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Shared-increment Euler coupling and payoff rates

Example

Work on a complete filtered probability space whose filtration is right-continuous and contains all $P$ -null subsets of $F$ in $F_{0}$ . Let $W$ be an adapted standard one-dimensional Brownian motion whose increments after time $s$ are independent of $F_{s}$ . For $d X_{t} = - X_{t} d t + d W_{t}$ , $X_{0} = 0$ , and $T = 1$ , couple the Euler paths with fine step $h = 1 / (2 n)$ and coarse step $2 h = 1 / n$ by summing each pair of fine Brownian increments for the coarse update. For $φ (x) = \cos x$ , the shared-Brownian theorem gives correction variance $O (h)$ , strong bias $O (h^{1 / 2})$ , and coupled path work $O (h^{- 1})$ .

Facts & Assumptions

Given: AC; the complete usual filtered Brownian setup above on $[0, 1]$ ; the $F_{0}$ -measurable initial value $X_{0} = 0$ ; $h = 1 / (2 n)$ for $n \geq 1$ ; the fine/coarse coupling described in the Example; and $φ (x) = \cos x$ .

[F1]

Under the stated nested-grid coupling and a globally Lipschitz payoff, the Euler correction has second moment $O (h)$ , its variance is $O (h)$ , the strong-route bias is $O (h^{1 / 2})$ , and fixed-refinement path work is $O (h^{- 1})$ . Shared-Brownian Euler level correction rates

[F2]

For a real continuous Brownian Itô process $X_{t} = X_{0} + \int_{0}^{t} b_{s} d s + \int_{0}^{t} σ_{s} d W_{s}$ , applying Itô's formula to $e^{t} X_{t}$ is valid and gives the displayed drift and diffusion terms. One-dimensional Ito formula

[F3]

For a square-integrable Brownian Itô integral $M_{t} = \int_{0}^{t} H_{s} d W_{s}$ , $E \sup_{t \leq T} ∣ M_{t} ∣^{2} \leq 4 E \int_{0}^{T} H_{s}^{2} d s$ . Doob maximal bound for the Ito integral

[F4]

A standard Brownian motion has independent normal increments of variance equal to the elapsed time. $d$ -dimensional Brownian motion

[A1]

The refinement factor is $M = 2$ ; each coarse increment is the sum of its two fine increments, and both paths start from the same $X_{0}$ and use the same Brownian path.

[A2]

The payoff is globally Lipschitz with constant $1$ , since $∣ \cos x - \cos y ∣ = ∣ \int_{y}^{x} - \sin s d s ∣ \leq ∣ x - y ∣$ .

[A3]

AC is inherited through the Brownian and Itô interfaces in [F2]–[F4]. The Axiom of Choice

Verification

1.1A3F2F3givenalgebra

The coefficients are $b (x) = - x$ and $σ (x) = 1$ . They are globally Lipschitz with constant $1$ , satisfy $∣ b (x) ∣^{2} + ∣ σ (x) ∣^{2} = x^{2} + 1$ , and are time independent, so the shared coefficient and time-modulus conditions hold. Applying [F2] to $e^{t} X_{t}$ gives $X_{t} = e^{- t} \int_{0}^{t} e^{s} d W_{s}$ . Applying [F2] to $e^{t} W_{t}$ then gives $X_{t} = \int_{0}^{t} e^{- (t - s)} d W_{s} = W_{t} - \int_{0}^{t} e^{- (t - s)} W_{s} d s .$ Hence $\sup_{t \leq 1} ∣ X_{t} ∣ \leq 2 \sup_{t \leq 1} ∣ W_{t} ∣$ , and [F3] gives $E \sup_{t \leq 1} ∣ X_{t} ∣^{2} \leq 16 < \infty$ .

1.2A1A2F4given

The payoff satisfies [A2]. The fine and coarse meshes obey $h \leq 1 / 2$ , $1 / h = 2 n \in N$ , and $1 / (2 h) = n \in N$ ; [A1] makes every coarse increment the corresponding Brownian increment over its full interval, [F4] gives its centered normal law with variance $2 h$ , and the given filtration-relative Brownian hypothesis supplies independence from the sigma-algebra at the start of that coarse interval. Thus the shared-rate theorem hypotheses hold.

2.1F1step 1.2∎

Applying [F1] yields $Var (\cos (Y_{1}^{h}) - \cos (Y_{1}^{2 h})) \leq C h, ∣ E \cos (X_{1}) - E \cos (Y_{1}^{h}) ∣ \leq C \sqrt{h},$ and work $O (h^{- 1})$ per coupled path pair. These rates correspond to $(α, β) = (1 / 2, 1)$ for this payoff.

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