Neural MJD: Neural Non-Stationary Merton Jump Diffusion for Time Series Prediction
Authors: Yuanpei Gao
Qi Yan
Yan Leng
Renjie Liao
Venue: NeurIPS 2025 poster ()
Keywords: time series
sequantial models
neural merton jump diffusion
Primary area: deep_learning
TL;DR: A novel neural Merton jump diffusion SDE for probabilistic time series prediction.

Abstract: While deep learning methods have achieved strong performance in time series prediction, their black-box nature and inability to explicitly model underlying stochastic processes often limit their robustness handling non-stationary data, especially in the presence of abrupt changes. In this work, we introduce Neural MJD, a neural network based non-stationary Merton jump diffusion (MJD) model. Our model explicitly formulates forecasting as a stochastic differential equation (SDE) simulation problem, combining a time-inhomogeneous Itô diffusion to capture non-stationary stochastic dynamics with a time-inhomogeneous compound Poisson process to model abrupt jumps. To enable tractable learning, we introduce a likelihood truncation mechanism that caps the number of jumps within small time intervals and provide a theoretical error bound for this approximation. Additionally, we propose an Euler-Maruyama with restart solver, which achieves a provably lower error bound in estimating expected states and reduced variance compared to the standard solver. Experiments on both synthetic and real-world datasets demonstrate that Neural MJD consistently outperforms state-of-the-art deep learning and statistical learning methods. Our code is available at https://github.com/DSL-Lab/neural-MJD.

Neural MJD: Neural Non-Stationary Merton Jump Diffusion for Time Series Prediction
Yuanpei Gao!
Qi Yan qi yaneecaneee... wwloeeee.bb...om ca.
yuanpeigostudent. www.nnt..bcco.mm ca
Yan Leng² yan. lww. utexas. edu
Renjie Liao 1244 rjliaoβece ubc. ca
University of British Columbia; Vector Institute University of Texas at Austin; *Canada CIFAR Al Chair
Abstract
While deep learning methods have achieved strong performance in time series prediction, their black-box nature and inability to explicitly model underlying stochastic processes often limit their robustness handling non-stationary data, es- pecially in the presence of abrupt changes. In this work, we introduce Neural MJD, neural network based non-stationary Merton jump diffusion (MJD) model. Our model explicitly formulates forecasting as stochastic differential equation (SDE) simulation problem, combining a time-inhomogeneous Itô diffusion to cap- non-stationary stochastic dynamics with time-inlie mamm-e-ioooeeeouu compound Poisson process to model abrupt jumps. To enable tractable learning, we intro- duce likelihood truncation mechanism that caps the number of jumps within small time intervals and provide theoretical eтror bound for this approxima- tion. Additionally,
propose Fuler-Maruyan achieves a provably lower error bound in estimating expected states and reduced variance compared the standard solver. Experiments both synthetic and real-world datasets demonstrate that Neural MJD consistently outperforms state- of-the-art deep learning and statistical learning methods. Our code is available Lab/ Lal--Lab//nnueeeee neural-MJD
with restart solver, which
pettp::/giithun com/DSL
Introduction
Real-world time series often exhibit a mix of continuous trends and abrupt changes (jumps) For example, stock prices generally follow steady patterns driven by macroeconomic factors but experience sudden jumps due to unexpected news or policy shifts 3 4 Similarly, retail revenue may
1
rise seasonally but jump abruptly due to sales promotions or supply chain disruptions [5] 6). These discontinuous changes pose significant challenges for temporal dynamics modeling.
general Lévy processes (7]. provide principled approach for modeling such data with jumps. They are effective for small datasets with well-understood statistical properties 8 10] However, their assumptions such as independent and stationary increments- often fail in real-world non-stationary settings. Additionally, these models struggle capture interdependencies across multiple time series, such competition effects among colocated businesses [11] or spillover dynamics in stock markets driven by investor attention shifts [12 [33. As result, they difficult scale effectively to large datasets. In contrast, deep learning approaches have demonstrated strong empirical performance by effectively learning time- varying patterns from data 14H19). Despite their success, these models are often black-box in nature and lack explicit mathematical formulations to describe the underlying dynamics. This limits their 39th Conference on Neural Information Processing Systems (NcurIPS 2025).
Classical statistical models, e.g., Merton jump diffusion (MJD) 3

