PaperlayerView exact revision on arXiv

Source-linked AI summary

Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with jumps

Ariel Neufeld, Tuan Anh Nguyen, Philipp Schmocker

arXiv:2610.10863v1math.NAcs.NAmath.PR

TL;DR

High-dimensional McKean–Vlasov SDEs with jumps have limited numerical coverage, motivating an approximation method that handles finite and infinite jump activity. The paper develops a jump-aware multilevel Picard algorithm and proves polynomial complexity under stated assumptions, with numerical examples reaching dimension 200.

  • Problem

    Numerical approximation of high-dimensional McKean–Vlasov SDEs with jumps is a relatively under-addressed problem compared with prior jump-free work.

    Source: §1 · p. 2

  • Method

    The paper develops a multilevel Picard algorithm for finite- and infinite-activity jumps, adding Monte Carlo jump-compensator approximations and small-jump truncation for infinite activity.

    Source: §1 · p. 3

  • Results

    The MLP algorithm overcomes the curse of dimensionality in the L2 sense, with computational cost polynomial in state-space dimension and reciprocal error tolerance.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Numerical examples demonstrate applicability to two different McKean–Vlasov SDEs in dimensions up to 200.

    Source: §1 · p. 3

    The stated result is conditioned on the common setting and its coefficient assumptions.

    Source: §2 · pp. 3–4

Abstract

from arXiv · show

We introduce multilevel Picard (MLP) approximations of McKean-Vlasov stochastic differential equations (SDEs) with jumps of finite or infinite activity. Under Lipschitz and integrability assumptions on the coefficients, we show that the MLP algorithm does not suffer from the curse of dimensionality when approximating the solution of the McKean-Vlasov SDE. The latter means that its computational cost grows at most polynomially in both the state-space dimension of the SDE and the reciprocal of the prescribed error tolerance. In two numerical experiments, we demonstrate the practical applicability of the MLP algorithm for two different McKean-Vlasov SDEs in dimensions up to 200.

1. Introduction

The introduction motivates numerical approximation of high-dimensional McKean–Vlasov SDEs with jumps and presents an MLP algorithm with polynomial complexity under stated conditions. It situates the work among existing jump-free and jump-inclusive methods and recalls the connection between these equations and interacting-particle systems.

1. Introduction

  • The target equations have coefficients depending on the process state and distribution, and finite interacting-particle systems replace coefficient expectations with empirical measures.Under propagation of chaos, each particle converges in law to the McKean–Vlasov solution.Source: §1 · pp. 1–2
  • For jump-free McKean–Vlasov SDEs, MLP approximations overcome the curse of dimensionality with constant diffusion, extended to nonconstant diffusion [68]; deep-learning experiments reached dimensions up to 50.Source: §1 · p. 2
  • The MLP approximation handles finite- and infinite-activity jumps and overcomes the curse of dimensionality in the L2 sense, with cost polynomial in dimension and reciprocal error tolerance.The paper also demonstrates applicability in two examples with dimensions up to 200.Source: §1 · p. 3
  • The jump-aware construction adds Monte Carlo approximations for the jump compensator and, for infinite activity, truncates small jumps.The analysis also develops new variance estimates for the resulting jump terms.Source: §1 · p. 3

1.1. Notations.

  • The notation section writes conditional expectations and variances given G as E_G[·] and V_G[·].This abbreviation is used to lighten the notation.Source: §1.1 · p. 3

2. Setting and main results

The section establishes existence and uniqueness of the exact solution and gives error and computational-complexity results for MLP approximations with finite and infinite jump activity. The scheme uses Monte Carlo approximations for jump terms and truncates small jumps in the infinite-activity case.

2. Setting and main results

  • The exact McKean–Vlasov SDE has a unique adapted solution under the stated common setting.The theorem asserts finite second moment for the solution.Source: §2 · pp. 3–4
  • The section states that the scheme covers both finite and infinite jump activity, with small-jump truncation required in the infinite-activity case.The finite-activity case uses δ = 0, while the infinite-activity case uses δ > 0.Source: §2 · p. 5

2.1. Finite jump-activity.

  • For finite jump activity, the section gives a global-error bound and shows polynomial computational complexity when coefficient evaluations and Lévy-measure growth satisfy polynomial-complexity conditions.No jump truncation is needed in this case, so δ = 0.Source: §2.1 · pp. 5–6

2.2. Infinite jump-activity.

  • For infinite jump activity, the section gives a global-error bound and establishes polynomial computational complexity under additional conditions on the Lévy measure and polynomial input-function costs.This case truncates jumps with δ > 0.Source: §2.2 · p. 6

