What are the main types of stochastic differential equations (SDEs), and how do they differ from each other?

Stochastic Differential Equations (SDEs) are a type of differential equations that involve random variables, meaning they are driven by random processes or noise terms. There are several main types of SDEs, and I will discuss the most important ones below: Scalar SDEs, System of SDEs, and Stochastic Partial Differential Equations.

1. Scalar SDEs:

Scalar SDEs involve a single stochastic process and are usually written as:

dX_t = μ(X_t, t)dt + σ(X_t, t)dW_t,

where X_t is the scalar stochastic process, μ(X_t, t) and σ(X_t, t) are functions representing the drift and diffusion terms, respectively, and W_t represents the standard Brownian motion or Wiener process. Scalar SDEs can be further classified into linear and nonlinear SDEs.

a. Linear Scalar SDEs:

Linear Scalar SDEs have linear drift and diffusion terms, meaning that μ(X_t, t) and σ(X_t, t) are linear functions in X_t. A simple and widely used example is the Geometric Brownian Motion (GBM) model, which describes a stock price S_t, assuming a constant drift rate μ and constant volatility σ:

dS_t = μS_tdt + σS_tdW_t.

b. Nonlinear Scalar SDEs:

Nonlinear Scalar SDEs have nonlinear drift and/or diffusion terms. A well-known example is the constant elasticity of variance (CEV) model:

dS_t = μS_tdt + σS_t^γdW_t,

where γ is the elasticity of the diffusion term.

2. System of SDEs:

System of SDEs involves multiple stochastic processes and is usually written as:

dX_t = μ(X_t, t)dt + Σ(X_t, t)dW_t,

where X_t is a vector of stochastic processes, μ(X_t, t) is a vector representing the drift terms for each process, Σ(X_t, t) is a matrix representing the diffusion terms, and W_t is a vector of independent standard Brownian motions. These can also be classified into linear and nonlinear systems of SDEs based on the nature of the drift and diffusion terms.

A well-known example of a system of SDEs is the Black-Scholes-Merton option pricing model, which models the evolution of the stock price S_t and the price of an option V(S_t, t) on the stock:

$$\begin{cases} dS_t = \mu S_t dt + \sigma S_t dW_t^1, \\ dV_t = \left( rV_t - rS_t \frac{\partial V_t}{\partial S_t} +\frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 V_t}{\partial S_t^2} \right)dt + \sigma S_t \frac{\partial V_t}{\partial S_t} dW_t^2, \end{cases}$$

where r is the risk-free rate, and W_t¹ and W_t² are two standard Brownian motions with correlation ρ (i.e., dW_t¹dW_t² = ρdt).

3. Stochastic Partial Differential Equations (SPDEs):

SPDEs are a more general formulation of SDEs that involve partial differential equations with stochastic terms. These equations involve multiple dimensions in both time and space, with randomness in their initial or boundary conditions, or in the coefficients. A well-known example of an SPDE is the heat equation with a random initial condition:

$$\begin{cases} \frac{\partial u(x,t)}{\partial t} = \alpha \frac{\partial^2 u(x,t)}{\partial x^2} + f(x,t), \quad x \in (0,1), t>0, \\ u(x,0) = \xi(x), \quad x \in (0,1), \\ u(0,t) = u(1, t) = 0, \quad t>0, \end{cases}$$

where u(x, t) represents the temperature at a point x and time t, α is the thermal diffusivity, ξ(x) is a random initial condition, and f(x, t) could also be a stochastic forcing term.

In summary, the main types of Stochastic Differential Equations are Scalar SDEs, System of SDEs, and Stochastic Partial Differential Equations, with further classifications based on the nature of their drift and diffusion terms (linear or nonlinear), and the presence of randomness in their coefficients, initial, or boundary conditions.