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interpretability and often results in poor generalization to non-stationary data with jumps. Notably, in previous deep learning time-series studies, "non-stationarity typically refers distributional shifts in the data over time. These works focus on mitigating such shifts using techniques like input-level normalization (e.g., DAIN (20), ST-norm (2]], ReviN (222,, or domain adaptation (e.g..
DDG-DA (3).. In contrast, our notion of "non-stationarity" centers on modeling MJD process with parameters that evolve over time.
To address these limitations, we propose Neural MJD, a neural parameterization of the non-stationary Merton jump diffusion model that combines the advantages of statistical and learning-baa -ased ap- proaches. In particular, our contributions are as follows:
Our Neural MJD integrates a time-inhomooennoou Itō diffusion to capture non-stationary stochastic dynamies and time-inhomogeneour compound Poisson process to model abrupt jumps. The parameters of the corresponding SDEs are predicted by a neural network conditioned on past data and contextual information.
To enable tractable learning, we present a likelihood truncation mechanism that caps the number of jumps within small time intervals and provide a theoretical error bound for this approximation. Additionally, inference, which achieves a provably lower error bound in estimating expected states and reduced variance compared the standard solver.
propose an Euler-Marayama with restart solver for
Extensive experiments on both synthetic and real-world datasets show that our model consistently outperforms deep learning and statistical baselines under both stochastic and deterministic evaluation protocols.
2
Related work
Many neural sequence models have been explored for time series prediction, e.g., long short-term
memory (LSTM) (24), transformers (25], and state space models (SSMs) (26). These models (2 36
have shown success across domains, including industrial production 37 disease prevention 38
and financial forecasting [40] 31] To handle more contextual information, extensions incorporating spatial context have been proposed, e.g.,
graph convolutional networks (STGCN) 41]
diffusion convolutional recurrent neural networks (DCRNN) 42). and graph message passing net- works (GMSDR) +3 However, these models remain fundamentally deterministic and do not explicitly model stochastic temporal dynamics. Generative models, e.g., deep auto-regressive mod- els 44 and diffusion/flow matching models 47) provide probabilistic modeling of the time series and generate diverse future scenarios 48 53 However, these models often face computational challenges, as either sampling or computing the likelihood can be expensive. Additionally, they do not explicitly model abrupt jumps, limiting their generalization ability to scenarios with discontinuities.
Another line of research integrates classical mathematical models, such as ordinary and stochastic differential equations (ODEs and SDEs), into deep learning frameworks [33] 59). In financial modeling, thyyics-informer neural networks (PINNs) (60) have been explored to incorporate hand- crafted Blhck-Schhles (BS) and MJD models as guidance to construct additional loss functions [6]] 62 However, these approaches differ from ours, as we directly parameterize the con-stationary MJD model using neural networks rather than imposing predefined model structures as constraints. Neural jump diffusion models have also been explored the context of temporal point processes (TPPs), such as Hawkes and Poisson processes (63 64) However, these methods primarily focus event- based modeling, where jumps are treated as discrete occurrences of events, thus requiring annotated jump labels during training. In contrast, our approach aims to predict time series values at any give time, irrespective of whether a jump occurs, without relying on labeled jump events. Morcover, since jump events unknown in our setting, our likelihood computation is more challenging since it requires summing over all possible number of jumps.
Finally, various extensions of traditional MJD have been proposed in financial mathematics to handle stationary data (65] 66), such as the stochastic volatility jump (SVJ) model [67]. affine jump models [68], and the Kou jump diffusion model [69]. However, these models rely on strong assumptions for analytical tractability, requiring manual design of parameter evolution and often being computationally expensive (70). For example, the SVJ model combines Heston's 's stochastic volatility with MJD under the assumption that volatility follows the Cox-Ingersoll-Ron --osss (CIR) process, meaning it reverts to a fixed long-tern term mean. Despite this, it lacks closed-fu likelihood function. Moreover,

