0.1 The Itô Integral
Suppose \(X_{t}\) is a diffusion process on the filtered probability space \((\Omega, F, (\mathcal{F}_{n}), \mathbb{P})\) and that \(f_{t}\) is another stochastic process that is adapted to the same filtration. Then the general Itô integral is defined as \[ Y_{t}=\int_{0}^t f_{s}dX_{s}. \] We are primarily interested in Itô integrals with respect to Brownian motion used in [[ksendal]] which have the integral form \[ X_{t+s}-X_{t}=\int_{t}^{t+s} \mu(X_{u}, u)du+\int_{t}^{t+s}\sigma(X_{u},u)dB_{u}, \] where our Itô integral is \[ \int_{t}^{t+s}\sigma(X_{u}, u) dB_{u}, \] where \(\sigma(X_{t}, t)\) is some function of time and the diffusion \(X_{t}\) and \(B_{t}\) is Brownian motion.
The construction of the Itô integral is detailed in Itô Integral Construction but is omitted from this note for efficiency. For clarity we do note the class of functions for which the Itô integral is defined \(\mathcal{V}=\mathcal{V}(S,T)\) which contains functions (random variables) \(f(t,\omega):[0, \infty)\times \Omega \to \mathbb{R}\) such that
- \((t,\omega)\to f(t,\omega)\) is \(\mathcal{B}\times \mathcal{F}\)-measurable, where \(\mathcal{B}\) denotes the Borel σ-Algebra on \([0,\infty)\),
- \(f(t,\omega)\) is \(\mathcal{F}_{t}\)-adapted, and
- \(\mathbb{E}\left[ \int_{S}^T f(t,\omega)^2~dt \right]<\infty\). ### Itô Isometry
The most important result for Itô integrals is the Itô isometry which states that for all \(f \in \mathcal{V}(S,T)\) we have \[ \mathbb{E}\left[ \left(\int_{S}^T f(t,\omega)~dB_{t}\right)^2 \right]=\mathbb{E}\left[ \int_{S}^T f^2(t,\omega)dt \right]. \] ### Itô Integral Properties
Let \(f,g\in \mathcal{V}(0,T)\) and let \(0 \leq S < U < T\), then the following hold:
- \(\int_{S}^T fdB_{t}=\int_{S}^U fdB_{t}+\int_{U}^T fdB_{t}\) for almost all \(\omega\),
- \(\int_{S}^T (cf+g)dB_{t}=c \cdot \int_{S}^T fdB_{t}+\int_{S}^T gdB_{t}\) where \(c\) is constant for almost all \(\omega\),
- \(\mathbb{E}\left[ \int_{S}^T fdB_{t} \right]=0\)
- \(\int_{S}^T fdB_{t}\) is \(\mathcal{F}_{T}\)-measurable.