1 Itô Process
1.1 Motivation
References: [[oksendal-pdf]]
Consider the probability space \((\Omega, \mathcal{F}, \mathbb{P})\) and let \((\boldsymbol{B}(t))=(B_{1}(t), \dots, B_{n}(t))\) be \(n\)-dimensional (standard) Brownian Motion. Define \(\boldsymbol{\mathcal{F}}(t)=\boldsymbol{\mathcal{F}}^{(n)}(t)\) to be the filtration generated by \((B_{i}(s))_{{1\leq i \leq n}, 0 \leq s \leq t}\). Recall that we call the natural filtration of \((\boldsymbol{B}(t))\) and interpret as the history of the process \(\boldsymbol{B}(t)\) up to time \(t\).
Our aim is to construct the integral \(I(f)=\int_{S}^{T}{f_{t}}~d{B_{t}}\) for a general process \(f_{t}\) in order to define an Itô process however this is somewhat challenging since \(B_{t}\) can be shown to be nowhere differentiable and have infinite variation meaning that we cannot use a Riemann Integral or Lebesgue Integral. Instead we construct the Itô integral as the limit in mean square (or probability) of simple processes (akin to Riemann sums) where the integrand is evaluated at the left endpoint of each subinterval. This ensures the process is non-anticipating (adapted to the filtration) which preserves martingale properties.
1.2 Itô Integral Construction
We begin by defining the space of all measurable (actually progressive), adapted and \(\mathcal{L}_{2}\) processes for which we can define the Itô integral. Let \(\mathcal{V}=\mathcal{V}(S,T)\) be the class of functions \[ \mathcal{V}(S,T):=\{f(t,\omega):[0,\infty) \times \Omega \to \mathbb{R}\}, \] such that: 1. \((t, \omega)\to f(t, \omega)\) is \(\mathcal{B}\times\mathcal{F}\)-measurable, where \(\mathcal{B}\) denotes the Borel σ-Algebra on \([0,\infty)\); 2. \(f(t,\omega)\) is \(\mathcal{F}_{t}\)-adapted; 3. \(\mathbb{E}[\int_{S}^T f(t,\omega)^2dt]<\infty\).
For (3) say \(f \in \mathcal{L}_{2}([S,T]\times \Omega)\) since \(\left< f_{t},g_{t} \right> = \mathbb{E}\left[ \int_{S}^T f_{t}g_{t} ~dt \right]\) is the inner product on such a space. Our goal is to define the stochastic integral \(I(f)=\int_{S}^T f_{t}dB_{t}\) for a general process \(f\) in such a Hilbert space.
Our construction process is as follows, first consider defining such an integral for an elementary process and then extend it to general processes in \(\mathcal{V}\). The main thought is to find a dense subset of the Hilbert space and define the stochastic integral on it to prove that it is actually an [[isometry|isometry]]. We are then able to extend it as the isometry to the whole Hilbert space.
A process \((X_{t})\) is called an elementary adapted process if it can be written in the form \[ \varphi_{t}(\omega)=\sum_{j \geq 0} e_{j}\mathbb{1}_{\left[\frac{j}{2^n}, \frac{{j+1}}{2^n}\right)}(t), \] where \(n\in \mathbb{N}\) and each \(e_{j}\in \mathcal{F}_{t}\) and \(e_{j}\in\mathcal{L}_{2}\).
This is a slightly changed step function where \(e_j\) are now a collection of \(\mathcal{F}\) measurable random variables with finite over the [[dyadic-number|dyadic number]] partition. For elementary functions we can define the stochastic integral \[ \int_{S}^T \varphi_{t}~dB_{t}=\sum_{j \geq 0}e_{j}[B_{t_{j+1}}-B_{t_{j}}], \] where we specify \[ t_{k}=t_{k}^{(n)}=\begin{cases} \frac{k}{2^n} & \text{if }S\leq \frac{k}{2^n}\leq T \\ S & \text{if } \frac{k}{2^n} < S \\ T & \text{if } \frac{k}{2^n}>T. \end{cases} \] At this point it is natural to approximate a given function \(f(t,\omega)\) by \[ \sum_{j}f_{t_{j}^*}(\omega)\cdot \mathbb{1}_{[t_{j},t_{j+1})}(t), \] where the points \(t_{j}^*\in[t_{j}, t_{j+1}]\) allowing us to define the integral \(\int_{S}^T f_{t}(\omega)dB_{t}(\omega)\) as the limit (in \(\mathcal{L}_{2}\)) of \(\sum_{j}f_{t_{j}^*}(\omega)[B_{t_{j+1}}-B_{t_{j}}](\omega)\) as \(n \to \infty\). However, unlike with the Riemann-Stieltjes Integral our choice of \(t_{j}^*\) impacts our final result, as shown with the following example.