2.3. Motivations of the scheme.

  • The scheme stores a discrete trajectory rather than only one random variable, allowing approximation of stochastic integrals with nonconstant diffusion coefficients.The time discretization uses K steps, and many applications require the entire solution trajectory.Source: §2.3 · p. 7
  • For jump cases, the scheme approximates expectations and integrals by Monte Carlo sums; when ν^d(R^d) < ∞, no jump truncation is needed and δ = 0.Source: §2.3 · pp. 7–8

3. Numerical experiments

The numerical section illustrates the MLP scheme in two McKean–Vlasov jump models: a finite-activity Ornstein–Uhlenbeck model and an infinite-activity Variance-Gamma model. Both experiments test dimensions through 200.

3. Numerical experiments

  • Algorithm 1 summarizes the MLP procedure, including path and compensator simulation and the truncation choice for finite- versus infinite-activity Lévy measures.The scheme outputs an approximation at the time-discretization points.Source: §3 · p. 9, §3.1 · p. 10

3.1. Finite jump-activity: Mean-field Ornstein-Uhlenbeck process with jumps.

  • The finite-activity experiment applies the MLP scheme to a mean-field Ornstein–Uhlenbeck model with drift, diffusion, and jump coefficients specified in the section.The model uses a finite Lévy measure and is checked against the paper’s assumptions for the finite-activity results.Source: §3.1 · pp. 9, 11

3. Numerical experiments

  • In both experiments, Algorithm 1 is run independently 10 times for d ∈ {10, 25, 50, 100, 200} and levels n = m ∈ {1, ..., 5}, with K = mn and T = 1.Table 1 reports (d−1/2-adjusted) L2-error, computational time, and computational cost. Both experiments approximate the true solution using Euler–Maruyama on a finer equidistant time grid.Source: §3.1 · p. 11, Table 1 · p. 12, §3.2 · p. 13

3.2. Infinite jump-activity: Mean-field Variance-Gamma model.

  • The infinite-activity experiment applies the MLP scheme to a mean-field Variance-Gamma model with a drift and jump coefficient and zero diffusion coefficient.The model’s assumptions are verified for the infinite-activity results, and the experiment uses a small-jump truncation threshold of δ = 0.02.Source: §3.2 · pp. 11–13

4. Proof of the main results

The proof separates approximation error into statistical and bias components, with additional truncation and time-discretization errors for infinite jump activity. Its preliminary estimates establish coefficient bounds and polynomial moment control for the exact solution.

4. Proof of the main results

  • For infinite jump activity, the proof tracks truncation, time-discretization, and MLP errors, while the MLP error itself splits into statistical error and bias.For finite activity, the global error separates into MLP and time-discretization errors.Source: §4 · p. 14

4.1. Some simple estimates.

  • Lemma 4.1 supplies integrability and growth estimates for the mean-field coefficients, including a Lipschitz bound for the diffusion coefficient in L2.The stated bound is at most 2c times the L2 distance between the random inputs.Source: §4.1 · pp. 14, 16

4.2. Polynomial moment bound for the exact solution.

  • Lemma 4.2 establishes a polynomial moment bound in the dimension for the exact solution, uniformly over the time interval.The supplied passages state the bound’s scope but do not expose its precise exponent or inequality.Source: §4.2 · pp. 16–17

4.3. Euler–Maruyama approximations and the corresponding polynomial moment bound.

This section introduces truncated Euler–Maruyama approximations and establishes moment control needed for the later error analysis.

4.3. Euler–Maruyama approximations and the corresponding polynomial moment bound.

  • Lemma 4.3 constructs a unique adapted Euler–Maruyama approximation with càdlàg paths for each dimension, discretization level, and truncation parameter.The construction covers δ ≥ 0, including the untruncated case.Source: §4.3 · p. 17
  • Lemma 4.4 establishes a dimension-polynomial bound on the L2 norm of the truncated Euler–Maruyama approximations.The bound is stated uniformly over dimension, discretization level, time, and truncation parameter.Source: §4.3 · pp. 17–18

4.4. Discretization error.

The discretization analysis establishes temporal regularity for the exact solution and controls the distance between truncated and untruncated Euler–Maruyama approximations.

4.4. Discretization error.

  • Lemma 4.5 establishes temporal regularity of the exact solution by bounding its change between two times in the interval.The bound is stated for every dimension and ordered pair of times.Source: §4.4 · p. 18
  • In the cited section, Lemma 4.7 establishes the distance between Euler–Maruyama approximations with truncation (δ > 0) and those without truncation (δ = 0).Source: §4.5 · p. 20

4.5. Truncation error.

Section 4.5 establishes a truncation-error estimate for infinite jump activity, with a constant uniform over the stated parameters. Its proof combines preliminary estimates and Grönwall’s lemma.

