A Stochastic process (or random process) is defined as a sequence of indexed random variables, where the index is interpreted as time. Formally, it is defined on a shared probability space \( (\Omega, \mathcal{F}, \mathcal{P}) \), where \(\Omega\) is the sample space, \(F\) is a \(\sigma\)-algebra, and \(P\) is a probability measure. The random variables are indexed by a set \(T\) and takes values in the same mathematical space \(S\). The latter must be measurable with respect to a \(\sigma\)-algebra \(\Sigma\). Therefore, given a probability space \( (\Omega, \mathcal{F}, \mathcal{P}) \) and a measurable space \( (S, \Sigma) \), a stocastic process can be represented as: \[\{ X_t : t \in T \}\] Traditionally, a point \(t\) from the set \(T\) has been interpreted as time, thus the R.V. \(X(t)\) represent a value observed at the time \(t\).
In stochastic processes, a crucial element is the increment, defined as the difference between two R.Vs of the same stochastic process. For the previous index set of time, an increment refers to the change of the stochastic process over a certain interval of time.
Stochastic differential equation (SDE)
A SDE is a differential equation in which one or more component is a stochastic process. A differential equation is described in terms of deterministic increments, while in a SDE the increment is a random variable.
The general form of a SDE is the following:
\[dX_t = \mu(t,X_t) \, dt + \sigma(t,X_t) \, dB_t
\]
where \(X_t\) is the state of the system at time \(t\); \(\mu(t,X_t)\) is the deterministic part, which represent the drift;
\(\sigma(t,X_t)\) is the stochastic part; \(dt\) is the differential of time, and \(B_t\) is a Wiener process (Brownian motion).
A function, or a path, \(X\) is a solution to the SDE above if it satisfies:
\[ X_T = \int_{0}^{T} \mu(t, X_t) \, dt + \int_{0}^{T} \sigma(t, X_t) \, dB_t \]
where the stochastic process \(X_t\) is called diffusion process.
an important example of SDE is the arihmetic Brownian motion: \[dX_t = \mu dt + \sigma \, dB_t\] But there are also SDE where \(\mu\) and \(\sigma\) not depend only by the present value of the process \(X_t\), but also by previous values of the process or others processes. In this case the solution, \(X\), is called an Itô process and is not a markov process.