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variants with time-depende parameters require calibrations on market data to obtain the functions of parameters [7T] In contrast, our model directly learns the parameters of the non-stationary MJD from data, which is similar to the classical SDE calibration for financial modeling, but provides better expressiveness and flexibility while still permitting closed-form likelihood evaluation.
3 Background
To better explain our method, we first introduce two prominent models in mathematical finance, the model and the Merton jump diffusion model.
Black-Scholes (BS) model. The
model was developed by Fischer Black and Myron
Scholes, assuming that asset prices follow a continuous stochastic process 722. Specifically, the dynamics of asset price St time is described by the following SDE:
dS, Sz Sj + odW.),
where is the drift rate, representing the expected return per unit of time, and is the volatility, indicating the magnitude of fluctuations in the asset price. W, refers to a standard Wiener process.
Merton jump diffusion (MJD) model. To account for discontinuities in asset price dynamics, Robert C. Merton extended the BS model by introducing the MJD model [3] This model incorporates an additional jump process that captures sudden and significant changes in asset prices, which cannot be explained by continuous stochastic processes alone.
The dynamics of the asset price St in the MJD model are described by the following SDE:
ds, S.((u Ak)dt + odW, + dQ:),
where Qi follows a compound Poisson process and captures the jump part. Specifically, Qr EN, (Y; - 1), where Y; is the price ratio caused by the i-th jump event occurring at the time 4i, Le., Y, /S, and N, is the total number of jumps up to time I. S., and St- are the prices after and before the jump at time t,, respectively. Y, captures the relative price jump size since dS, (S., - S )/S = Y, 1. The price ratio Y, follows a log-normal distribution, i.e., In Y ~T,((x,))) where u and y2 are the mean and the variance respectively. N₂ denotes the number of jumps that occur up to time t and follows Poisson process with intensity X, which is the expected number jumps per unit time. To make the expected relative price change B|dS,,/ /S..] remain the same as in the BS model in Eq. (1. MJD introduces an additional adjustment in the drift term of the diffusion, ie... Akdé in Eq. 2 particular, we have,
E[dQu] = IE[(N; N. - 1)dNN] = E|YN, L]E[d,N,),
where the assumption of MJD that "how much it jumps" (captured by YN,) and "when it jumps" " (captured by N₂)) are independent. For the log-normal distributed YN., we can compute the expected jump magnitude E[(,, 1] exp (v + +22/2) define k: == EYN,, - 1]. For the Poisson process N,, we have E(dN,] = Xdt. Therefore, we have E[dQ.] = Akdt, which justifies the adjustment term - -dedt in Eq. (
1. To simplify the notation,
The MJD model has an explicit solution for its asset price dynamics, given by:
S₁ S.
o² o² 1 1+ Σ In.
In
-u-
Ak
oW
Based on this solution, the conditional probability of the log- log-return at time t, given the initial price So and the number of jumps N, n, be derived
P (In S. N₂ = n) = (an; 12))
where a, So (N Ak -// + and by = oz Therefore, we obtain the likelihood.
ny
(In Σ P(N. P S. N; n) (Ae)"
(In St a..
exp
(5)
V
26%
Here we use the fact that P(N, = n) follows Poisson distribution. One can then perform the maximum likelihood estimation (MLE) to learn the parameters {u, A, 7),. Additionally, the conditional expectation has a closed-form.
E | Sb) = So exp(μt).
(6)
The derivations of the above formulas are provided in App. A
3

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with intensity measure:
Atf y)dydt where ( 1f)
ds, S,((4 Ayki)dt + a,dWi
Midt dy))
Sampling Time f
RR
Contextual Information
Neural MJD
Sequence
Jump
C
Ground Truth
Ww
Prediction
T
Ty
T
Simulation
Real Sequence
(pr: Co Ae Try
The overview of Neural MJD. Our model captures discontinuous jumps in time-series data and uncovers the underlying non-stationary SDEs from historical sequences and context information. Our method enables numerical simulations for future forecasting along time.
Figure 1:
4 Methods
We consider the general time series prediction problem where, given the observed past data {Sp.S-1... S........... S-z, J. and optional contextual information (e.g., additional features) C, the goal is to predict the future values (Si...., Sr, J. Here 7 and 7, denote the past and future time horizons, respectively. While we use integer indices for denoting time, sampling time is not restricted to integers. Our model is built upon a diffusion process, which is inherently continuous in time and compatible with arbitrary sampling mechanism. An overview of our method shown in Fig.
4.1 Neural Non-Stationary Merton Jump Diffusion
In the vanilla MJD model, the increments from the Wiener and Compound Poisson processes are independent and stationary, i.e., odW, iid N( ddd)), d.,, lid Pois(Adt), and In Y, lid (N,, The stationary assumption may be too strong in the real-world applications. For example, the stock prices of certain companies, such Nvidia, experienced significantly larger jumps in the past decade compared to the previous one. Specifically, we allow independent but non-stationary increments in MJD by introducing tmm-inhomogeneous parameters (14, ot, At, Te, M JoSSST, the original SDE. Thus, at any future time i, the modified SDE follows
d.S Sc ((rr dcki)ddt o,dW,+ (pe-eee))
(y (1-1))((d,,d))
Here Å IR and oe, Me, he Rd. while We denotes d-dimensional standard Wiener process. With a slight abuse notation, o,dW, means element-wise product between two size-d vectors. N (dt, dy) Poisson random measure on [0, Ty] x IRd,, which encodes both timing and magnitude of jumps. Intuitively, Poisson random measure extends the idea of Poisson process to random events distributed over both time and magnitude spaces, and N (dl, dy) counts the number jumps occurring within the infinitesimal time interval [t, l di] whose sizes fall within [y. y + dy). The jump component S; (g 1)N (dt, dy) introduces abrupt discontinuities in the process, accounting for sudden shifts in data.
The statistical properties of the Poisson random measure are uniquely determined by its intensity measure Aefy (t. d)ddt In our model, the intensity measure controls imm-inhhoooeenouu jump magnitudes and frequencies. Namely, jump times follow Poisson process with deppedden intensity X, and jump magnitudes follow a time-depende dependent lor-normal distribution, i.e., jump Y, at
time follows In Y, (N,,, 7,), where we denote the log-normal density Y, as (v (1.3). Similarly,
E|Y, 1] exp(e: + 2 -1 in the drift term. This makes Eq. 7 equivalent to using the compensated Poisson measure (dt, dy) == (dtt, dy) - >efy (t., y)ddtt to remove the expected contribution of jumps. Note Eq. (7) includes ke, that fge (v f y)dydt
define
A,E 1]dl = Atldd Namely, it can be rewritten as,
ds, S₁ (1,t+ +o, ((a
This preserves the martingale property of the process induced by (dtt dy), egg. R(dS./S, matches the drift term in the non-stationary Black-Scholes model without jumps.

