Stochastic differential equation

Differential equations
Scope
Fields Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics List of named differential equations
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Coupled / Decoupled Exact Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay
Solution
Existence and uniqueness Picard–Lindelöf theorem Peano existence theorem Carathéodory's existence theorem Cauchy–Kowalevski theorem
General topics Initial conditions Boundary values Dirichlet Neumann Robin Cauchy problem Wronskian Phase portrait Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Green's function Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters
People
List Isaac Newton Gottfried Leibniz Jacob Bernoulli Leonhard Euler Józef Maria Hoene-Wroński Joseph Fourier Augustin-Louis Cauchy George Green Carl David Tolmé Runge Martin Kutta Rudolf Lipschitz Ernst Lindelöf Émile Picard Phyllis Nicolson John Crank
v t e

A stochastic differential equation (SDE) is a

.

SDEs have a random differential that is in the most basic case random

Background

Stochastic differential equations originated in the theory of

, 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

Stratonovich

, leading to a calculus similar to ordinary calculus.

Terminology

The most common form of SDEs in the literature is an

. Another construction was later proposed by Russian physicist

Stratonovich

, leading to what is known as the Stratonovich integral. The 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

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

.

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,^[10] leading to a N=2 supersymmetric model closely related to supersymmetric quantum mechanics. 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

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[11] include the Euler–Maruyama method, Milstein method, Runge–Kutta method (SDE), Rosenbrock method,^[12] and methods based on different representations of iterated stochastic integrals.^[13][14]

Use in physics

In physics, SDEs have wide 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 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:

${\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=F(x(t))+\sum _{\alpha =1}^{n}g_{\alpha }(x(t))\xi ^{\alpha }(t),\,}$

where ${\displaystyle x\in X}$ is the position in the system in its phase (or state) space, ${\displaystyle X}$, assumed to be a differentiable manifold, the ${\displaystyle F\in TX}$ is a flow vector field representing deterministic law of evolution, and ${\displaystyle g_{\alpha }\in TX}$ is a set of vector fields that define the coupling of the system to Gaussian white noise, ${\displaystyle \xi ^{\alpha }}$. If ${\displaystyle X}$ is a linear space and ${\displaystyle g}$ 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 ${\displaystyle g(x)\propto x}$.

For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.

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

]

Use in probability and mathematical finance

The notation used in

numerical methods

for solving stochastic differential equations. This notation makes the exotic nature of the random function of time ${\displaystyle \eta _{m}}$ in the physics formulation more explicit. In strict mathematical terms, ${\displaystyle \eta _{m}}$ cannot be chosen as an ordinary function, but only as a generalized function. The mathematical formulation treats this complication with less ambiguity than the physics formulation.

A typical equation is of the form

${\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} B_{t},}$

where ${\displaystyle B}$ denotes a Wiener process (standard Brownian motion). This equation should be interpreted as an informal way of expressing the corresponding integral equation

${\displaystyle X_{t+s}-X_{t}=\int _{t}^{t+s}\mu (X_{u},u)\mathrm {d} u+\int _{t}^{t+s}\sigma (X_{u},u)\,\mathrm {d} B_{u}.}$

The equation above characterizes the behavior of the

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[1] 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 (${\displaystyle \Omega ,\,{\mathcal {F}},\,P}$). 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

${\displaystyle \mathrm {d} X_{t}=\mu X_{t}\,\mathrm {d} t+\sigma X_{t}\,\mathrm {d} B_{t}.}$

which is the equation for the dynamics of the price of a stock in the Black–Scholes options pricing model^[2] of financial mathematics.

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.^{[2][16][17][18]} This leads to models that can deal with the volatility smile in financial mathematics.

The simpler SDE called arithmetic Brownian motion^[3]

${\displaystyle \mathrm {d} X_{t}=\mu \,\mathrm {d} t+\sigma \,\mathrm {d} B_{t}}$

was used by Louis Bachelier as the first model for stock prices in 1900, known today as Bachelier model.

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.

