📂Stochastic Differential Equations

Ito Process

Definition ¹

Given a probability space $( \Omega , \mathcal{F} , P)$ and a filtration $\left\{ \mathcal{F}_{t} \right\}_{t \ge 0}$, suppose that a Wiener process $\left\{ W_{t} \right\}_{t \ge 0}$ is $\mathcal{F}_{t}$-adapted, and for $f \in \mathcal{L}^{1} [0 , \infty)$ and $g \in \mathcal{L}^{2} [0 , \infty)$, we define a $1$-dimensional continuous $\mathcal{F}_{t}$-adapted stochastic process $\left\{ X_{t} \right\}_{t \ge 0}$ as a $1$-dimensional Itô Process. $$X (t) := X_{0} + \int_{0}^{t} f(s) ds + \int_{0}^{t} g(s) d W_{s}$$

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• $\mathcal{L}^{p} (E)$ is the Lebesgue space consisting of functions with domain $E$.

Explanation

Normally, because there are many integral symbols, it is inconvenient to use the above definition as it is, so it is often expressed using Stochastic Differential. $$d X(t) = f(t) dt + g(t) d W_{t}$$

Generalization ²

Suppose a $i \ne j \implies W_{i} (t) \perp W_{j}$-dimensional Brownian motion $\left\{ \mathbf{W}_{t} \right\}_{t \ge 0} := \left( W_{1} (t) , \cdots , W_{m} (t) \right)$ is $\mathcal{F}_{t}$-adapted $$\begin{align*} \mathbf{f} (t) = \left( f_{1} (t) , \cdots , f_{d} (t) \right) \in & \mathcal{L}^{1} \left( [0, \infty)^{d} \right) \\ \mathbf{g} (t) = \begin{bmatrix} g_{11} (t) & \cdots & g_{1m} (t) \\ \vdots & \ddots & \vdots \\ g_{d1} (t) & \cdots & g_{dm} (t) \end{bmatrix} \in & \mathcal{L}^{2} \left( [0, \infty)^{d \times m} \right) \end{align*}$$ For vector functions $\mathbf{f} : [0, \infty) \to \mathbb{R}^{d}$ and matrix functions $\mathbf{g} : [0, \infty) \to \mathbb{R}^{d \times m}$, we define a $d$-dimensional continuous $\mathcal{F}_{t}$-adapted stochastic process $\left\{ \mathbf{X}_{t} \right\}_{t \ge 0}$ as a $d$-dimensional Itô Process. $$\mathbf{X} (t) := \mathbf{X}_{0} + \int_{0}^{t} \mathbf{f}(s) ds + \int_{0}^{t} \mathbf{g}(s) d \mathbf{W}_{s}$$ Of course, it can also be written in the form of the following stochastic differential. $$d \mathbf{X}(t) = \mathbf{f}(t) dt + \mathbf{g}(t) d \mathbf{W}_{t}$$