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Geometric Brownian motion and the Milstein update

Example

For $μ, ν, x_{0} \in R$ , consider the geometric Brownian SDE $d X_{t} = μ X_{t} d t + ν X_{t} d W_{t}, X_{0} = x_{0} .$ A solution is $X_{t} = x_{0} \exp ((μ - ν^{2} / 2) t + ν W_{t})$ . On a grid step of length $h$ , the scalar Milstein update is $Y_{n + 1} = Y_{n} (1 + μ h + ν Δ W_{n} + \frac{1}{2} ν^{2} ((Δ W_{n})^{2} - h)) .$ This example verifies the exact-solution form and the update algebra; it makes no global rate claim under the bounded-coefficient hypotheses of the strong-order theorem.

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; real constants $μ, ν, x_{0}$ ; and a finite interval.

[F1]

Milstein updates, iterated Brownian integrals, and commutative noise: the scalar update is $Y_{n + 1} = Y_{n} + b (Y_{n}) h + σ (Y_{n}) Δ W_{n} + \frac{1}{2} σ (Y_{n}) σ^{'} (Y_{n}) ((Δ W_{n})^{2} - h)$ .

[F2]

Multidimensional Ito formula for Brownian-driven processes: for a smooth function of a Brownian It^o process, the drift includes one half of the Hessian applied to the instantaneous covariance.

[A1]

The Axiom of Choice: AC is inherited from the Brownian and It^o interface [F2].

Verification

Given: The assumptions above.

1.1F2A1given

Define $G (t, w) = x_{0} e^{(μ - ν^{2} / 2) t + ν w}$ . Its derivatives are $G_{t} = (μ - ν^{2} / 2) G$ , $G_{w} = ν G$ , and $G_{w w} = ν^{2} G$ . It^o's formula [F2] therefore gives $d G (t, W_{t}) = (μ - ν^{2} / 2) G d t + \frac{1}{2} ν^{2} G d t + ν G d W_{t} = μ G d t + ν G d W_{t}$ , with $G (0, W_{0}) = x_{0}$ . Thus the displayed process solves the SDE, including $x_{0} = 0$ .

2.1F1givenalgebra∎

For $b (x) = μ x$ and $σ (x) = ν x$ , $σ^{'} (x) = ν$ and $σ (x) σ^{'} (x) = ν^{2} x$ . Substituting these coefficients into [F1] and factoring $Y_{n}$ gives the displayed Milstein update. If $ν = 0$ , its correction vanishes and it reduces to the deterministic Euler step.

Depends on

Used by

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Sources

Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review 43 (2001), Sections 4 and 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)