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Shared-Brownian Euler level correction rates

Statement

Assume AC and the finite- $L^{2}$ coefficient and initial-value hypotheses of Euler–Maruyama strong rate for square-integrable initial data. Fix an integer refinement factor $M \geq 2$ and a fine step $h$ with $0 < h \leq 1 / M$ , $T / h \in N$ , and $T / (M h) \in N$ . Construct the fine and coarse Euler paths with the same initial value and Brownian motion. The coarse Brownian increment over each coarse interval is, by definition, the sum of the $M$ fine Brownian increments in that interval. Let $X_{T}$ be the common exact solution, and let $φ : R^{d} \to R$ be globally Lipschitz with constant $L_{φ}$ . Then constants independent of $h$ satisfy $E ∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq C_{M} h,$ $Var (φ (Y_{T}^{h}) - φ (Y_{T}^{M h})) \leq C_{M} L_{φ}^{2} h, ∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C L_{φ} h^{1 / 2} .$ Thus the Lipschitz-payoff Euler level pair has correction variance exponent $β = 1$ and strong-error bias exponent $α = 1 / 2$ . If, in addition, $φ$ is bounded and its backward value function satisfies the hypotheses of Euler–Maruyama weak order one under backward regularity, then the bias bound improves to $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ , so $α = 1$ while $β = 1$ remains. With fixed dimension, fixed $M$ , and unit-cost coefficient and payoff evaluations, one coupled fine/coarse sample has work $O (h^{- 1})$ .

Facts & Assumptions

Given: AC; the stated finite- $L^{2}$ SDE assumptions; the fixed refinement factor and nested grids; a common exact solution and Brownian path; and the Lipschitz payoff (plus, for the weak branch, the bounded-payoff backward contract).

[F1]

Under the finite- $L^{2}$ coefficient, time-modulus, Brownian and initial-value hypotheses, the continuous Euler path satisfies $E \sup_{t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h$ , uniformly in $0 < h \leq 1$ with $T / h \in N$ . Euler–Maruyama strong rate for square-integrable initial data

[F2]

For the weak-bias branch, retain finite $d, m$ , the Euclidean/Frobenius norm convention and $A : B = tr (A^{T} B)$ , and the same finite-horizon, complete usual filtered Brownian and grid setup as in the Statement. Assume jointly Borel coefficients that are globally Lipschitz in state, have uniform linear growth, and satisfy the half-Hölder time modulus. Require an $F_{0}$ -measurable initial value independent of future Brownian increments with $E ∣ X_{0} ∣^{p} < \infty$ for finite real $q \geq 0$ and finite $p \geq 2 q + 8$ (using $∣ x ∣^{0} = 1$ ), and uniform pointwise order- $p$ moments of the exact solution, Euler interpolation, and Euler step process on $[0, T]$ . The payoff is bounded Borel, and its backward value function $u (t, x) = E [φ (X_{T}^{t, x})]$ , using the unique shifted strong solutions supplied by the global well-posedness theorem, satisfies $u (T, x) = φ (x)$ , $(\partial_{t} + L_{t}) u = 0$ for $L_{t} v = b (t, \cdot) \cdot \nabla v + \frac{1}{2} (σ σ^{T}) (t, \cdot) : D_{x}^{2} v$ , and continuous mixed derivatives $\partial_{t}^{j} D_{x}^{α} u$ for $0 \leq j \leq 2$ , $∣ α ∣ \leq 4$ , bounded by $C_{u} (1 + ∣ x ∣^{q})$ . Under this full contract of Euler–Maruyama weak order one under backward regularity, $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ .

[A1]

The refinement factor $M$ is fixed as $h \to 0$ ; both mesh sizes obey $0 < h \leq 1$ and have $T / h$ and $T / (M h)$ integral. The coarse path uses the sum of each corresponding block of $M$ fine Brownian increments, so both Euler paths use the same Brownian path and initial value.

[A2]

$∣ φ (x) - φ (y) ∣ \leq L_{φ} ∣ x - y ∣$ for all states. In the weak branch, the same $φ$ is bounded and satisfies the additional backward regularity contract [F2].

