|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''Runge–Kutta method''' is a technique for the approximate [[numerical analysis|numerical solution]] of a [[stochastic differential equation]]. It is a generalization of the [[Runge–Kutta methods|Runge–Kutta method]] for [[ordinary differential equation]]s to stochastic differential equations.
| | Emilia Shryock is my name but you can contact me anything you like. California is our beginning location. One of the very best things in the world for him is to gather badges but he is struggling to discover time for it. Hiring is her day occupation now but she's always wanted her own company.<br><br>Here is my page - [http://indianrajneet.com/groups/best-strategies-for-keeping-candida-albicans-from-increasing/ std home test] |
| | |
| Consider the [[Itō diffusion]] ''X'' satisfying the following Itō stochastic differential equation
| |
| | |
| :<math>{\mathrm{d} X_{t}} = a(X_{t}) \, \mathrm{d} t + b(X_{t}) \, \mathrm{d} W_{t},</math>
| |
| | |
| with [[initial condition]] ''X''<sub>0</sub> = ''x''<sub>0</sub>, where ''W''<sub>''t''</sub> stands for the [[Wiener process]], and suppose that we wish to solve this SDE on some interval of time [0, ''T'']. Then the '''Runge–Kutta approximation''' to the true solution ''X'' is the [[Markov chain]] ''Y'' defined as follows:
| |
| | |
| * partition the interval [0, ''T''] into ''N'' equal subintervals of width ''δ'' = ''T'' ⁄ ''N'' > 0:
| |
| | |
| :<math>0 = \tau_{0} < \tau_{1} < \dots < \tau_{N} = T;</math>
| |
| | |
| * set ''Y''<sub>0</sub> = ''x''<sub>0</sub>;
| |
| | |
| * recursively define ''Y''<sub>''n''</sub> for 1 ≤ ''n'' ≤ ''N'' by
| |
| | |
| :<math>Y_{n + 1} = Y_{n} + a(Y_{n}) \delta + b(Y_{n}) \Delta W_{n} + \frac{1}{2} \left( b(\hat{\Upsilon}_{n}) - b(Y_{n}) \right) \left( (\Delta W_{n})^{2} - \delta \right) \delta^{-1/2},</math>
| |
| | |
| : where | |
| | |
| :<math>\Delta W_{n} = W_{\tau_{n + 1}} - W_{\tau_{n}}</math>
| |
| | |
| : and
| |
| | |
| :<math>\hat{\Upsilon}_{n} = Y_{n} + a(Y_n) \delta + b(Y_{n}) \delta^{1/2}.</math>
| |
| | |
| Note that the [[random variables]] Δ''W''<sub>''n''</sub> are [[independent and identically distributed]] [[normal distribution|normal random variables]] with [[expected value]] zero and [[variance]] ''δ''.
| |
| | |
| This scheme has strong order 1, meaning that the approximation error of the actual solution at a fixed time scales with the time step ''δ''. It has also weak order 1, meaning that the error on the statistics of the solution scales with the time step ''δ''. See the references for complete and exact statements.
| |
| | |
| The functions ''a'' and ''b'' can be time-varying without any complication. The method can be generalized to the case of several coupled equations; the principle is the same but the equations become longer. Higher-order schemes also exist, but become increasingly complex.
| |
| | |
| ==References==
| |
| | |
| * {{cite book | author=Kloeden, P.E., & Platen, E. | title=Numerical Solution of Stochastic Differential Equations | publisher=Springer | location=Berlin | year=1999 | isbn=3-540-54062-8 }}
| |
| | |
| {{DEFAULTSORT:Runge-Kutta Method (Sde)}}
| |
| [[Category:Numerical differential equations]]
| |
| [[Category:Stochastic differential equations]]
| |
Emilia Shryock is my name but you can contact me anything you like. California is our beginning location. One of the very best things in the world for him is to gather badges but he is struggling to discover time for it. Hiring is her day occupation now but she's always wanted her own company.
Here is my page - std home test