A generalization of stochastic differential equations with the Fisk-Stratonovich integral to semimartingales with jumps are the SDEs of Marcus type. The Marcus integral is an extension of McShane's stochastic calculus.^[19]

An innovative application in stochastic finance derives from the usage of the equation for Ornstein–Uhlenbeck process

${\displaystyle \mathrm {d} R_{t}=\mu R_{t}\,\mathrm {d} t+\sigma _{t}\,\mathrm {d} B_{t}.}$

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]}

SDEs on manifolds

More generally one can extend the theory of stochastic calculus onto

and for this purpose one uses the Fisk-Stratonovich integral. Consider a manifold ${\displaystyle M}$, some finite-dimensional vector space ${\displaystyle E}$, a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in \mathbb {R} _{+}},P)}$ with ${\displaystyle ({\mathcal {F}}_{t})_{t\in \mathbb {R} _{+}}}$ satisfying the and let ${\displaystyle {\widehat {M}}=M\cup \{\infty \}}$ be the and ${\displaystyle x_{0}}$ be ${\displaystyle {\mathcal {F}}_{0}}$-measurable. A stochastic differential equation on ${\displaystyle M}$ written

${\displaystyle \mathrm {d} X=A(X)\circ dZ}$

is a pair ${\displaystyle (A,Z)}$, such that

• ${\displaystyle Z}$ is a continuous ${\displaystyle E}$-valued semimartingale,
• ${\displaystyle A:M\times E\to TM,(x,e)\mapsto A(x)e}$ is a homomorphism of vector bundles over ${\displaystyle M}$.

For each ${\displaystyle x\in M}$ the map ${\displaystyle A(x):E\to T_{x}M}$ is linear and ${\displaystyle A(\cdot )e\in \Gamma (TM)}$ for each ${\displaystyle e\in E}$.

A solution to the SDE on ${\displaystyle M}$ with initial condition ${\displaystyle X_{0}=x_{0}}$ is a continuous ${\displaystyle \{{\mathcal {F}}_{t}\}}$-adapted ${\displaystyle M}$-valued process ${\displaystyle (X_{t})_{t<\zeta }}$ up to life time ${\displaystyle \zeta }$, s.t. for each test function ${\displaystyle f\in C_{c}^{\infty }(M)}$ the process ${\displaystyle f(X)}$ is a real-valued semimartingale and for each stopping time ${\displaystyle \tau }$ with ${\displaystyle 0\leq \tau <\zeta }$ the equation

${\displaystyle f(X_{\tau })=f(x_{0})+\int _{0}^{\tau }(\mathrm {d} f)_{X}A(X)\circ \mathrm {d} Z}$

holds ${\displaystyle P}$-almost surely, where ${\displaystyle (df)_{X}:T_{x}M\to T_{f(x)}M}$ is the differential at ${\displaystyle X}$. It is a maximal solution if the life time is maximal, i.e.,

${\displaystyle \{\zeta <\infty \}\subset \left\{\lim \limits _{t\nearrow \zeta }X_{t}=\infty {\text{ in }}{\widehat {M}}\right\}}$

${\displaystyle P}$-almost surely. It follows from the fact that ${\displaystyle f(X)}$ for each test function ${\displaystyle f\in C_{c}^{\infty }(M)}$ is a semimartingale, that ${\displaystyle X}$ is a semimartingale on ${\displaystyle M}$. Given a maximal solution we can extend the time of ${\displaystyle X}$ onto full ${\displaystyle \mathbb {R} _{+}}$ and after a continuation of ${\displaystyle f}$ on ${\displaystyle {\widehat {M}}}$ we get

${\displaystyle f(X_{t})=f(X_{0})+\int _{0}^{t}(\mathrm {d} f)_{X}A(X)\circ \mathrm {d} Z,\quad t\geq 0}$

up to indistinguishable processes.^[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. A theory of Ito calculus on manifolds was first developed by

As rough paths

Usually the solution of an SDE requires a probabilistic setting, as the integral implicit in the solution is a stochastic integral. If it were possible to deal with the differential equation path by path, one would not need to define a stochastic integral and one could develop a theory independently of probability theory. This points to considering the SDE

