Math 2.0

Numerical analysis Ordinary differential equations Theorem

Runge–Kutta method for stochastic differential equations

Builds on Stochastic differential equationWiener processMilstein methodRunge–Kutta methods

Statement

Let WW be a one-dimensional Wiener process. Consider the scalar Itô stochastic differential equation dXt=a(Xt) dt+b(Xt) dWt,X0=x0∈ℝ,0≤t≤T.dX_t = a(X_t)\, dt + b(X_t)\, dW_t, \qquad X_0 = x_0 \in \R, \qquad 0 \le t \le T. Assume, as for the Milstein method, that aa and bb are twice continuously differentiable with bounded first and second derivatives. Assume also that bb′b b' is globally Lipschitz continuous. Let h=T/Nh = T/N, tn=nht_n = n h and ΔWn=Wtn+1−Wtn\Delta W_n = W_{t_{n+1}} - W_{t_n}. The derivative-free scheme of strong order 11 sets Y0=x0Y_0 = x_0, Υn=Yn+a(Yn) h+b(Yn)h,\Upsilon_n = Y_n + a(Y_n)\, h + b(Y_n) \sqrt{h}, Yn+1=Yn+a(Yn) h+b(Yn) ΔWn+b(Υn)−b(Yn)2h((ΔWn)2−h).Y_{n+1} = Y_n + a(Y_n)\, h + b(Y_n)\, \Delta W_n + \frac{b(\Upsilon_n) - b(Y_n)}{2 \sqrt{h}} \left( (\Delta W_n)^2 - h \right). Then there is a constant CC, independent of NN, such that max⁡0≤n≤N(𝔼∣Xtn−Yn∣2)1/2≤Ch.\max_{0 \le n \le N} \left( \mathbb{E} \abs{X_{t_n} - Y_n}^2 \right)^{1/2} \le C h.

Remarks

By Taylor's theorem, b(Υn)−b(Yn)h=b′(Yn) b(Yn)+O(h)\frac{b(\Upsilon_n) - b(Y_n)}{\sqrt{h}} = b'(Y_n)\, b(Y_n) + O(\sqrt{h}). So the scheme differs from the Milstein scheme by a term of size O(h)⋅O(h)O(\sqrt{h}) \cdot O(h) with mean zero, and it keeps the strong order 11. As in deterministic Runge–Kutta methods, extra evaluations of a coefficient replace its derivative. A naive transfer of a deterministic Runge–Kutta method to the stochastic case does not in general give a convergent or high-order scheme, because Itô calculus changes the order conditions. Example: for dX=σX dWdX = \sigma X\, dW, one has Υn=Yn(1+σh)\Upsilon_n = Y_n (1 + \sigma \sqrt{h}), and the correction term equals σ22Yn((ΔWn)2−h)\frac{\sigma^2}{2} Y_n ((\Delta W_n)^2 - h), exactly the Milstein term.

Going deeper

For a student

Stochastic Runge–Kutta methods are built to match the Itô–Taylor expansion up to a given order, the way deterministic ones match the Taylor expansion. Their order conditions are described with rooted trees, as for deterministic methods, but with several kinds of nodes for the drift and the noise. For several noise sources, strong order 11 again needs iterated integrals or a commutativity condition on the noise coefficients.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectEuler–Maruyama method, concept, no evidence levelEuler–Maruyama methodconceptFunction, concept, no evidence levelFunctionconceptGeometric Brownian motion, concept, no evidence levelGeometric Brownian mo…conceptItô calculus, concept, no evidence levelItô calculusconceptQuadratic variation, concept, no evidence levelQuadratic variationconceptRunge–Kutta methods, concept, no evidence levelRunge–Kutta methodsconceptStochastic differential equation, concept, no evidence levelStochastic differenti…conceptWiener process, concept, no evidence levelWiener processconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptItô's lemma, statement, no evidence levelItô's lemmastatementMilstein method, statement, no evidence levelMilstein methodstatementRunge–Kutta method for stochastic differential equations, statement, no evidence levelRunge–Kutta method fo…statement
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  1. Euclidean space object
  2. Euler–Maruyama method concept
  3. Function concept
  4. Geometric Brownian motion concept
  5. Itô calculus concept
  6. Quadratic variation concept
  7. Runge–Kutta methods concept
  8. Stochastic differential equation concept
  9. Wiener process concept
  10. Function of several variables concept
  11. Partial derivative concept
  12. Itô's lemma statement
  13. Milstein method statement
  14. Runge–Kutta method for stochastic differential equations statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_method_(SDE)
About this page
TagJ8ET1M5V10
URIm2://J8ET1M5V10
Kindstatement, theorem
WritingW0 stub
MSC65C30, 60H35
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AboutStochastic differential equation
UsesStochastic differential equation, Wiener process, Milstein method, Runge–Kutta methods

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