Stochastic Differential Equations
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A stochastic differential equation (SDE) is a
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
in which one or more of the terms is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
, resulting in a solution which is also a stochastic process. SDEs are used to
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
various phenomena such as stock prices or physical systems subject to
thermal fluctuations In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.In statistical mechanics they are often simply referred to as fluctuations. All thermal fluctuations b ...
. Typically, SDEs contain a variable which represents random
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
calculated as the derivative of
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
or the
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
. However, other types of random behaviour are possible, such as
jump process A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process. In finance, various stochastic mod ...
es.
Random differential equation In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
s are conjugate to stochastic differential equations.


Background

Stochastic differential equations originated in the theory of
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, in the work of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
and Smoluchowski. These early examples were linear stochastic differential equations, also called 'Langevin' equations after French physicist Langevin, describing the motion of a harmonic oscillator subject to a random force. The mathematical theory of stochastic differential equations was developed in the 1940s through the groundbreaking work of Japanese mathematician Kiyosi Itô, who introduced the concept of
stochastic integral Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
and initiated the study of nonlinear stochastic differential equations. Another approach was later proposed by Russian physicist Stratonovich, leading to a calculus similar to ordinary calculus.


Terminology

The most common form of SDEs in the literature is an
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
with the right hand side perturbed by a term dependent on a
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
variable. In most cases, SDEs are understood as continuous time limit of the corresponding
stochastic difference equation In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
s. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral. Such a mathematical definition was first proposed by Kiyosi Itô in the 1940s, leading to what is known today as the
Itô calculus Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The centra ...
. Another construction was later proposed by Russian physicist Stratonovich, leading to what is known as the
Stratonovich integral In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in a ...
. The Itô integral and
Stratonovich integral In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in a ...
are related, but different, objects and the choice between them depends on the application considered. The
Itô calculus Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations. The centra ...
is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time. The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion on
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
. An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the
Smoluchowski equation Marian Smoluchowski (; 28 May 1872 – 5 September 1917) was a Polish physicist who worked in the Polish territories of the Austro-Hungarian Empire. He was a pioneer of statistical physics, and an avid mountaineer. Life Born into an upper- ...
or the
Fokker–Planck equation In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
, an equation describing the time evolution of
probability distribution function Probability distribution function may refer to: * Probability distribution * Cumulative distribution function * Probability mass function * Probability density function In probability theory, a probability density function (PDF), or density ...
s. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of stochastic evolution operator. In physical science, there is an ambiguity in the usage of the term "Langevin SDEs". While Langevin SDEs can be of a more general form, this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure, leading to a N=2 supersymmetric model closely related to
supersymmetric quantum mechanics In theoretical physics, supersymmetric quantum mechanics is an area of research where supersymmetry are applied to the simpler setting of plain quantum mechanics, rather than quantum field theory. Supersymmetric quantum mechanics has found ap ...
. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e., (overdamped) Langevin SDEs are never chaotic.


Stochastic calculus

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
or the
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
was discovered to be exceptionally complex mathematically. The
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
is almost surely nowhere differentiable; thus, it requires its own rules of calculus. There are two dominating versions of stochastic calculus, the Itô stochastic calculus and the
Stratonovich stochastic calculus In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in ap ...
. Each of the two has advantages and disadvantages, and newcomers are often confused whether the one is more appropriate than the other in a given situation. Guidelines exist (e.g. Øksendal, 2003) and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again. Still, one must be careful which calculus to use when the SDE is initially written down.


Numerical solutions

Numerical methods for solving stochastic differential equations include the
Euler–Maruyama method In Itô calculus, the Euler–Maruyama method (also called the Euler method) is a method for the approximate numerical solution of a stochastic differential equation (SDE). It is an extension of the Euler method for ordinary differential equations ...
,
Milstein method In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published it in 1974. Description Consider the autonomous Itō stoch ...
and Runge–Kutta method (SDE).


Use in physics

In physics, SDEs have widest applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the
dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called ' ...
to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence. There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs: :\frac = F(x(t)) + \sum_^ng_\alpha(x(t))\xi^\alpha(t),\, where x\in X is the position in the system in its phase (or state) space, X, assumed to be a differentiable manifold, the F\in TX is a flow vector field representing deterministic law of evolution, and g_\alpha\in TX is a set of vector fields that define the coupling of the system to Gaussian white noise, \xi^\alpha. If X is a linear space and g are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. This term is somewhat misleading as it has come to mean the general case even though it appears to imply the limited case in which g(x) \propto x. For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition. Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a
stochastic difference equation In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a uniquely defined mathematical object that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation. 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 In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as ...
(FPE). The Fokker–Planck equation is a deterministic
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
. It tells how the probability distribution function evolves in time similarly to how the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
gives the time evolution of the quantum wave function or the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's la ...
gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by
Monte Carlo Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
simulation. Other techniques include the
path integration Path integration is the method thought to be used by animals for dead reckoning. History Charles Darwin first postulated an inertially-based navigation system in animals in 1873.quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(for example, the Fokker-Planck equation can be transformed into the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
by rescaling a few variables) or by writing down
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
for the statistical moments of the probability distribution function.


