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Computational solution of stochastic differential equations

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Stochastic differential equations (SDEs) provide accessible mathematical models that combine deterministic and probabilistic components of dynamic behavior. This article is an overview of numerical solution methods for SDEs. The solutions are stochastic processes that represent diffusive dynamics, a common modeling assumption in many application areas. We include a description of fundamental numerical methods and the concepts of strong and weak convergence and order for SDE solvers. In addition, we briefly discuss the extension of SDE solvers to coupled systems driven by correlated noise. WIREs Comput Stat 2013. doi: 10.1002/wics.1272 This article is categorized under: Applications of Computational Statistics > Computational Finance Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Algorithms and Computational Methods > Numerical Methods
Solution to the Black–Scholes stochastic differential equation . The exact solution (5) is plotted as a gray curve. The Euler–Maruyama approximation with time step Δt = 1/8 is plotted as a dark curve. The drift and diffusion parameters are set to μ = 0.2 and σ = 1, respectively.
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The mean error of the estimation of E(X(T)) for SDE (15). The plot compares the Euler–Maruyama method (circles) which has weak order 1, and the weak order 2 Runge–Kutta type method (squares) given in (19). Parameters used were X(0) = 10, T = 1, μ = − 3, σ = 0.2.
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Solution to Langevin equation . The path is the solution approximation for parameters μ = 10, σ = 1, computed by the Euler–Maruyama method with stepsize Δti = 0.01 for all i.
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Algorithms and Computational Methods > Numerical Methods
Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods
Applications of Computational Statistics > Computational Finance

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