Example (Choice of \(t^*\) on interval) We define \[ \varphi^{(1)}_{t}(\omega)=\sum_{j \geq 0} B_{\frac{j}{2^n}}(\omega)\mathbb{1}_{[\frac{j}{2^n}, \frac{{j+1}}{2^n})}(t)\quad \& \quad \varphi_{t}^{(2)}(\omega) = \sum_{j \geq 0} B_{\frac{j+1}{2^n}}(\omega)\mathbb{1}_{[\frac{j}{2^n}, \frac{{j+1}}{2^n})}(t). \] Then from independent increments we have \[ \mathbb{E}\left[ \int_{0}^T \varphi _{t}^{(1)}(\omega)~dB_{t}(\omega)\right]=\sum_{j \geq 0 }\mathbb{E}[B_{t_{j}}(B_{t_{j+1}}-B_{t_{j}})]=0. \] However \[ \mathbb{E}\left[ \int_{0}^T \varphi^{(2)}_{t}(\omega)~dB_{t}(\omega) \right]=\sum_{j \geq 0 }\mathbb{E}[B_{t_{j+1}}\cdot(B_{t_{j+1}}-B_{t_{j}})]=T. \] The two most useful choices have been found to be the left end point \(t_{j}^* = t_{j}\) leading to the Itô integral (the primary consideration in this note and throughout this topic) and the mid point \(t_{j}^*=(t_{j}+t_{j+1})/2\) leading to the Stratonovich integral.
To begin our extension from the elementary functions we make the following important observation.
If \(\varphi_{t}(\omega)\) is bounded and elementary then we have \[ \mathbb{E}\left[ \left( \int_{S}^T \varphi_{t}(\omega )dB_{t} \right) ^2 \right] =\mathbb{E}\left[ \int_{S}^T \varphi_{t}^2(\omega )dt \right]. \]
PROOF: Define \(\Delta B_{j}=B_{t_{j+1}}-B_{t_{j}}\) and then note that \[ \mathbb{E}[e_{i}e_{j}\Delta B_{i}\Delta B_{j}]=\begin{cases}0 & \text{if }i \neq j \\ \mathbb{E}[e_{j}^2]\cdot(t_{j+1}-t_{j}) & \text{if }i = j, \end{cases} \] from independent increments. Thus we have \[ \begin{align}\mathbb{E}\left[ \int_{S}^T \varphi_{t}~dB_{t} \right]=\sum_{i,j}\mathbb{E}[e_{i}e_{j}\Delta B_{i}\Delta B_{j}]=\sum_{j}\mathbb{E}[e_{j}^2]\cdot(t_{j+1}-t_{j})=\mathbb{E}\left[ \int_{S}^T \varphi_{t}^2~dt \right],\end{align} \] completing the proof. \(\square\)
1.2.1 Step 1: Bounded Continuous Processes
Letting \(g \in \mathcal{V}\) be bounded and \(g_{\cdot}(\omega)\) continuous for each \(\omega\). Then there exist elementary functions \(\varphi^{(n)}\in \mathcal{V}\) such that \[ \mathbb{E}\left[ \int_{S}^T (g_{t} - \varphi_{t})^2dt \right]\to 0\quad \text{as}\quad n \to \infty. \] To see this define \(\varphi_{t}^{(n)}(\omega)=\sum_{j}g_{t_{j}}(\omega)\cdot \mathbb{1}_{[t_{j},t_{j+1})}(t)\) which is elementary since \(g \in\mathcal{V}\) and \[ \int_{S}^T(g-\varphi_{n})^2dt\to 0\quad \text{as}\quad n \to \infty, \] for each \(\omega\) (pointwise) since \(g_{\cdot}(\omega)\) is pointwise continuous. Hence the result follows by bounded convergence.
