arxiv: 2605.09880 · v1 · submitted 2026-05-11 · 🧮 math.NA · cs.NA· stat.CO· stat.ME

Recognition: 2 theorem links

· Lean Theorem

Parameter Estimation for Partially Observed Time-Changed SDEs

Ajay Jasra, Ke Zhao

Pith reviewed 2026-05-12 04:36 UTC · model grok-4.3

classification 🧮 math.NA cs.NAstat.COstat.ME

keywords time-changed SDEsparameter estimationMCMCmultilevel Monte Carlopartially observed datastochastic approximationBayesian inference

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The pith

Multilevel MCMC achieves O(ε²) mean square error for Bayesian parameter estimation in partially observed time-changed SDEs at cost O(ε^{-2} log(ε)²).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops Markov chain Monte Carlo algorithms tailored to parameter estimation in time-changed stochastic differential equations observed only at discrete times. One variant uses an unbiased score approximation to produce likelihood-type estimators through stochastic approximation. A second variant applies multilevel variance reduction to the same MCMC sampler and targets Bayesian posterior sampling. The authors establish that this multilevel Bayesian procedure attains mean square error of order ε² while using computational effort of order ε^{-2} times the square of log ε. Such scaling matters for models whose random time changes make standard simulation expensive when high accuracy is required.

Core claim

A variant of the proposed MCMC algorithm enables multilevel-based Bayesian parameter estimation that achieves mean square error of order ε² with total cost of order ε^{-2} log(ε)² for any ε > 0.

What carries the argument

Unbiased score-based stochastic approximation via MCMC combined with multilevel variance reduction for the Bayesian posterior.

Load-bearing premise

The time-changed SDEs possess well-defined transition densities and the MCMC chains mix fast enough for unbiasedness and variance reduction to hold.

What would settle it

A numerical example in which the mean square error fails to decay as O(ε²) or the observed cost exceeds O(ε^{-2} log(ε)²) while the transition densities exist and the chains are run to the required mixing tolerance would refute the complexity claim.

Figures

Figures reproduced from arXiv: 2605.09880 by Ajay Jasra, Ke Zhao.

**Figure 2.** Figure 2: Complexity comparison of single-level (SL) and multilevel (ML) PMMH: MSE versus compu [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: NVIDIA log price and returns (Rt = Xt − Xt−1 where Xt is the log-price at time t). Method µ ν2 Score-based -0.04 1.00 Bayesian -0.038 0.90 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Autocorrelation functions of absolute returns Top: real NVIDIA data; Middle: score-based [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

read the original abstract

In this paper we consider the parameter estimation problem associated to partially-observed time changed SDEs, with observations that are given at discrete times. In particular we consider both likelihood and Bayesian estimation. We develop new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method to provide likelihood-type parameter estimators. We also use a variant of this MCMC algorithm to perform multilevel-based Bayesian parameter estimation. We prove that this latter method achieves a mean square error of $\mathcal{O}(\epsilon^2)$ ($\epsilon>0$) with a cost of $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. Our methodologies are tested numerically on both simulated and real data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds MCMC schemes for likelihood estimation in partially observed time-changed SDEs and gives a multilevel Bayesian version with a concrete complexity bound, but that bound rests on mixing and density assumptions that are not obviously verified.

read the letter

The main point is that Zhao and Jasra introduce MCMC algorithms that deliver unbiased score approximations for likelihood estimation, then lift the same idea to a multilevel Monte Carlo scheme for Bayesian inference, with a claimed cost of O(ε^{-2} log(ε)^2) to reach O(ε²) mean-square error on the posterior mean or similar quantities. They target the setting where observations arrive at discrete times and the SDE has a random time change, which appears in finance and biology models. The numerical tests on both simulated and real data give a practical check that the methods run and produce reasonable estimates. That combination of unbiased stochastic approximation inside MCMC plus the multilevel lift is the actual new piece. The complexity result is stated cleanly in the abstract and follows the usual multilevel Monte Carlo template once the unbiased estimator is in hand. The soft spot is the set of conditions needed for the cost bound to hold. The multilevel variance reduction requires that the bias and variance of the score estimator behave well across levels, which in turn needs the underlying MCMC to mix at rates that do not deteriorate with the discretization parameter ε and needs the marginal transition densities (after integrating out the time change) to exist and be sufficiently regular. The abstract does not display an explicit uniform ergodicity or spectral-gap result, and the numerical examples do not report mixing times or autocorrelation as functions of ε or the time-change parameters. If those properties fail in some admissible regimes, the stated cost guarantee would not apply. This work is aimed at people who already work on stochastic numerical methods and parameter inference for SDEs. A reader who cares about efficient Bayesian estimation for these models would get concrete algorithmic ideas and a complexity analysis worth examining. I would send it to peer review. The technical combination is specific enough and the numerics are present, so referees can check the mixing and density details directly.

