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A superlinear drift outside the Euler–Maruyama moment contract

Facts & Assumptions

Given: AC, a scalar state, a scalar Brownian increment, and the coefficient pair $b (x) = - x^{3}$ , $σ (x) = 1$ .

[F1]

The shared contract requires globally Lipschitz coefficients in state and a uniform linear-growth bound. Euler–Maruyama recursion

[F2]

The uniform-moment theorem applies only under the coefficient and initial-value hypotheses of the shared contract. Euler–Maruyama uniform moments and interpolation increments

[A1]

AC is inherited from the Brownian interface; no choice is made in the coefficient calculation. The Axiom of Choice

Example

For the scalar coefficient pair $b (x) = - x^{3}$ and $σ (x) = 1$ , the drift is superlinear: its magnitude grows faster than $1 + ∣ x ∣$ . This is an example about the hypotheses of the Euler–Maruyama moment theorem; it makes no claim about strong existence, instability, or convergence for this particular equation.

Verification

1.1F1algebra

The diffusion $σ = 1$ is globally Lipschitz and has linear growth. For the drift, $∣ b (x) ∣ / (1 + ∣ x ∣) = ∣ x ∣^{3} / (1 + ∣ x ∣) \to \infty$ as $∣ x ∣ \to \infty$ , so no finite linear-growth constant exists. Also $∣ b (x) - b (0) ∣ / ∣ x ∣ = ∣ x ∣^{2} \to \infty$ , so no finite global Lipschitz constant exists.

2.1A1F1F2algebra∎

Since both required drift conditions fail, the uniform-moment theorem [F2] cannot be invoked for this coefficient pair. The conclusion is only that its stated hypotheses are absent; no contrary convergence or moment claim follows. AC in [A1] is inherited and no choice is used.

Depends on

Used by

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Dependency tree · two levels

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Sources

Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, §2 (standard reference, not scraped)
Simo Särkkä and Arno Solin, Applied Stochastic Differential Equations, Algorithm 8.1 (standard reference, not scraped)