4.5. Truncation error.

  • For infinite jump activity, Lemma 4.7 establishes a truncation-error estimate with a constant κ.The supplied passage does not show the estimate’s full formula.Source: §4.5 · p. 20
  • The estimate holds for all d, K, t, and δ in the ranges stated in Lemma 4.7.Source: §4.5 · p. 20
  • The proof combines Jensen’s inequality, Itô’s isometry, earlier lemmas, the triangle inequality, and Grönwall’s lemma.Source: §4.5 · pp. 20–21

4.6. Integrability of the approximations.

The section establishes L2-integrability of the MLP approximations in both finite- and infinite-activity cases. Its key claims are proved inductively by showing the required integrals are well-defined.

4.6. Integrability of the approximations.

  • Lemma 4.9 states two properties for both activity cases across the listed indices and parameters.The supplied passages identify the properties as (i) and (ii), but do not show their full statements.Source: §4.6 · p. 22
  • The proof proceeds by induction on n and establishes that the integrals in (ii) are well-defined, after which (i) follows from (7).Source: §4.6 · pp. 22–23

4.7. Statistical error.

The section estimates statistical error for both finite- and infinite-activity cases. It bounds conditional variances for the constituent sums and combines those estimates in Corollary 4.14.

4.7. Statistical error.

  • Lemmas 4.10–4.13 estimate conditional variances for the first, second, third, and last sums in both activity cases.Source: §4.7 · pp. 23–25
  • The conditional-variance argument uses independence and distributional properties, and identifies one integral as a discrete sum.Source: §4.7 · pp. 26–29
  • Corollary 4.14 gives a statistical-error estimate for both finite- and infinite-jump-activity cases.The bound’s full formula is not visible in the supplied passage.Source: §4.7 · p. 29

4.8. Bias.

The section treats the bias of the MLP approximations in both activity cases and combines it with statistical error to obtain an MLP error estimate.

4.8. Bias.

  • Lemma 4.15 states a conditional-expectation property for the approximations in both activity cases.Source: §4.8 · p. 30
  • Lemma 4.16 establishes a bias estimate for both activity cases over the stated parameter ranges.The full bound is not visible in the supplied passages.Source: §4.8 · p. 31
  • The paper combines the statistical-error estimate and bias to obtain an MLP error estimate in both activity cases.Source: §4.8 · pp. 31–32

4.9. Completion of the proofs of Theorems 2.5 and 2.6.

The proofs complete Theorems 2.5 and 2.6 by combining earlier estimates with the triangle inequality. They also account for the computational costs of the MLP construction, including its Poisson-randomness representation.

Theorem 2.5

  • Theorem 2.5’s proof combines Corollary 4.17, Lemma 4.6, and the triangle inequality to obtain its stated bound.The proof concludes by establishing a bound for all d, n, m, K and t in the stated ranges.Source: §4.9 · p. 33

Theorem 2.6

  • Theorem 2.6 represents each time-sliced random measure using a Poisson count and independent jump samples.The number of stored random variables can be unbounded, but its expectation is νd(Rd)T/K + 1.Source: §4.9 · p. 33
  • The stated computational-cost bound is Cd_n,m,K ≤ κd^κK^4n m^n.The proof accounts for process generation and evaluations of the coefficients and jump function when deriving the cost.Source: §4.9 · pp. 33–34

4.10. Completion of the proofs of Theorems 2.8 and 2.9.

The proofs complete Theorems 2.8 and 2.9, with bounds stated for the setting indexed by δ. They account for discretized random measures and the associated MLP computational costs.

Theorem 2.8

  • Theorem 2.8’s proof combines Corollary 4.17, Lemmas 4.6 and 4.7, and the triangle inequality to establish its bound.The bound is stated for d, n, m, K, δ, and t in the ranges specified in the passage.Source: §4.10 · pp. 34–35

Theorem 2.9

  • Theorem 2.9’s proof analyzes the cost of representing each discretized random-measure part with Poisson-generated random variables.The number required can be unbounded, while its expectation is given in terms of νd(Ad_δ)T/K + 1.Source: §4.10 · p. 35
  • Theorem 2.9’s cost accounting includes generating Brownian and Poisson randomness and evaluating the coefficients and jump function.The proof separately counts preparation of paired processes, a scalar random variable, and the required function evaluations.Source: §4.10 · p. 35
Loading 2610.10863v1…