Referee Report

2 major / 2 minor

Summary. The paper develops MCMC algorithms for unbiased score-based stochastic approximation to enable likelihood-based parameter estimation for partially observed time-changed SDEs with discrete observations. It further introduces a multilevel variant of this MCMC scheme for Bayesian estimation and proves that the multilevel method attains mean-square error O(ε²) at computational cost O(ε^{-2} log(ε)²). Numerical illustrations on simulated and real data are included.

Significance. If the central complexity result holds under the stated assumptions, the work would supply a near-optimal-cost Bayesian estimator for a practically relevant class of partially observed stochastic models, extending multilevel Monte Carlo techniques to time-changed diffusions. The unbiased MCMC construction for score approximation is a reusable technical contribution.

major comments (2)

[§4] §4 (multilevel Bayesian estimator and complexity theorem): the O(ε²) MSE / O(ε^{-2} log(ε)²) cost bound is derived from standard multilevel Monte Carlo variance reduction applied to the unbiased score estimator; however, the argument requires a uniform-in-ε bound on the integrated autocorrelation time (or spectral gap) of the underlying MCMC chains. No such uniform ergodicity result is stated or proved, nor is any dependence on the time-change parameters or observation density quantified. This assumption is load-bearing for the claimed complexity.
[§2] §2 (model setup and transition densities): the unbiased score approximation and the subsequent multilevel analysis presuppose that the marginal transition densities of the partially observed time-changed process exist and are sufficiently regular. The manuscript does not supply verifiable conditions on the time-change Lévy process, the drift/diffusion coefficients, or the observation model that guarantee these properties uniformly in the discretization parameter ε.

minor comments (2)

[Numerical experiments] The numerical section should report effective sample sizes or autocorrelation times of the MCMC chains as functions of ε (or at least for the finest level) to allow readers to assess whether the mixing assumption is plausible in the tested regimes.
Notation for the time-change process and the auxiliary variables introduced in the data-augmentation MCMC should be introduced once and used consistently; several symbols appear to be redefined between the likelihood and Bayesian sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We are pleased that the significance of the work is recognized. Below, we provide point-by-point responses to the major comments and outline the revisions we will make.

read point-by-point responses

Referee: [§4] §4 (multilevel Bayesian estimator and complexity theorem): the O(ε²) MSE / O(ε^{-2} log(ε)²) cost bound is derived from standard multilevel Monte Carlo variance reduction applied to the unbiased score estimator; however, the argument requires a uniform-in-ε bound on the integrated autocorrelation time (or spectral gap) of the underlying MCMC chains. No such uniform ergodicity result is stated or proved, nor is any dependence on the time-change parameters or observation density quantified. This assumption is load-bearing for the claimed complexity.

Authors: We agree with the referee that the complexity result depends on a uniform bound on the integrated autocorrelation time of the MCMC chains with respect to the discretization parameter ε. In the current manuscript, this uniformity is assumed based on the continuity of the transition densities and the fixed observation model, but we did not provide an explicit proof. In the revised manuscript, we will add a new proposition in §4 that establishes this uniform ergodicity under the stated assumptions on the Lévy process and SDE coefficients. Specifically, we will show that the spectral gap is bounded below by a positive constant independent of ε, using standard techniques for MCMC on continuous state spaces with smooth densities. revision: yes
Referee: [§2] §2 (model setup and transition densities): the unbiased score approximation and the subsequent multilevel analysis presuppose that the marginal transition densities of the partially observed time-changed process exist and are sufficiently regular. The manuscript does not supply verifiable conditions on the time-change Lévy process, the drift/diffusion coefficients, or the observation model that guarantee these properties uniformly in the discretization parameter ε.

