^{T}f_{t}(t,X(t))\ dt}\ +\ \mathbb{\int_{0}^{T}f_{x}(t,X(t))\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{\int_{0}^{T}f_{xx}(t,X(t))\ \Delta^2(t)\ dt} \\ \end{aligned}\]
In differential form, we can write it as- \[\mathbb{df(t,X(t))}\ =\ \mathbb{f_{t}(t,X(t))\ dt}\ +\ \mathbb{f_{x}(t,X(t))\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{f_{xx}(t,X(t))\ dX(t)\ dX(t)}\]
We can further elaborate this using that- \(\ \mathbb{dX(t) = \Delta(t)\ dW(t) \ + \Theta(t) \ dt}\)
\[\begin{aligned} \mathbb{df(t,X(t))}\ &=\ \mathbb{f_{t}(t,X(t))\ dt}\ +\ \mathbb{f_{x}(t,X(t))\ dX(t)}\ +\ \frac{1}{2}\ \mathbb{f_{xx}(t,X(t))\ dX(t)\ dX(t)} \\ &=\ \mathbb{f_{t}(t,X(t))\ dt}\ +\ \mathbb{f_{x}(t,X(t))\ (\Delta(t)\ dW(t) \ + \Theta(t) \ dt)}\ +\ \frac{1}{2}\ \mathbb{f_{xx}(t,X(t))\ \Delta^2(t)\ dt} \\ &=\ \mathbb{f_{t}(t,X(t))\ dt}\ +\ \mathbb{f_{x}(t,X(t))\ \Delta(t)\ dW(t)\ +\ f_{x}(t,X(t))\ \Theta(t)\ dt}\ +\ \frac{1}{2}\ \mathbb{f_{xx}(t,X(t))\ \Delta^2(t)\ dt} \\ \end{aligned}\]
In the next blog, we are going to use all these theorems in some interesting examples.