확률 미분 방정식

이 글은 일반적인 참고문헌 목록을 포함하고 있지만, 그에 상응하는 인라인 인용구가 충분하지 않기 때문에 대체로 검증되지 않은 상태로 남아 있다. 좀 더 정밀한 인용구를 도입하여 이 기사를 개선할 수 있도록 도와주십시오. (2013년 7월) (이 템플릿 메시지를 제거하는 방법과 시기 알아보기)

미분 방정식
장애물 주위의 공기 흐름을 시뮬레이션하는 데 사용되는 Navier-Stokes 미분 방정식
범위
필드 자연과학 공학 천문학 물리학 화학 생물학 지질학 응용 수학 연속체 역학 혼돈 이론 다이너믹 시스템 사회과학 경제학 인구역학
분류
종류들 보통 부분적 차동알지브라질 인터그로 차동 분수 선형 비선형 변수 유형별 종속변수와 독립변수 자율적 콤플렉스 결합/분리됨 정확한 동종/비동종 특징들 주문 연산자 표기법
프로세스와의 관계 차이(구체 아날로그) 스토카스틱 확률적 부분 지연
해결책
존재와 고유성 피카르-린델뢰프 정리 페아노 존재 정리 카라테오도리의 존재 정리 카우치-코왈레프스키 정리
일반 주제 룬스키안 위상 초상화 위상공간 랴푸노프 / 점근증 / 지수 안정성 수렴율 시리즈/적분 솔루션 수치적 통합 디라크 델타 함수
솔루션 방법 감사 특성방법 오일러 지수 반응식 유한차이 (크랭크-니콜슨) 유한요소 무한요소 유한 부피 갤러킨 페트로프-갈러킨 집적인자 적분 변환 섭동 이론 룬게쿠타 변수의 분리 결정되지 않은 계수 모수의 변동
사람
리스트 아이작 뉴턴 고트프리드 라이프니즈 레온하르트 오일러 에밀 피카르 요제프 마리아 회네브로이스키 에른스트 린델뢰프 루돌프 립시츠 아우구스틴루이코치 존 크랭크 필리스 니콜슨 칼 데이비드 톨메 룬지 Martin Kutta
v t

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as unstable stock prices or physical systems subject to thermal fluctuations. Typically, SDEs contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process. However, other types of random behaviour are possible, such as jump processes. Random differential equations are conjugate to stochastic differential equations.^[1]

Background

Stochastic differential equations originated in the theory of Brownian motion, in the work of Albert Einstein 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 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 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. 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.

SSE에 대한 대안적인 견해는 차이점형식의 확률적 흐름이다. 이러한 이해는 명확하며 확률적 차이 방정식의 연속적 시간 제한의 스트라토노비치 버전과 일치한다. SDE와 연관된 것은 확률 분포 함수의 시간 진화를 설명하는 방정식인 스몰루코스키 방정식 또는 Fokker-Planck 방정식이다. 포커-플랑크 진화의 일반화는 확률적 진화 연산자의 개념에 의해 제공된다.

물리과학에서는 "랜지빈 SDE"라는 용어의 용어가 모호하다. Langevin SSE는 더 일반적인 형태일 수 있지만, 이 용어는 일반적으로 구배 유량 벡터 필드를 가진 좁은 등급의 SDE를 가리킨다. 이 등급의 SDE는 파리-파리의 출발점이기 때문에 특히 인기가 있다.Sourlas 확률적 정량화 절차,^[2] N=2 초대칭 양자 역학과 밀접하게 관련된 초대칭 모델을 유도한다. 그러나 물리적인 관점에서 보면 이 SSE의 등급은 위상학적 초대칭의 자발적 분해를 결코 보여주지 않기 때문에, 즉 (과대칭) 란제빈 SDE는 결코 혼란스럽지 않기 때문에 그다지 흥미롭지 않다.

확률 미적분학

브라운 운동이나 위너 과정은 수학적으로 예외적으로 복잡한 것으로 밝혀졌다. Wiener 과정은 거의 확실히 다른 곳이 없다. 따라서, 그것은 그것 자체의 미적분학 규칙을 필요로 한다. 확률론적 미적분학에는 Itô 확률론적 미적분과 Stratonovich 확률론적 미적분학의 두 가지 지배적인 버전이 있다. 두 사람은 각각 장단점이 있고, 신인은 주어진 상황에서 하나가 상대방보다 적절한지 헷갈리는 경우가 많다. 가이드라인이 존재하며(예: 익센달, 2003) 편리하게도 Itô SDE를 동등한 스트라토노비치 SDE로 쉽게 변환했다가 다시 되돌릴 수 있다. 그래도 처음에 SDE를 적었을 때 어떤 미적분을 사용할지 주의해야 한다.

수치해결

확률적 미분 방정식을 풀기 위한 수치적 방법으로는 오일러-마루야마 방법, 밀슈타인 방법, 룬게-쿠타 방법(SSE)이 있다.

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 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:

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

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.^[3] 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 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 (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.^{[citation needed]}

Use in probability and mathematical finance

The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different. It is also the notation used in publications on 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 continuous time stochastic process X_t as the sum of an ordinary Lebesgue integral and an Itô 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 satisfies the 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 (${\displaystyle \Omega ,\,{\mathcal {F}},\,P}$). 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 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-dimensional Euclidean space Rⁿ and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2).

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-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

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

Some explicitly solvable SDEs^[4]

Linear SDE: general case

${\displaystyle dX_{t}=(a(t)X_{t}+c(t))dt+(b(t)X_{t}+d(t))dW_{t}}$

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

where

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

Reducible SDEs: Case 1

${\displaystyle dX_{t}={\frac {1}{2}}f(X_{t})f'(X_{t})dt+f(X_{t})dW_{t}}$

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

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

which has a general solution

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

where

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

Reducible SDEs: Case 2

${\displaystyle dX_{t}=\left(\alpha f(X_{t})+{\frac {1}{2}}f(X_{t})f'(X_{t})\right)dt+f(X_{t})dW_{t}}$

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

${\displaystyle dX_{t}=\alpha f(X_{t})dt+f(X_{t})\circ W_{t}}$

which is reducible to

${\displaystyle dY_{t}=\alpha dt+dW_{t}}$

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 differential forms on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological supersymmetry 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, 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. The theory also offers a resolution of the Ito–Stratonovich dilemma in favor of Stratonovich approach.

See also

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

References

Further reading

• 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.
• Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3.
• 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. pp. 523–527.CS1 maint: extra text: authors list (link)
• 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.
• P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations. Springer. ISBN 0-387-54062-8.
• Higham., Desmond J. (January 2001). "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations". SIAM Review. 43 (3): 525–546. Bibcode:2001SIAMR..43..525H. CiteSeerX 10.1.1.137.6375. 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).