Stochastic: refers to the property of being well described by a random probability distribution.

Stochastic differential equation (SDE): differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process.

Note:

• Ordinary differential equations (ODE): derivatives w.r.t one variable.

• Partial differential equations (PDE): derivatives w.r.t several variables.

Typical SDE in finance

The typical SDE in finance represents the dynamic of an Itô process.

“Informal” equation: \(dX_t = \mu (X_t, t) dt + \sigma (X_t, t) dB_t\)

Note: the informal form is characterized by the use of $d[…]$. It’s important to understand that such notation implies an integral i.e. $f(X_t,t)dt := \int_0^t f(X_u,u)du$.

The “formal” equation is obtained in integrating each term over the required interval:

$\int_{t}^{t+s}dX_t = [x]_{X_t}^{X_{t+s}} = X_{t+s}-X_t$

Similarly with the RHS, the formal equation is thus:

\[X_{t+s}-X_t = \int_{t}^{t+s} \mu(X_u, u)du + \int_{t}^{t+s} \sigma(X_u, u)dB_u\]

In other words, the price value change of an asset is the addition of the return and the risk at all instant.