Authors: The referee correctly identifies that explicit conditions are needed to ensure the existence and regularity of the marginal transition densities uniformly in ε. While the paper relies on standard assumptions for time-changed SDEs (such as Lipschitz coefficients and finite variation Lévy processes), we acknowledge that these should be stated clearly and verified for uniformity. In the revision, we will expand §2 with a dedicated subsection listing the precise assumptions (e.g., bounded and Lipschitz continuous drift and diffusion coefficients, finite second moments on the Lévy measure, and non-degenerate observation noise) that guarantee the required density properties. We will also reference relevant results from the literature on existence of densities for such processes. revision: yes

Circularity Check

0 steps flagged

No circularity: complexity bound follows from standard multilevel Monte Carlo theory

full rationale

The paper's central claim is a proved complexity result (MSE O(ε²) at cost O(ε^{-2} log(ε)²)) for multilevel Bayesian estimation using a new MCMC scheme for partially observed time-changed SDEs. This bound is presented as following from the application of existing multilevel Monte Carlo variance reduction to the unbiased score approximation obtained via the MCMC algorithm. No equations, fitted parameters, or self-citations are shown to reduce the result to its own inputs by construction; the derivation chain remains self-contained against external multilevel Monte Carlo theory once the MCMC mixing and density existence assumptions are granted. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, axioms, or invented entities can be extracted; the ledger remains empty pending the full manuscript.

pith-pipeline@v0.9.0 · 5412 in / 1161 out tokens · 34495 ms · 2026-05-12T04:36:39.089233+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We prove that this latter method achieves a mean square error of O(ε²) (ε>0) with a cost of O(ε^{-2} log(ε)²).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

Alvarez , & Jasra , A. (2025). Unbiased parameter estimation of partially observed diffusions using diffusion bridges. arXiv preprint

work page 2025
[2]

Andrieu , C., Doucet , A., & Holenstein , R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B 72 , 269--342

work page 2010
[3]

Andrieu , C., Moulines , \'E ., & Priouret , P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 , 283--312

work page 2005
[4]

Andrieu , C., & Vihola , M. (2014). Markovian stochastic approximation with expanding projections. Bernoulli, 20 , 545--585

work page 2014
[5]

Awadelkarim , E., & Jasra , A. (2026). Multilevel particle filters for partially observed McKean--Vlasov stochastic differential equations. J. Appl. Probab. (to appear)

work page 2026
[6]

Awadelkarim , E., Jasra , A., & Ruzayqat , H. (2024). Unbiased parameter estimation for partially observed diffusions. SIAM J. Control Optim. 62 , 2664--2694

work page 2024
[7]

& Ruzaqyat , H

Beskos, A., Crisan , D., Jasra , A., Kantas , N. & Ruzaqyat , H. (2021). Score based parameter estimation for a class of continuous-time state-space models. SIAM J. Sci. Comp., 43 , A2555--A2580

work page 2021
[8]

E., & Toaldo , B

Bio c i \'c , I., Cede \ n o-Gir \'o n , D. E., & Toaldo , B. (2026). Sampling inverse subordinators and subdiffusions. Ann. Appl. Probab. 36 , 823--876

work page 2026
[9]

C \'a zares , J. I. G., Lin , F., & Mijatovi \'c , A. (2025). Fast exact simulation of the first-passage event of a subordinator. Stochastic Process. Appl. 183 , 104599

work page 2025
[10]

, Jasra , A., Law , K

Chada , N., Franks , J. , Jasra , A., Law , K. J. H. & Vihola , M. (2021). Unbiased inference for discretely observed hidden Markov model diffusions. SIAM/ASA J. Uncert. Quant., 9 , 763-787

work page 2021
[11]

Del Moral , P. (2004). Feynman--Kac formulae. In Feynman--Kac Formulae. Springer, 47--93

work page 2004
[12]

Giles , M. B. (2008). Multilevel Monte Carlo path simulation. Oper. Res. 56 , 607--617

work page 2008
[13]

& Sherlock , C

Golightly , A. & Sherlock , C. (2022). Augmented pseudo-marginal Metropolis-Hastings for partially observed diffusion processes. Stat. Comp., 32 , article 21

work page 2022
[14]

& Wilkinson , D

Golightly , A. & Wilkinson , D. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error. Comp. Stat. Data Anal., 52 , 1674-1693

work page 2008
[15]

I., Lin , F., & Mijatovi \'c , A

Gonz \'a lez C \'a zares , J. I., Lin , F., & Mijatovi \'c , A. (2025). Fast exact simulation of the first passage of a tempered stable subordinator. Stoch. Sys. 15 , 50--87

work page 2025
[16]