Use in probability and mathematical finance

The notation used in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
(and in many applications of probability theory, for instance
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
) is slightly different. It is also the notation used in publications on
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
for solving stochastic differential equations. This notation makes the exotic nature of the random function of time \eta_m in the physics formulation more explicit. In strict mathematical terms, \eta_m cannot be chosen as an ordinary function, but only as a
generalized function In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
. The mathematical formulation treats this complication with less ambiguity than the physics formulation. A typical equation is of the form : \mathrm X_t = \mu(X_t,t)\, \mathrm t + \sigma(X_t,t)\, \mathrm B_t , where B denotes a
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
(standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding
integral equation In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
: X_ - X_ = \int_t^ \mu(X_u,u) \mathrm u + \int_t^ \sigma(X_u,u)\, \mathrm B_u . The equation above characterizes the behavior of the
continuous time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
''X''''t'' as the sum of an ordinary
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
and an Itô integral. A
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
(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 In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
''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 In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
, and satisfies the
Markov property In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
. 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 In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
(\Omega,\, \mathcal,\, 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 A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...
:\mathrm X_t = \mu X_t \, \mathrm t + \sigma X_t \, \mathrm B_t. which is the equation for the dynamics of the price of a
stock In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...
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 Itô 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 SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in ''n''-
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R''n'' and driven by an ''m''-dimensional Brownian motion ''B''; the proof may be found in Øksendal (2003, §5.2). Let ''T'' > 0, and let :\mu : \mathbb^ \times
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
\to \mathbb^; :\sigma : \mathbb^ \times
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
\to \mathbb^; be
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...
s for which there exist constants ''C'' and ''D'' such that :\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'' ∈  , ''T''and all ''x'' and ''y'' ∈ R''n'', where :, \sigma , ^ = \sum_^ , \sigma_ , ^. Let ''Z'' be a random variable that is independent of the ''σ''-algebra generated by ''B''''s'', ''s'' ≥ 0, and with finite
second moment In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total ma ...
: :\mathbb \big Z , ^ \big< + \infty. Then the stochastic differential equation/initial value problem :\mathrm X_ = \mu (X_, t) \, \mathrm t + \sigma (X_, t) \, \mathrm B_ \mbox t \in
, T The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
:X_ = Z; has a P-
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
unique ''t''-continuous solution (''t'', ''ω'') ↦ ''X''''t''(''ω'') such that ''X'' is
adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
to the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
''F''''t''''Z'' generated by ''Z'' and ''B''''s'', ''s'' ≤ ''t'', and :\mathbb \left X_ , ^ \, \mathrm t \right< + \infty.


Some explicitly solvable SDEsKloeden 1995, pag.118


Linear SDE: general case

:dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t :X_t=\Phi_\left(X_+\int_^t\Phi^_(c(s)-b(s)d(s))ds+\int_^t\Phi^_d(s)dW_s\right) where :\Phi_=\exp\left(\int_^t\left(a(s)-\frac\right)ds+\int_^tb(s)dW_s\right)


Reducible SDEs: Case 1

:dX_t=\frac12f(X_t)f'(X_t)dt+f(X_t)dW_t for a given differentiable function f is equivalent to the Stratonovich SDE :dX_t=f(X_t)\circ W_t which has a general solution :X_t=h^(W_t+h(X_0)) where :h(x)=\int^\frac


Reducible SDEs: Case 2

:dX_t=\left(\alpha f(X_t)+\frac12 f(X_t)f'(X_t)\right)dt+f(X_t)dW_t for a given differentiable function f is equivalent to the Stratonovich SDE :dX_t=\alpha f(X_t)dt + f(X_t)\circ W_t which is reducible to :dY_t=\alpha dt+dW_t where Y_t=h(X_t) where h is defined as before. Its general solution is :X_t=h^(\alpha t+W_t+h(X_0))


SDEs and supersymmetry

In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as
chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (''Kinnikuman'') * Chaos (''Sailor Moon'') * Chaos (''Sesame Park'') * Chaos (''Warhammer'') * Chaos, in ''Fabula Nova Crystallis Final Fantasy'' * Cha ...
,
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
,
self-organized criticality Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase ...
etc. and the Goldstone theorem explains the associated long-range dynamical behavior, i.e.,
the butterfly effect ''The Butterfly Effect'' is a 2004 American science fiction thriller film written and directed by Eric Bress and J. Mackye Gruber. It stars Ashton Kutcher, Amy Smart, Eric Stoltz, William Lee Scott, Elden Henson, Logan Lerman, Ethan Suplee, and ...
, 1/f and crackling noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.


See also

*
Langevin dynamics In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by French physicist Paul Langevin. The approach is characterized by the use of simplified models while acco ...
*
Local volatility A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level S_t and of time t . As such, it is a generalisation of the Black–Sch ...
*
Stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
*
Stochastic volatility In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name d ...
*
Stochastic partial differential equations Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations. They have ...
*
Diffusion process In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
*
Stochastic difference equation In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...


References


Further reading

* * * * * * * * * * * * Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM, (2021). {{Authority control Differential equations Stochastic processes