Stochastic differential equation

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.[1][3] Such a mathematical definition was first proposed by Kiyosi Itô in the 1940s, leading to what is known today as the Itô calculus. Another construction was later proposed by Russian physicist Stratonovich, leading to what is known as the Stratonovich integral. The Itô integral and Stratonovich integral are related, but different, objects and the choice between them depends on the application considered. 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.[9]

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 or the Fokker–Planck equation, an equation describing the time evolution of probability distribution functions. 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,[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.

Brownian motion or the Wiener process was discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable;[1][3] 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. 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)[3] and conveniently, one can readily convert an Itô SDE to an equivalent Stratonovich SDE and back again.[1][3] Still, one must be careful which calculus to use when the SDE is initially written down.

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].

More generally one can extend the theory of stochastic calculus onto differential manifolds 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 usual conditions and let ${\displaystyle {\widehat {M}}=M\cup \{\infty \}}$ be the one-point compactification 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 Laurent Schwartz through the concept of Schwartz morphism,[6] see also the related 2-jet interpretation of Ito SDEs on manifolds based on the jet-bundle.[8] This interpretation is helpful when trying to optimally approximate the solution of an SDE given on a large space with the solutions of an SDE given on a submanifold of that space,[9] in that a Stratonovich based projection does not result to be optimal. This has been applied to the filtering problem, leading to optimal projection filters.[9]

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 predictable stopping time 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.

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

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

where

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

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

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

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

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

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

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