Motivation

Let $(X_{t})_{t\ge 0},(Y_{t})_{t\ge 0}$ be stochastic processes on $(\Omega,\mathcal{F},P)$. Our goal is to make sense of $$\int\limits Y \, dX$$

1. Lebesgue-Stieltjes Pathwise Integral

We first tried to use the LS integral to do this. We first noted that the Lebesgue-Stieltjes measure, $\mu_{g}$ only worked with $g$ increasing and that $\mu_{g}$ was the difference between two increasing functions.

We then saw that for any $g=g_{1}-g_{2}$ where both $g_{1},g_{2}$ are increasing that $g$ then must have finite variation on $\mathbb{R}^{+}$.

We then put the final nail in the coffin by showing Continuous Martingale with finite variation is constant

To summarize, if we wished to use the LS integral to define $$\int\limits Y \, dX$$ we needed $X$ to be a continuous martingale with finite variaiton but this form meant that $X$ was constant which is pretty useless!

2. Riemann-Stieltjes Pathwise Integrals

We then thought, ok how about RS integrals? i.e. $$\forall\omega\in\Omega\quad\lim_{ |\pi| \to 0 } \sum_{i=0}^{N-1}Y_{t_{i}}(\omega)(X_{t_{i+1}}(\omega)-X_{t_{i}}(\omega))$$where $\pi=(t_{0},\dots,t_{N})$ is a subdivision of $[0,1]$ (for simplicity). We then defined the following: Let $g:[0,1]\to \mathbb{R}$ be continuous. Define $S_{\pi}\in C^{0}([0,1]:\mathbb{R})$ to be $$S_{\pi}(f)=\sum_{i=0}^{N-1}f(t_{i})(g(t_{i+1})-g(t_{i}))$$Assume $\lim_{ |\pi| \to 0 }S_{\pi}(f)$ exists $\forall f\in C^{0}([0,1]:\mathbb{R})$. We then applied the principle of uniform boundedness to find that $g$ also needs to have finite variation on $[0,1]$ in order for this to be defined i.e. $$\lVert S_{\pi} \rVert _{op}=\sup_{\lVert f \rVert _{\infty}\le 1}|S_{\pi}(f)|=V(g;\pi)$$and by the theorem we found that $$\sup_{n\in\mathbb{N}}\lVert S_{\pi} \rVert _{op}=\sup_{n\in\mathbb{N}}V(g;\pi)<\infty$$ Then considering the standard Brownian motion $(B_{t})_{t\ge 0}$ we computed $$\int\limits _{0}^{t}B_{s} \, dB_{s}=\frac{1}{2}(B_{t}^{2}-t)$$in probability by applying the framework of the RS integral.

Motivation Trying Again!

Let now $(M_{t})_{t\ge 0}$ be a right continuous $L^{2}$-martingale and $(X_{t})_{t\ge 0}$ be a stochastic process on $(\Omega,\mathcal{F},P)$. Our goal is to make sense of $$\int\limits X \, dM$$in a way that it agrees with the LS pathwise integrals whenever they’re defined.

Itô Isometry (0)

Now, we take $g$ increasing and right continuous meaning we can redefine the Lebesgue-Stieltjes measure $\mu_{g}$ strictly on intervals of the type $(a,b]$ i.e. since $g$ right continuous $\implies g(b^{+})=g(b)$ and $g(a^{+})=g(a)$ Hence $$\mu_{g}((a,b])=g(b)-g(a)$$ Hence for $0\le s<t$ we define $$\int\limits \mathbb{1}_{(s,t]\times \Omega} \, dM=M_{t}-M_{s}$$Letting $F\subset\Omega$ we then consider $$\mathbb{1}_{(s,t]\times F}\begin{cases} \mathbb{1}_{(s,t]} & \omega\in F \\ 0 & \text{otherwise} \end{cases}$$Giving us $$\int\limits \mathbb{1}_{(s,t]\times F} \, dM =\mathbb{1}_{F}(M_{t}-M_{s})$$

Itô Isometry (1)

Let now $(M_{t})_{t\ge 0}$ be a right continuous $L^{2}$-martingale. We denote $\mu_{M}$ as the Doléans measure on $(\mathbb{R}^{+}\times\Omega,\mathscr{P})$ associated with $M$. where $\mathscr{P}$ is the predictable σ-algebra generated by $\mathcal{R}$. Now let $F\in\mathcal{F}_{s}$ $$E\left[ \left( \int\limits \mathbb{1}_{(s,t]\times F} \, dM \right)^{2} \right]=E[\mathbb{1}_{F}(M_{t}-M_{s})^{2}]=\underbrace{ E[\mathbb{1}_{F}(M_{t}^{2}-M_{s}^{2})] }_{ \mu_{M}({(s,t]\times F}) }$$which is our first version of the isometry. More formally: $\forall 0\le s<t, \forall F\in\mathcal{F}_{s}$ $$E\left[ \left( \smash[b]{\underbrace{ \int\limits \mathbb{1}_{(s,t]\times F} \, dM }_{ \mathclap{\text{Itô Stochastic Integral}} }}\right)^{2} \right]=\mu_{M}({(s,t]\times F}) =\underbrace{ \int\limits _{\mathbb{R}^{+}\times\Omega}\mathbb{1}_{(s,t]\times F} \, d\mu_{M} }_{ \text{Lebesgue Integral} }$$Which we can think of as a bridge from our Itô stochastic integral to the Lebesgue integral or Lebesgue-Stieltjes integral.

Itô Isometry (2)

We then introduce some sets/σ-algebras:

• $\mathcal{R}$, the set of predictable rectangles
• $\mathcal{A}$, the ring generated by $\mathcal{R}$
• $\mathscr{P}$, predictable $\sigma$-algebra We then define $\mathcal{E}$ to be the set of set of simple predictable processes where any $X\in\mathcal{E}$ can be represented like so: $$\tag{★}X=\sum_{i=1}^{n}a_{i}\mathbb{1}_{(s_{i},t_{i}]\times F_{i}}+\sum_{k=1}^{n}d_{k}\mathbb{1}_{\{ 0 \}\times F_{0k}}$$where $0\le s_{i}<t_{i}, F_{i}\in\mathcal{F}_{s_{i}}, \forall i=1,\dots,n$ and $F_{0k}\in\mathcal{F}_{0}, \forall k=1,\dots,n$. We then use this to define the new isometry:

Itô Isometry (3)

Now we extend the previous isometry by proving

This allows us to get to our final destination.