Ito Calculus

Here we denote a stochastic process as \(f(t, \omega),\)f: I \times \Omega \longrightarrow \mathbb{R}$$.

Ito Integral

Let X_t be a stochastic process and \(W_t\) be a Brownian motion. The Ito integral of \(X_t\) with respect to \(W_t\) over the time interval \([0,t]\) is defined as:

\[\int_0^t X_s dW_s = \lim_{n \to \infty} \sum_{i=0}^{n-1} X_{t_i} (W_{t_{i+1}} - W_{t_i})\]

where \(0=t_0 < t_1 < \ldots < t_n = t\) is a partition of the interval \([0, t]\).

Note that the limit is defined in a probabilistic sense

Stochastic Differential Equation

Let \((\Omega, \mathbb{F}, \mathbb{P})\) be a probability space. Let \(X_t: [0, +\infty[ \times \Omega \longrightarrow \mathbb{R}\) be a stochastic process. Let \(\mu: [0, +\infty[ \times \mathbb{R} \longrightarrow \mathbb{R}\) and \(\sigma: [0, +\infty[ \times \mathbb{R} \longrightarrow \mathbb{R}\) be continuous functions. If \(X_t\) satisfies the equation:

\[X_t = x_0 + \int_0^t \mu(s, X_s) ds + \int_0^t \sigma(s, X_s) dW_s\]

then we say that \(X_t\) is a strong solution of the stochastic differential equation denoted by:

\[dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t\]

with initial initial condition \(X_0=x_0\).