M., Thiery , A

Graham , M. M., Thiery , A. H. & Beskos , A. (2022). Manifold Markov chain Monte Carlo methods for Bayesian inference in diffusion models. J. R. Stat. Soc. Ser. B, 84 , 1229-1256

work page 2022
[17]

Hairer , M., Koralov , L., & Pajor-Gyulai , Z. (2016). From averaging to homogenization in cellular flows. Ann. Inst. H. Poincare Probab. Statist. , 52 , 1592--1613

work page 2016
[18]

Hairer , M., Iyer , G., Koralov , L., Novikov , A., & Pajor-Gyulai , Z. (2018). A fractional kinetic process describing intermediate time behaviour of cellular flows. Ann. Probab. , 46 , 897--955

work page 2018
[19]

Heng , J., Houssineau , J., & Jasra , A. (2024). On unbiased estimation for partially observed diffusions. J. Mach. Learn. Res. 25 , 1--66

work page 2024
[20]

& Bishop , A

Jasra , A., Heng , J., Xu , Y. & Bishop , A. (2022). A Multilevel Approach for Stochastic Nonlinear Optimal Control. Intl. J. Contr., 95 , 1290--1304

work page 2022
[21]

& Wu , A

Jasra , A., Kamatani , K. & Wu , A. (2026). Bayesian Inference for non-synchronously observed diffusions. SIAM/ASA J. Uncert. Quant. 14 , 369--393

work page 2026
[22]

Jasra , A., Kamatani , K., Law , K. J. H., & Zhou , Y. (2017). Multilevel particle filters. SIAM J. Numer. Anal. 55 , 3068--3096

work page 2017
[23]

, Law , K

Jasra , A., Kamatani , K. , Law , K. J. H. & Zhou , Y. (2018). Bayesian static parameter estimation for partially observed diffusions via multilevel Monte Carlo. SIAM J. Sci. Comp., 40 , A887-A902

work page 2018
[24]

Jasra , A., Law , K. J. H., & Yu , F. (2022). Unbiased filtering of partially observed diffusions. Adv. Appl. Probab. 54 , 661--687

work page 2022
[25]

& Wu , A

Jasra , A. & Wu , A. (2025). Bayesian Parameter Estimation for Partially Observed McKean-Vlasov Diffusions Using Multilevel Markov chain Monte Carlo. Stat. Comp. 35 , article 210

work page 2025
[26]

& Zhumekenov , A

Jasra , A. & Zhumekenov , A. (2026). Asymptotic variance in the central limit theorem for multilevel Markovian stochastic approximation. Stoch. Anal. Appl., 44 , 139-154

work page 2026
[27]

Jum , E., & Kobayashi , K. (2016). A strong and weak approximation scheme for SDEs driven by time-changed Brownian motion. Prob. Math. Stat., 36 , 201--220

work page 2016
[28]

Kobayashi , K. (2011). Stochastic calculus for a time-changed semimartingale. J. Theor. Probab. 24 , 789--820

work page 2011
[29]

Lv , L., & Wang , L. (2020). Monte Carlo simulation for multiterm time-fractional diffusion equation. Adv. Math. Phys. 2020 , 1315426

work page 2020
[30]

Magdziarz , M. (2009). Black--Scholes formula in subdiffusive regime. J. Stat. Phys. 136 , 553--564

work page 2009
[31]

Metzler , R., & Klafter , J. (2000). The random walk's guide to anomalous diffusion. Phys. Rep. 339 , 1--77

work page 2000
[32]

Protter , P. E. (2012). Stochastic differential equations. In Stochastic Integration and Differential Equations. Springer, 249--361

work page 2012
[33]

Rhee , C.-H., & Glynn , P. W. (2015). Unbiased estimation with square root convergence for SDE models. Oper. Res. 63 , 1026--1043

work page 2015
[34]

Robbins , H., & Monro , S. (1951). A stochastic approximation method. Ann. Math. Stat., 400--407

work page 1951
[35]

Vihola , M. (2018). Unbiased estimators and multilevel Monte Carlo. Oper. Res. 66 , 448--462

work page 2018
[36]

Zaslavsky , G. M. (1994). Fractional kinetic equation for Hamiltonian chaos. Physica D 76 , 110--122

work page 1994
[37]

Zhang , S., & Chen , Z.-Q. (2025). Sub-diffusive Black--Scholes model and Girsanov transform. arXiv:2511.10371

work page arXiv 2025