1.2.2 Step 2: Bounded Processes
Let \(h \in \mathcal{V}\) be bounded. Then there exist bounded functions \(g^{(n)} \in \mathcal{V}\) such that \(g_{t}^{(n)}(\omega)\) is continuous for all \(\omega\) and \(n\), and \[ \mathbb{E}\left[ \int_{S}^T (h_{t} - g_{t}^{(n)})^2dt \right]\to 0. \] To see this suppose \(\lvert h_{t}(\omega) \rvert\leq M\) for all \((t, \omega)\). For each \(n\) let \(\phi^{(n)}\) be a non-negative, continuous function on \(\mathbb{R}\) such that: 1. \(\phi^{(n)}(x)=0\) for \(x \leq -\frac{1}{n}\) and \(x \geq 0\); and 2. \(\int_{\mathbb{R}}\phi^{(n)}(x)~dx =1.\) Define \[ g_{t}^{(n)}(\omega)=\int_{0}^t\phi^{(n)}_{t}(s-t)h_{s}(\omega)ds. \] Then \(g_{\cdot}^{(n)}(\omega)\) is continuous for each \(\omega\) (pointwise) and \(\lvert g_{t}^{(n)}(\omega) \rvert\leq M\). Since \(h \in \mathcal{V}\) we show that \(g^{(n)}_{t}(\cdot)\) is adapted (e.g. Karatzas and Shreve (1991); pg 133). Moreover \[ \int_{S}^T(g^{(n)}_{s}(\omega)-h_{s}(\omega))^2ds\to 0 \quad \text{pointwise as}\quad n\to \infty, \] since \(\{ \phi_{n} \}\) constitutes an approximate identity (e.g. Hoffman (1962); pg 22). Hence, by bounded convergence we obtain the result.
1.2.3 Step 3: General Process
Let \(f \in\mathcal{V}\). Then there exists a sequence \(\{ h^{(n)} \}\subset\mathcal{V}\) such that \(h_{n}\) is bounded for each \(n\) and \[ \mathbb{E} \left[ \int_{S}^T(f-h^{(n)})^2 dt \right]\to 0 ~~\text{as}~~ n \to \infty. \] To see this we simply put \[ h^{(n)}_{t}(\omega) = \begin{cases} -n & \text{if } f_{t}(\omega)<-n \\ f_{t}(\omega) & \text{if } -n \leq f_{t}(\omega)\leq n \\ n & \text{if } f_{t}(\omega)>n, \end{cases} \] and the conclusion follows by dominated convergence.
1.3 Itô Integral
1.3.1 Itô Integral
For \(f\in\mathcal{V}(S,T)\) the Itô integral from \(S\) to \(T\) is defined by \[ \int_{S}^Tf(t, \omega)dB_{t}(\omega) =\lim_{ n \to \infty } \int_{S}^T \phi_{n}(t, \omega)dB_{t}(\omega), \] where the limit is taken in \(L^2\) and \(\{ \phi_{n} \}\) is a sequence of elementary functions such that \[ \mathbb{E}\left[ \int_{S}^T (f(t, \omega)-\phi_{n}(t, \omega))^2dt \right] \to 0, \] as \(n \to \infty\).
1.3.2 Itô Isometry
For \(f \in\mathcal{V}(S,T)\) we have that \[ \mathbb{E}\left[ \left( \int_{S}^T f(t,\omega )dB_{t} \right) ^2 \right] =\mathbb{E}\left[ \int_{S}^T f^2(t,\omega )dt \right]. \]
1.3.3 Itô Integral Properties
1.4 Itô Process
A Stochastic Differential Equation (SDE) written in differential form is \[ dX_{t}=\mu(t, X_{t})dt + \sigma(t, X_{t})dB_{t}, \] with initial value \(X_{0}=x\). This is not rigorous since we have not formally defined \(dB_{t}\). Instead this expression derives its meaning from the integral form of the SDE which is \[ X_{t}=x+\int_{0}^{t}{\mu(s, X_{s})}~d{s} + \int_{0}^{t}{\sigma(s,X_{s})}~d{B_{s}}, \] where we say that \(\mu\) is the drift coefficient (controls the growth of the process) and \(\sigma\) is the diffusion coefficient (controls the size of the noise).