[5][6][7][8] Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein and Marian Smoluchowski in 1905, although Louis Bachelier was the first person credited with modeling Brownian motion in 1900, giving a very early example of a stochastic differential equation now known as Bachelier model.
Some of 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 most common form of SDEs in the literature is an ordinary differential equation with the right hand side perturbed by a term dependent on a white noise variable.
In most cases, SDEs are understood as continuous time limit of the corresponding stochastic difference equations.
This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral.
The Itô calculus 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, although it is possible and in some cases preferable to model random motion on manifolds through Itô SDEs,[6] for example when trying to optimally approximate SDEs on submanifolds.
This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations.
Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically.
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 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.
is a set of vector fields that define the coupling of the system to Gaussian white noise,
For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.
[15] Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations.
It is also the notation used in publications on numerical methods for solving stochastic differential equations.
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.
Generalizing the geometric Brownian motion, it is also possible to define SDEs admitting strong solutions and whose distribution is a convex combination of densities coming from different geometric Brownian motions or Black Scholes models, obtaining a single SDE whose solutions is distributed as a mixture dynamics of lognormal distributions of different Black Scholes models.
A generalization of stochastic differential equations with the Fisk-Stratonovich integral to semimartingales with jumps are the SDEs of Marcus type.
[19] An innovative application in stochastic finance derives from the usage of the equation for Ornstein–Uhlenbeck process which is the equation for the dynamics of the return of the price of a stock under the hypothesis that returns display a Log-normal distribution.
Under this hypothesis, the methodologies developed by Marcello Minenna determines prediction interval able to identify abnormal return that could hide market abuse phenomena.
[20] [21] More generally one can extend the theory of stochastic calculus onto differential manifolds and for this purpose one uses the Fisk-Stratonovich integral.
[22] Although Stratonovich SDEs are the natural choice for SDEs on manifolds, given that they satisfy the chain rule and that their drift and diffusion coefficients behave as vector fields under changes of coordinates, there are cases where Ito calculus on manifolds is preferable.
However, a direct path-wise interpretation of the SDE is not possible, as the Brownian motion paths have unbounded variation and are nowhere differentiable with probability one, so that there is no naive way to give meaning to terms like
However, motivated by the Wong-Zakai result[23] for limits of solutions of SDEs with regular noise and using rough paths theory, while adding a chosen definition of iterated integrals of Brownian motion, it is possible to define a deterministic rough integral for every single
[24] 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-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2).
[3] Let T > 0, and let be measurable functions for which there exist constants C and D such that for all t ∈ [0, T] and all x and y ∈ Rn, where Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment: Then the stochastic differential equation/initial value problem has a P-almost surely unique t-continuous solution (t, ω) ↦ Xt(ω) such that X is adapted to the filtration FtZ generated by Z and Bs, s ≤ t, and The stochastic differential equation above is only a special case of a more general form where More generally one can also look at stochastic differential equations on manifolds.
The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as chaos, turbulence, self-organized criticality etc.
and the Goldstone theorem explains the associated long-range dynamical behavior, i.e., the butterfly effect, 1/f and crackling noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.