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Multilevel Monte Carlo algorithms for L\'{e}vy-driven SDEs with Gaussian correction

Steffen Dereich

arXiv:1101.1369v1math.PR

TL;DR

Estimating path-functional expectations for Lévy-driven SDEs is complicated by declining MLMC efficiency when small jumps are frequent. The paper simulates large jumps and Gaussian-approximates small ones, deriving worst-case error bounds of at most order τ^-1/6 as runtime grows.

  • Problem

    For Lévy-driven SDEs, approximating increments can worsen MLMC performance when the BG-index exceeds one because small jumps occur more frequently.

    Source: §1 · pp. 2–3

  • Method

    The paper analyzes multilevel Monte Carlo schemes that simulate large Lévy jumps individually and replace small jumps with a Gaussian approximation.

    Source: §1 · p. 3

  • Results

    τ^-1/6: worst-case errors are at most of this order as runtime τ tends to infinity.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The analysis covers general path-dependent Lipschitz functionals, including most continuous payoffs appearing in finance.

    Source: Visualization of the results and… · p. 7

    The results are upper bounds only, and no lower bounds are known.

    Source: Visualization of the results and… · p. 7

Abstract

from arXiv · show

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where $Y=(Y_t)_{t\in[0,1]}$ is the solution of a multidimensional L\'{e}vy-driven stochastic differential equation and $f$ is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the L\'{e}vy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions $f$ that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the L\'{e}vy measure around zero. In particular, one can derive upper bounds from the Blumenthal--Getoor index of the L\'{e}vy process. In the case where the Blumenthal--Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the L\'{e}vy process does not incorporate a Wiener process or if the Blumenthal--Getoor index $\beta$ is larger than $\frac{4}{3}$, then the upper bound is of order $\tau^{-({4-\beta})/({6\beta})}$ when the runtime $\tau$ tends to infinity. Whereas in the case, where $\beta$ is in $[1,\frac{4}{3}]$ and the L\'{e}vy process has a Gaussian component, we obtain bounds of order $\tau^{-\beta/(6\beta-4)}$. In particular, the error is at most of order $\tau^{-1/6}$.

1. Introduction.

The paper studies multilevel Monte Carlo estimation for path-functional expectations of Lévy-driven SDEs, a problem relevant to path-dependent option payoffs. It addresses declining MLMC efficiency for processes with frequent small jumps by approximating those jumps with a Gaussian distribution.

1. Introduction.

  • The BG-index measures small-jump frequency; for β≤1, earlier algorithms have worst-case errors at most order τ^-1/2, while efficiency declines for larger indices.A larger BG-index corresponds to small jumps occurring at higher frequencies.Source: §1 · p. 2
  • The schemes simulate large Lévy jumps individually and approximate small jumps with a normal distribution.The Gaussian approximation follows the approach of Asmussen and Rosiński.Source: §1 · p. 3
  • τ^-1/6: worst-case error is at most of this order as runtime grows; for BG-index above one, adjusted algorithms yield significantly smaller errors than prior approaches.The comparison is with algorithms that do not use a Gaussian approximation.Source: §1 · p. 3
  • Approximating the small jumps requires a uniform ellipticity assumption.The stated assumption restricts the support of the small-jump Lévy measure to a subspace for sufficiently small jump sizes.Source: Notation and universal assumptions · pp. 4–5
  • The runtime model assumes constant-time sampling of large jumps and uniform times, evaluation of the coefficient function, and evaluation of functionals on piecewise-constant paths proportional to their breakpoints.Under these assumptions, average runtime is bounded by a constant multiple of the defined cost.Source: Main results · p. 5
  • The analysis targets worst-case error over measurable supremum-norm Lipschitz path functionals, including most continuous payoffs appearing in finance.The motivating application includes payoffs of possibly path-dependent options.Source: §1 · p. 2, Visualization of the results and… · p. 7

2. Multilevel Monte Carlo.

The section defines a parameterized multilevel Monte Carlo algorithm that approximates expectations by estimating level-wise differences. Its error analysis uses Wasserstein bounds and a runtime cost model.

