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A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, [1] resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices , [ 2 ] random ...
In Itô calculus, the Euler–Maruyama method (also simply 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 named after Leonhard Euler and Gisiro Maruyama. The ...
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 ...
A stochastic process S t is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): = + where is a Wiener process or Brownian motion, and ('the percentage drift') and ('the percentage volatility') are constants.
An integro-differential equation (IDE) is an equation that combines aspects of a differential equation and an integral equation. A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of ...
Consider the autonomous Itō stochastic differential equation: = + with initial condition =, where denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [,]. Then the Milstein approximation to the true solution X {\displaystyle X} is the Markov chain Y {\displaystyle Y} defined as follows:
In numerical analysis, the Runge–Kutta methods (English: / ˈ r ʊ ŋ ə ˈ k ʊ t ɑː / ⓘ RUUNG-ə-KUUT-tah [1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. [2]
In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. [1] [2] The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. [3]