[A3]

AC is included because [F1] carries that assumption from the Doob and Brownian Itô interfaces. The Axiom of Choice

Proof

[F1] Under the finite- $L^{2}$ coefficient, time-modulus, Brownian and initial-value hypotheses, the continuous Euler path satisfies $E \sup_{t \leq T} ∣ X_{t} - Y_{t}^{h} ∣^{2} \leq C h$ , uniformly in $0 < h \leq 1$ with $T / h \in N$ . Euler–Maruyama strong rate for square-integrable initial data

[F2] For the weak-bias branch, retain finite $d, m$ , the Euclidean/Frobenius norm convention and $A : B = tr (A^{T} B)$ , and the same finite-horizon, complete usual filtered Brownian and grid setup as in the Statement. Assume jointly Borel coefficients that are globally Lipschitz in state, have uniform linear growth, and satisfy the half-Hölder time modulus. Require an $F_{0}$ -measurable initial value independent of future Brownian increments with $E ∣ X_{0} ∣^{p} < \infty$ for finite real $q \geq 0$ and finite $p \geq 2 q + 8$ (using $∣ x ∣^{0} = 1$ ), and uniform pointwise order- $p$ moments of the exact solution, Euler interpolation, and Euler step process on $[0, T]$ . The payoff is bounded Borel, and its backward value function $u (t, x) = E [φ (X_{T}^{t, x})]$ , using the unique shifted strong solutions supplied by the global well-posedness theorem, satisfies $u (T, x) = φ (x)$ , $(\partial_{t} + L_{t}) u = 0$ for $L_{t} v = b (t, \cdot) \cdot \nabla v + \frac{1}{2} (σ σ^{T}) (t, \cdot) : D_{x}^{2} v$ , and continuous mixed derivatives $\partial_{t}^{j} D_{x}^{α} u$ for $0 \leq j \leq 2$ , $∣ α ∣ \leq 4$ , bounded by $C_{u} (1 + ∣ x ∣^{q})$ . Under this full contract of Euler–Maruyama weak order one under backward regularity, $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq C h$ .

[A1] The refinement factor $M$ is fixed as $h \to 0$ ; both mesh sizes obey $0 < h \leq 1$ and have $T / h$ and $T / (M h)$ integral. The coarse path uses the sum of each corresponding block of $M$ fine Brownian increments, so both Euler paths use the same Brownian path and initial value.

[A2] $∣ φ (x) - φ (y) ∣ \leq L_{φ} ∣ x - y ∣$ for all states. In the weak branch, the same $φ$ is bounded and satisfies the additional backward regularity contract [F2].

[A3] AC is included because [F1] carries that assumption from the Doob and Brownian Itô interfaces. The Axiom of Choice

1.1F1A1algebra

By [A1], the fine and coarse paths are driven by nested increments of this common Brownian path. Since $h \leq 1 / M$ , both meshes lie in the range of [F1]. Therefore $∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq 2 ∣ Y_{T}^{h} - X_{T} ∣^{2} + 2 ∣ Y_{T}^{M h} - X_{T} ∣^{2} .$ Taking expectations and applying [F1] at step sizes $h$ and $M h$ gives $E ∣ Y_{T}^{h} - Y_{T}^{M h} ∣^{2} \leq 2 C h + 2 C M h = 2 C (1 + M) h .$ This is the correction second-moment bound.

1.2A2F1algebra

The same Lipschitz bound and Cauchy–Schwarz give $∣ E φ (X_{T}) - E φ (Y_{T}^{h}) ∣ \leq L_{φ} E ∣ X_{T} - Y_{T}^{h} ∣ \leq L_{φ} (E ∣ X_{T} - Y_{T}^{h} ∣^{2})^{1 / 2} \leq C^{1 / 2} L_{φ} h^{1 / 2} .$ Thus $β = 1$ and the strong route gives $α = 1 / 2$ .

2.1A2step 1.1algebra

Let $D = φ (Y_{T}^{h}) - φ (Y_{T}^{M h})$ . Since variance is at most second moment and [A2] gives $∣ D ∣ \leq L_{φ} ∣ Y_{T}^{h} - Y_{T}^{M h} ∣$ , step 1.1 yields $Var (D) \leq E D^{2} \leq 2 C (1 + M) L_{φ}^{2} h .$

3.1F2step 2.1

If the additional bounded-payoff backward contract holds, [F2] applied to the same fine Euler scheme gives bias at most $C h$ . It changes only the bias exponent to $α = 1$ ; the correction-variance bound of step 2.1 is unchanged.

4.1A1A3step 3.1∎

A fine path takes $T / h$ Euler steps and the coarse path takes $T / (M h)$ steps. The coarse increments are sums of already generated fine increments, so they add no Brownian draws; for fixed $M, d$ and unit-cost coefficient/payoff evaluations, total path work is proportional to $T / h + T / (M h) = O (h^{- 1})$ . AC is inherited only through [A3].

Remarks

This is the solver-rate interface for multilevel examples; it makes no claim about learned-drift or learned-score bias. The variance-rate and work statements do not import the abstract MLMC complexity theorem.

Depends on

Used by

Finite-L2 strong and conditional weak Euler MLMC cost Corollary
Shared-increment Euler coupling and payoff rates Example

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Sources

Michael B. Giles, Multilevel Monte Carlo Path Simulation, §2 and Theorem 3.1 (standard reference, not scraped)
Desmond J. Higham, Xuerong Mao, and Andrew M. Stuart, Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, §4 (standard reference, not scraped)