${\displaystyle \mathrm {d} X_{t}(\omega )=\mu (X_{t}(\omega ),t)\,\mathrm {d} t+\sigma (X_{t}(\omega ),t)\,\mathrm {d} B_{t}(\omega )}$

as a single deterministic differential equation for every ${\displaystyle \omega \in \Omega }$, where ${\displaystyle \Omega }$ is the sample space in the given probability space (${\displaystyle \Omega ,\,{\mathcal {F}},\,P}$). 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 ${\displaystyle \mathrm {d} B_{t}(\omega )}$, precluding also a naive path-wise definition of the stochastic integral as an integral against every single ${\displaystyle \mathrm {d} B_{t}(\omega )}$. However, motivated by the Wong-Zakai result

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 ${\displaystyle \omega \in \Omega }$ that coincides for example with the Ito integral with probability one for a particular choice of the iterated Brownian integral.[23] Other definitions of the iterated integral lead to deterministic pathwise equivalents of different stochastic integrals, like the Stratonovich integral. This has been used for example in financial mathematics to price options without probability.^[24]

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-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

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

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

be measurable functions for which there exist constants C and D such that

${\displaystyle {\big |}\mu (x,t){\big |}+{\big |}\sigma (x,t){\big |}\leq C{\big (}1+|x|{\big )};}$

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

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

${\displaystyle |\sigma |^{2}=\sum _{i,j=1}^{n}|\sigma _{ij}|^{2}.}$

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

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

Then the stochastic differential equation/initial value problem

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

${\displaystyle X_{0}=Z;}$

has a P-

F_t^Z generated by Z and B_s, s ≤ t, and

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

General case: local Lipschitz condition and maximal solutions

The stochastic differential equation above is only a special case of a more general form

${\displaystyle \mathrm {d} Y_{t}=\alpha (t,Y_{t})\mathrm {d} X_{t}}$

where

• ${\displaystyle X}$ is a continuous semimartingale in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle Y}$ is a continuous semimartingal in ${\displaystyle \mathbb {R} ^{d}}$
• ${\displaystyle \alpha :\mathbb {R} _{+}\times U\to \operatorname {Lin} (\mathbb {R} ^{n};\mathbb {R} ^{d})}$ is a map from some open nonempty set ${\displaystyle U\subset \mathbb {R} ^{d}}$, where ${\displaystyle \operatorname {Lin} (\mathbb {R} ^{n};\mathbb {R} ^{d})}$ is the space of all linear maps from ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} ^{d}}$.

More generally one can also look at stochastic differential equations on manifolds.

Whether the solution of this equation explodes depends on the choice of ${\displaystyle \alpha }$. Suppose ${\displaystyle \alpha }$ satisfies some local Lipschitz condition, i.e., for ${\displaystyle t\geq 0}$ and some compact set ${\displaystyle K\subset U}$ and some constant ${\displaystyle L(t,K)}$ the condition

${\displaystyle |\alpha (s,y)-\alpha (s,x)|\leq L(t,K)|y-x|,\quad x,y\in K,\;0\leq s\leq t,}$

where ${\displaystyle |\cdot |}$ is the Euclidean norm. This condition guarantees the existence and uniqueness of a so-called maximal solution.

Suppose ${\displaystyle \alpha }$ is continuous and satisfies the above local Lipschitz condition and let ${\displaystyle F:\Omega \to U}$ be some initial condition, meaning it is a measurable function with respect to the initial σ-algebra. Let ${\displaystyle \zeta :\Omega \to {\overline {\mathbb {R} }}_{+}}$ be a

with ${\displaystyle \zeta >0}$ almost surely. A ${\displaystyle U}$-valued semimartingale ${\displaystyle (Y_{t})_{t<\zeta }}$ is called a maximal solution of

${\displaystyle dY_{t}=\alpha (t,Y_{t})dX_{t},\quad Y_{0}=F}$

with life time ${\displaystyle \zeta }$ if

• for one (and hence all) announcing ${\displaystyle \zeta _{n}\nearrow \zeta }$ the stopped process ${\displaystyle Y^{\zeta _{n}}}$ is a solution to the stopped stochastic differential equation

${\displaystyle \mathrm {d} Y=\alpha (t,Y)\mathrm {d} X^{\zeta _{n}}}$

• on the set ${\displaystyle \{\zeta <\infty \}}$ we have almost surely that ${\displaystyle Y_{t}\to \partial U}$ with ${\displaystyle t\to \zeta }$.^[25]

${\displaystyle \zeta }$ is also a so-called explosion time.

