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. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ...

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:

where 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 dependant 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 Schrodinger equation by rescaling a few variables). The Fokker-Planck equation (also known as the Kolmogorov Forward equation) describes the time evolution of the probability density function of position and velocity of a particle. ... Monte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. ... This article is about a formulation of quantum mechanics. ... Fig. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

Use in probability and financial mathematics

The notation used in the context of financial mathematics as well as in probability theory is slightly different. A typical equation is of the form Financial mathematics is the branch of applied mathematics concerned with the financial markets. ... Probability theory is the mathematical study of probability. ...

This equation should be interpreted as a slightly colloquial way of expressing the corresponding integral equation In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ...

The equation above characterises the behavior of the continuous time stochastic process X_t as the sum of an ordinary Lebesgue integral and an Itô integral. Continuous time occurs when time is sampled continuously. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...

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 . 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. In mathematics, a probability space or probability measure is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...

An important example is the equation for geometric Brownian motion A geometric Brownian motion (GBM) (occasionally, exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, or, perhaps more precisely, a Wiener process. ...

which is the equation for the dynamics of the price of a stock in the Black Scholes options pricing model of financial mathematics. A stock, also referred to as a share, is commonly a share of ownership in a corporation. ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...