NationMaster - Encyclopedia: Stochastic differential equation

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. Typically, SDEs incorporate white noise which can be thought of as the derivative of Brownian motion (or the Wiener Process); however, it should be mentioned that other types of random fluctuations are possible, such as jump processes (see [1]). A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... In the mathematics of probability, a stochastic process is a random function. ... Calculated spectrum of a generated approximation of white noise White noise is a random signal (or process) with a flat power spectral density. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ...

1 Background
2 Use in Physics
- 2.1 Note on "the Langevin equation"
3 Use in probability and financial mathematics
4 Existence and uniqueness of solutions
5 References
6 See also
7 External links

Background

The earliest work on SDEs was done to describe Brownian motion in Einstein's famous paper. This work was followed upon by Langevin. Later Ito and Stratonovich put SDEs on more solid mathematical footing.

Terminology

In physical science, SDEs are usually written as Langevin equations. These are sometimes confusingly called "the Langevin equation" even though there are many possible forms. These consist of an oridinary differential equation containing a deterministic part and an additonal random white noise term. A second form is the Fokker-Planck equation. The Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability distribution function. The third form is the stochastic differential equation. This is similar to the Langevin form, but it is usually written in differential form. This form is used frequently by mathematicians and in quantitative finance. SDEs come in two varieties, corresponding to two versions of stochastic calculus. Calculated spectrum of a generated approximation of white noise White noise is a random signal (or process) with a flat power spectral density. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

Stochastic Calculus

Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is non-differentiable; thus, it requires its own rules of calculus. Two versions of stochastic calculus are used, the Ito stochastic calculus and the Stratonovich stochastic calculus. It is somewhat ambiguous when one should use one or the other. Conveniently, one can readily convert an Ito SDE to an equivalent Stratonovich SDE and back again to aid in solution; however, one must be careful which calculus to use when the SDE is initially written down. Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ... A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ... ItÅ calculus, named after Kiyoshi ItÅ, treats mathematical operations on stochastic processes. ... In probability theory, a branch of mathematics, the Stratonovich integral (developed simultaneously by Ruslan L. Stratonovich and D. L. Fisk) is a stochastic integral, the most common alternative to the ItÅ integral. ...

Numerical Solutions

Numerical solution of stochastic differential equations and especially stochastic partial differential equations is a young field relatively speaking. Almost all algorithms that are used for the solution of ordinary differential equations will work very poorly for SDEs, having very poor numerical convergence. Stochastic partial differential equations (SPDEs) are similar to ordinary stochastic differential equations. ...

Use in Physics

In Physics, one typically writes down an SDE in the Langevin form and refers to it as "the Langevin equation." For example, a general coupled set of first-order SDEs 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 $mathbf{x}={x_i|1le ile k}$ 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". This form is usually usable because there are standard techniques for transforming a higher-order equations into several coupled first-order equations by introducing new unknowns. 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. Additive noise is the simpler of the two cases. One can often find the correct solution using ordinary calculus. In particular, one can use the ordinary chain rule of calculus. However, in the case of mulitplicative noise, the Langevin equation is not a well-defined intenty on it's own, and one must specify whether the Langevin equation should be interpreted as an Ito SDE or a Stratonovich SDE. Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

In Physics, the main method of solution is to find the probability distribution function as a function of time using the equivalent Fokker-Planck equation (FPE). The Fokker-Planck equation is a deterministic partial differential equation. It tells how the probability distribution function evolves in time similarly to how the Schrödinger equation gives the time evolution of the quantum wave function or the diffusion equation gives the time evolution of chemical concentration. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques include the path integration that draws 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.) or by writing down ordinary differential equations for the statistical moments of the probability distribution function. The Fokker-Planck equation (named after Adriaan Fokker and Max Planck; also known as the Kolmogorov Forward equation) describes the time evolution of the probability density function of position and velocity of a particle. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems, and for other computations. ... This article or section is in need of attention from an expert on the subject. ... Fig. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ... -1...

Note on "the Langevin equation"

"the" in "the Langevin equation" is somewhat ungrammatical nomenclature. Each individual physical model has has its own Langevin equation. Perhaps, "a Langevin equation" or "the associated Langevin equation" would conform better with common English usage.

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. Probability theory is the branch of mathematics concerned with analysis of random phenomena. ... Mathematical finance is the branch of applied mathematics concerned with the financial markets. ... In mathematics, generalized functions are objects generalizing the notion of functions. ...

