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A bounded scalar multiplicative-noise Milstein step

Example

Consider the bounded scalar multiplicative-noise SDE $d X_{t} = \sin (X_{t}) d W_{t}$ with $X_{0} = π / 4$ , and initialize the numerical method with $Y_{0} = X_{0}$ . Fix $T > 0$ and an integer $N \geq 1$ with $h = T / N \leq 1$ , and use the uniform grid $t_{n} = n h$ for $0 \leq n \leq N$ . At the state $Y_{0} = π / 4$ , its first Milstein step is $Y_{1} = \frac{π}{4} + \frac{1}{\sqrt{2}} Δ W_{0} + \frac{1}{4} ((Δ W_{0})^{2} - h) .$ Unlike additive noise, the diffusion derivative contributes a nonzero centered correction. The hypotheses of the strong-order theorem apply to this coefficient field.

Facts & Assumptions

Given: AC; a filtered probability space whose filtration satisfies the usual conditions; and a standard Brownian motion $W$ adapted to that filtration such that $W_{t} - W_{s}$ is independent of $F_{s}$ and has law $N (0, t - s)$ for every $0 \leq s < t$ (for example, the usual augmentation of the natural filtration of $W$ ), as required by Continuous Brownian Ito processes; the SDE initial value $X_{0} = π / 4 \in L^{2}$ and matching numerical initial value $Y_{0} = X_{0}$ ; and a fixed horizon $T > 0$ with an integer $N \geq 1$ , the uniform grid $t_{n} = n h$ for $0 \leq n \leq N$ , and $h = T / N \leq 1$ .

[F1]

Milstein updates, iterated Brownian integrals, and commutative noise: for scalar noise, the increment contains $\frac{1}{2} σ (Y_{n}) σ^{'} (Y_{n}) ((Δ W_{n})^{2} - h)$ .

[F2]

Standard normal and normal laws: if $Δ W_{n} \sim N (0, h)$ , then its first, second, third and fourth moments are $0, h, 0, 3 h^{2}$ .

[L1]

Strong order one for scalar and commutative-noise Milstein schemes: for coefficients in $C_{b}^{3}$ under the stated Brownian hypotheses, the grid root-mean-square error is $O (h)$ .

[A1]

The Axiom of Choice: AC is inherited through the Brownian/strong-Milstein interface [L1].

Verification

Given: The assumptions above.

1.1F1L1givenalgebra

Here $b (x) = 0$ , $σ (x) = \sin x$ , and $σ^{'} (x) = \cos x$ . At $Y_{0} = π / 4$ , both $σ (Y_{0})$ and $σ^{'} (Y_{0})$ equal $1 / \sqrt{2}$ , so [F1] gives the stated update with correction coefficient $\frac{1}{2} (1 / \sqrt{2})^{2} = 1 / 4$ . The function $\sin x$ and its derivatives through order three are bounded and continuous, so it meets the regularity premise of [L1].

2.1F2step 1.1algebra

By [F2], the correction has conditional mean $\frac{1}{4} (h - h) = 0$ and variance $\frac{1}{16} (3 h^{2} - 2 h^{2} + h^{2}) = h^{2} / 8$ . Its covariance with the linear noise term is $\frac{1}{4 \sqrt{2}} E [(Δ W_{0})^{3} - h Δ W_{0}] = 0$ . Hence the one-step increment variance is $h / 2 + h^{2} / 8$ , and the derivative correction is a separate nonzero fluctuation.

3.1L1A1given∎

Applying [L1] to the SDE with $b = 0$ and $σ (x) = \sin x$ yields $\max_{0 \leq n \leq N} E ∣ Y_{n} - X_{t_{n}} ∣^{2} \leq C_{T} h^{2}$ , with $C_{T}$ independent of $N$ , on the supplied uniform grid. Taking square roots yields grid root-mean-square error $O (h)$ on every fixed finite horizon. This is the theorem's bounded-coefficient multiplicative-noise case under the inherited AC convention [A1]; it supplies no weak-order-two statement for Heun.

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Sources

Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43 (2001), Section 6 (standard reference, not scraped)
Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Section 10.3, printed pp. 345-351 (standard reference, not scraped)