Some explicitly solvable examples

Explicitly solvable SDEs include:^[11]

Linear SDE: General case

${\displaystyle \mathrm {d} X_{t}=(a(t)X_{t}+c(t))\mathrm {d} t+(b(t)X_{t}+d(t))\mathrm {d} W_{t}}$

${\displaystyle X_{t}=\Phi _{t,t_{0}}\left(X_{t_{0}}+\int _{t_{0}}^{t}\Phi _{s,t_{0}}^{-1}(c(s)-b(s)\mathrm {d} (s))\mathrm {d} s+\int _{t_{0}}^{t}\Phi _{s,t_{0}}^{-1}\mathrm {d} (s)\mathrm {d} W_{s}\right)}$

where

${\displaystyle \Phi _{t,t_{0}}=\exp \left(\int _{t_{0}}^{t}\left(a(s)-{\frac {b^{2}(s)}{2}}\right)\mathrm {d} s+\int _{t_{0}}^{t}b(s)\mathrm {d} W_{s}\right)}$

Reducible SDEs: Case 1

${\displaystyle \mathrm {d} X_{t}={\frac {1}{2}}f(X_{t})f'(X_{t})\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}}$

for a given differentiable function ${\displaystyle f}$ is equivalent to the Stratonovich SDE

${\displaystyle \mathrm {d} X_{t}=f(X_{t})\circ W_{t}}$

which has a general solution

${\displaystyle X_{t}=h^{-1}(W_{t}+h(X_{0}))}$

where

${\displaystyle h(x)=\int ^{x}{\frac {\mathrm {d} s}{f(s)}}}$

Reducible SDEs: Case 2

${\displaystyle \mathrm {d} X_{t}=\left(\alpha f(X_{t})+{\frac {1}{2}}f(X_{t})f'(X_{t})\right)\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}}$

for a given differentiable function ${\displaystyle f}$ is equivalent to the Stratonovich SDE

${\displaystyle \mathrm {d} X_{t}=\alpha f(X_{t})\mathrm {d} t+f(X_{t})\circ W_{t}}$

which is reducible to

${\displaystyle \mathrm {d} Y_{t}=\alpha \mathrm {d} t+\mathrm {d} W_{t}}$

where ${\displaystyle Y_{t}=h(X_{t})}$ where ${\displaystyle h}$ is defined as before. Its general solution is

${\displaystyle X_{t}=h^{-1}(\alpha t+W_{t}+h(X_{0}))}$

SDEs and supersymmetry

In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the

noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc.

See also

• Backward stochastic differential equation
• Langevin dynamics
• Local volatility
• Stochastic process
• Stochastic volatility
• Stochastic partial differential equations
• Diffusion process
• Stochastic difference equation

References

Further reading

• Evans, Lawrence C (2013). An Introduction to Stochastic Differential Equations American Mathematical Society.
• 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.
  ISBN 978-0-12-044375-8
  .
• Adomian, George (1989). Nonlinear stochastic systems theory and applications to physics. Mathematics and its Applications (46). Dordrecht: Kluwer Academic Publishers Group.
• Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315.
  ISBN 978-981-4678-93-3
  .
• Teugels, J.; Sund, B., eds. (2004). Encyclopedia of Actuarial Science. Chichester: Wiley. pp. 523–527.
• C. W. Gardiner (2004). Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer. p. 415.
• Thomas Mikosch (1998). Elementary Stochastic Calculus: with Finance in View. Singapore: World Scientific Publishing. p. 212.
  ISBN 981-02-3543-7
  .
• Seifedine Kadry (2007). "A Solution of Linear Stochastic Differential Equation". Wseas Transactions on Mathematics. USA: WSEAS TRANSACTIONS on MATHEMATICS, April 2007.: 618.
  ISSN 1109-2769
  .
• Higham., Desmond J. (January 2001). "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations". SIAM Review. 43 (3): 525–546.
  doi:10.1137/S0036144500378302
  .
• Desmond Higham and Peter Kloeden: "An Introduction to the Numerical Simulation of Stochastic Differential Equations", SIAM,
  ISBN 978-1-611976-42-7
  (2021).