A typical equation is of the form

$mathrm{d} X_t = mu(X_t,t), mathrm{d} t + sigma(X_t,t), mathrm{d} B_t ,$

where $B$ denotes a Wiener process (Standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding integral equation A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ...

$X_{t+s} - X_{t} = int_t^{t+s} mu(X_u,u) mathrm{d} u + int_t^{t+s} sigma(X_u,u), mathrm{d} B_u .$

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 normally distributed with expectation μ(X_t, t) δ and variance σ(X_t, t)² δ and is independent of the past behavior of the process. This is so because the increments of a Wiener process are independent and normally distributed. The function μ is referred to as the drift coefficient, while σ is called the diffusion coefficient. The stochastic process X_t is called a diffusion process, and is usually a Markov process. Continuous time occurs when time is sampled continuously. ... In the mathematics of probability, a stochastic process is 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. ... ItÅ calculus, named after Kiyoshi ItÅ, treats mathematical operations on stochastic processes. ... Look up Heuristic in Wiktionary, the free dictionary. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... A diffusion process is in probability theory the solution to a stochastic differential equation. ... It has been suggested that this article or section be merged with Markov property. ...

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, Pr). 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, the definition of the probability space is the foundation of probability theory. ...

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 a Wiener process. ...

$mathrm{d} X_t = mu X_t , mathrm{d} t + sigma X_t , mathrm{d} B_t.$

which is the equation for the dynamics of the price of a stock in the Black Scholes options pricing model of financial mathematics. This article does not cite any references or sources. ... The Blackâ€“Scholes model is a model that posits that a stock price evolves according to geometric Brownian motion and that there is a well defined risk free instrument that pays a constant interest rate. ...

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.

Existence and uniqueness of solutions

As with deterministic ordinary and partial differential equations, it is important to know whether a given a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itō SDEs taking values in n-dimensional Euclidean space Rⁿ and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2). 2-dimensional renderings (ie. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

Let T > 0, and let

$mu : mathbb{R}^{n} times [0, T] to mathbb{R}^{n};$

$sigma : mathbb{R}^{n} times [0, T] to mathbb{R}^{n times m};$

be measurable functions for which there exist constants C and D such that In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

$big| mu (x, t) big| + big| sigma (x, t) big| leq C big( 1 + | x | big);$

$big| mu (x, t) - mu (y, t) big| + big| sigma (x, t) - sigma (y, t) big| leq D | x - y |;$

for all t ∈ [0, T] and all x and y ∈ Rⁿ, where

$| sigma |^{2} = sum_{i, j = 1}^{n} | sigma_{ij} |^{2}.$

Let Z be a random variable that is independent of the σ-algebra generated by B_s, s ≥ 0, and with finite second moment:-1...

$mathbb{E} big[ | Z |^{2} big] < + infty.$

Then the stochastic differential equation/initial value problem

$mathrm{d} X_{t} = mu (X_{t}, t) , mathrm{d} t + sigma (X_{t}, t) , mathrm{d} B_{t} mbox{ for } t in [0, T];$

X t = Z;

has a Pr-almost surely unique t-continuous solution (t, ω) |→ X_t(ω) such that X is adapted to the filtration F_t^Z generated by Z and B_s, s ≤ t, and In probability theory, an event happens almost surely (a. ... In the study of stochastic processes, an adapted process (or non-anticipating process) is one that cannot see into the future. It is essential, for instance, in the definition of the ItÅ integral, which only makes sense if the integrand is an adapted process. ... In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ...

$mathbb{E} left[ int_{0}^{T} | X_{t} |^{2} , mathrm{d} t right] < + infty.$

References

Adomian, George (1983). Stochastic systems, Mathematics in Science and Engineering (169). Orlando, FL: Academic Press Inc..
Adomian, George (1986). Nonlinear stochastic operator equations. Orlando, FL: Academic Press Inc..
Adomian, George (1989). Nonlinear stochastic systems theory and applications to physics, Mathematics and its Applications (46). Dordrecht: Kluwer Academic Publishers Group.
Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1.
Teugels, J.and Sund B. (eds.) (2004). Encyclopedia of Actuarial Science. Chichester: Wiley, 523–527.

Bernt Karsten Ã˜ksendal (born 1945-04-10, Fredrikstad, Norway) is a Norwegian mathematician. ...

External links

Archive of well-known stochastic differential equations, sitmo.com

Categories: Differential equations | Stochastic differential equations | Stochastic processes

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