2. Multilevel Monte Carlo.

  • The algorithm is parameterized by a maximum level, sample counts, thresholds, and time steps.The parameters are denoted m, n1,...,nm, ε1,...,εm, and h1,...,hm.Source: §2 · p. 8
  • The algorithm estimates each level’s expectation difference from independent samples and sums the resulting averages.It also samples the coarsest-level expectation directly.Source: §2 · p. 9
  • The analysis targets the Wasserstein distance between the genuine solution and its approximation to control the mean squared error.The section states that estimates for this distance are its main task, while other error terms use estimates from.Source: The mean squared error of an algorithm · p. 9
  • Simulating a level pair requires jumps of size at least h_k and Wiener increments on the two levels’ time skeletons.The cost model treats runtime as proportional to the number of entries in the finer-level time grid.Source: The cost function · p. 10

3. Weak approximation.

This section estimates the Wasserstein distance between the genuine SDE solution and a time-discretized approximation with a Gaussian replacement for small jumps. Its proof separates approximation errors and uses a coupling of the small-jump process with a Wiener process.

3. Weak approximation.

  • The section’s central result is an estimate for the Wasserstein distance between the genuine solution and the Gaussian-approximated solution.The approximation retains compensated jumps larger than h and adds a Wiener component for the small-jump variance.Source: §3 · p. 11
  • The approximation uses random update times and is identically distributed to the multilevel approximation when ε and h match its level parameters.The update time is the latest random time in the grid not exceeding the current time.Source: §3 · p. 11

3.1. Strategy of the proof of Theorem 3.1 and main notation.

  • The proof splits the discrepancy into separate terms, estimating each before combining the bounds for the theorem.Only the final two terms in the stated decomposition depend on the coupling choice.Source: §3.1 · p. 13
  • The coupling result pairs a bounded-jump Lévy martingale with a Wiener process over a finite time interval.The theorem assumes the Lévy measure is supported on a ball of radius h and imposes a condition involving ϑ.Source: §6 · pp. 17–18

7. Coupling the Gaussian approximation.

The section couples the small-jump Lévy component with a Wiener process on intervals determined by random times. It uses a finite bound on the number of intervals to carry the coupling estimates into the proposition bounds.

7. Coupling the Gaussian approximation.

  • Given the random times and the other driving processes, the small-jump component is coupled with Wiener increments separately on each interval.Each paired process segment is coupled according to Theorem 6.1.Source: §7 · p. 21
  • The section combines the component estimates to obtain the stated proposition bounds for the coupled approximation.The bounds are stated in Propositions 7.1 and 7.2.Source: §7 · pp. 21, 23, 24
  • The number of random-time grid entries in [0,1] is bounded by 2/ε′ + 1.This count lets the estimates for deterministic grids extend to the random-grid setting.Source: §7 · p. 24

8. Proof of the main results.

The proof selects multilevel parameters and sample counts under a runtime budget to establish the main error bounds. It then specializes the result using a power-law bound on the Lévy measure to obtain bounds expressed through the Blumenthal–Getoor index.

8. Proof of the main results.

  • In case (I), the proof chooses the maximum level so the algorithm’s cost is at most τ.The selected level satisfies 2^m approximately (τ log τ)^(2/3).Source: §8 · p. 26
  • In case (II), the proof selects the maximum level using g*(τ), then bounds log g*(τ) by log τ.The construction assumes τ is sufficiently large and uses a fixed lower level l.Source: §8 · pp. 26–27
  • For the corollary, the proof bounds the Lévy-measure behavior by a power law and applies Theorem 1.1.It chooses g(h)=κ2 h^-β′ for h in (0,1], with β′ selected in relation to the Blumenthal–Getoor index β.Source: §8 · p. 27

APPENDIX

The appendix collects auxiliary martingale, process, and Lévy-measure estimates. It gives an Itô-isometry justification, a Gronwall-type argument, and filtration assumptions.

  • Lemma A.1’s martingale statement follows from the Itô isometry for Lévy-driven stochastic differential equations.Source: APPENDIX · pp. 27–28
  • A bound for the Section 3.1 processes uses a constant depending only on K, and its proof is described as a Gronwall-type argument.Source: APPENDIX · p. 28
  • Auxiliary estimates are stated under second-moment and Lévy-measure tail conditions or conditions on a decreasing function g, with constants depending only on g.Source: APPENDIX · p. 28
  • A filtration-based lemma assumes U_j is measurable at step j and V_j is measurable at the next step and independent of G_j.Source: APPENDIX · p. 29
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