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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 to stochastic differential equations. It is named after Leonhard Euler and Gisiro Maruyama. Unfortunately, the same generalization cannot be done for any arbitrary deterministic method. Consider the stochastic differential equation (see Itô calculus) :\mathrm X_t = a(X_t, t) \, \mathrm t + b(X_t, t) \, \mathrm W_t, with initial condition ''X''0 = ''x''0, where ''W''''t'' stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time , ''T'' Then the Euler–Maruyama approximation to the true solution ''X'' is the Markov chain ''Y'' defined as follows: * partition the interval , ''T''into ''N'' equal subintervals of width \Delta t>0: ::0 = \tau_ float: """ Sample a ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equation
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 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. 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 stochasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Differential Equations
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 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. 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 stochasti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leimkuhler–Matthews Method
In mathematics, the Leimkuhler-Matthews method (or LM method in its original paper ) is an algorithm for finding discretized solutions to the Brownian dynamics :\mathrm X = -\nabla V(X ) \, \mathrm t + \sigma \, \mathrm W, where \sigma>0 is a constant, V(X) is an energy function and W(t) is a Wiener process. This stochastic differential equation has solutions (denoted X(t) \in \mathbb^N at time t ) distributed according to \pi(X) \propto \exp(-V(x)) in the limit of large-time, making solving these dynamics relevant in sampling-focused applications such as classical molecular dynamics and machine learning. Given a time step \Delta t>0, the Leimkuhler-Matthews update scheme is compactly written as :X_ = X_t -\nabla V(X_t) \Delta t + \sigma\frac2 \, (R_t+R_), with initial condition X_0 := X(0) , and where X_t \approx X(t) . The vector R_t is a vector of independent normal random numbers redrawn at each step so \text R_t \cdot R_N\delta_ (where \textbullet/math> denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Method (SDE)
In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation. It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs). Importantly, the method does not involve knowing derivatives of the coefficient functions in the SDEs. Most basic scheme Consider the Itō diffusion X satisfying the following Itō stochastic differential equation : = a(X_) \, t + b(X_) \, W_, with initial condition X_0=x_0, where W_t stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time ,T/math>. Then the basic Runge–Kutta approximation to the true solution X is the Markov chain Y defined as follows: * partition the interval ,T/math> into N subintervals of width \delta=T/N > 0: 0 = \tau_ < \tau_ < \dots < \tau_ = T; * set ; * recursively compute for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 stochastic differential equation: \mathrm X_t = a(X_t) \, \mathrm t + b(X_t) \, \mathrm W_t with initial condition X_ = x_, where W_ stands for the Wiener process, and suppose that we wish to solve this SDE on some interval of time ,T/math>. Then the Milstein approximation to the true solution X is the Markov chain Y defined as follows: * partition the interval ,T/math> into N equal subintervals of width \Delta t>0: 0 = \tau_0 < \tau_1 < \dots < \tau_N = T\text\tau_n:=n\Delta t\text\Delta t = \frac * set * recursively define for by: |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and starte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ornstein–Uhlenbeck Process
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck. The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over time, the process tends to drift towards its mean function: such a process is called mean-reverting. The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ranks as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
