Stochastic differential equation

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SDE redirects here; for the video display issue known as SDE, see screen door effect.

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.

1 Use in physics
2 Use in probability and financial mathematics
3 Reference
4 See also
5 External links

[edit] Use in physics

For example a general, coupled set of first-order SDEs (note that there are standard techniques for transforming a higher-order equation into several coupled first-order equations by introducing new unknowns) is often written in the form:

$\dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum_{m=1}^ng_i^m(\mathbf{x})\eta_m(t),\,$

where $\mathbf{x}=\{x_i|1\le i\le k\}$ is the set of unknowns, the $f i$ and $g i$ are arbitrary functions and the $η m$ are random functions of time, often referred to as "noise terms". If the $g i$ are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise.

The main method of solution is by use of the Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependent probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integration have also been used, drawing on the analogy between statistical physics and quantum mechanics (for example the Fokker-Planck equation can be transformed into the Schrödinger equation by rescaling a few variables).

[edit] Use in probability and financial mathematics

The notation used in probability theory (and in many applications of probability theory, for instance financial mathematics) is slightly different. The reason is that the random function of time $η m$ in the physics formulation can typically not be chosen as a usual function, but only as a generalized function. The following formulation avoids this mathematical complication.

A typical equation is of the form

$dX_t = \mu(X_t,t)\, dt + \sigma(X_t,t)\, dB_t\, ,$

where $B$ denotes a Wiener process (Brownian motion). This equation should be interpreted as a slightly colloquial way of expressing the corresponding integral equation

$X_{t+s} - X_{t} = \int_t^{t+s} \mu(X_u,u) du + \int_t^{t+s} \sigma(X_u,u)\, dB_u\, .$

The equation above characterizes the behavior of the continuous time stochastic process X_t as the sum of an ordinary Lebesgue integral and an Itō integral. A heuristic (but very helpful) interpretation of the stochastic differential equation is that in a small time interval of length $δ$ the stochastic process $X t$ changes its value by an amount that is normal distributed with expectation $μ(X t, t)δ$ and variance $σ 2 (X t, t)δ$ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normal distributed. The function $μ(x, t)$ is referred to as the drift coefficient, while $σ(x, t)$ is called the diffusion coefficient. The stochastic process $X t$ is called a diffusion process, and is usually a Markov process.

The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution. Both require the existence of a process X_t that solves the integral equation version of the SDE. The difference between the two lies in the underlying probability space $(Ω, F, P)$ . A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space.

An important example is the equation for geometric Brownian motion

$dX_t = \mu X_t\, dt + \sigma X_t\, dB_t.$

which is the equation for the dynamics of the price of a stock in the Black Scholes options pricing model of financial mathematics.

There are also more general stochastic differential equations where the coefficients $μ$ and $σ$ depend not only on the present value of the process $X t$ , but also on previous values of the process and possibly on present or previous values of other processes too. In that case the solution process, $X$ , is not a Markov process, and it is called an Itō process and not a diffusion process. When the coefficients depends only on present and past values of $X$ , the defining equation is called a stochastic delay differential equation.