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More importantly, inspired by the amortized inference in VAEs [73,, we use neural networks to predict these inmomoonooogee parameters based on the historical data and the contextual information C,
μt,, ot. Ae: Yt. 14 [So. S-1..... S-r,, C.t),
(8)
where f is neural network parameterized by 0. To simplify the notation, we denote the set of all observed data as C = {So. 5_...... S-r,, C} from now on. Importantly, we only train a single network across all series and optimize the conditional log-likelihood as the training objective, which differs from standard statistical inference. At test time, the network produces context-dependen estimates (Jtz, oe. Ae, V1, 7+)) across all future time in single forward pass.
The stochastic process described by the SDE in Eq. 7 is formally an instance of an additive process [65] Ch. 14], characterized by independent but non-stationary increments. If Tt M4 E L22 ie.. they are square-integrable functions, and max, f |y2 (v (T, y)dy) < cc, then our non-stationary MJD has unique solution for every So > 0 65 Theorem 14.1]. As prediction time horizon is closed domain E [0. Ty these conditions are easily satisfied as long as the neural network fo does not produce unbounded values. At any future time T,, the explicit solution of the SDE given by,
ST = [ (mu dsk In Su
of jdt + L o,dW, (-' In (dt, dy).
2
Next, we model the conditional probability of log-retum In St given initial price So and past data C,
P S( = exp(- L"d A,
n=0 n! Pn.
(10)
where
On = f. / 1o.1y 1e oln ST; un bū) bl))dl1+ dl1 dln:
On Sn + 1-" (Me ( Acke
σi
dt
Ut. 6n
l- oi + Σ 9
Here @In ST; on, 62) ) is the density of a normal distribution with mean and variance 62 t denote the timing of jumps. Further, we compute the conditional expectations as,
= Sy E = exp( [' p,dl).
(11)
Please refer to App. B for derivations. Evaluating Eq. (9) and Eq. (10) is non-trivial due to time inhomogeneity and jumps, typically requiring Monte Carlo methods or partial atfegr-diifferentall equation techniques for approximate solutions [65] Ch. 6, Ch. 12].
4.2 Tractable Learning Method
While Eq. (TOp provides the exact likelihood function, evaluating precisely is impractical due to 1) integrals with time-dependent parameters lacking closed-fom solutions and 2) the infinite series over the number of jumps. To learn the model via the maximum likelihood principle, we propose computationally tractable learning objective with parameter bootstrapping and series truncation.
First, given a finite number of future time steps {1,..., T we discretize the continuous time in SDEs to construct a piecewise non-stationary MJD. Our model predicts time-varying parameters
{Ur, T7, Ar, Tr, Mr / For any time < Tr, we map it to integer index via Pt == [t] + 1. Thus, likelihood the data at given the data at I, where Pt - 1 5
Pt. is given by:
00
P PIn S,,C)= Σ P P(AN P
P(n
= =n)
8t exp (-> 8) N n!!
(In S an 62.6)
(12)
n=0
5

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Algorithm Neural MJD Training
Algorithm 2 Fuler-Maruyama with Restart Inference
1: repeat
Require: Solver step size 1/M
(C,TT, ~ Drain
I: C ~ Dust with C = 1S-Tp.0: C]
with C 1S- C]
Se(C) 2: {ur, (rr,ar,M, vr, Yo F
3: {ur, 0r, Or, MT, 1r) /,t,,,)
+ Fo(C)
to f 0,
M x T's
4:
So t So
4: for i = 1, , do
5:
for = 1,..... TT do
5:
ti + ti-1 + 1/M, pt; + [t:] +1
6:
Fr In P(In
ST-1 - C)
6:
(N': Ape, kip, pp: /2)/M
Drift
+ Eq. 12)
t op, 21 /// /MM with 21 NV(0,))
Diffusion
7:
S- + T[S+ |C]
Eq. 13)
8:
Ç + KVM, + //7p1. 22
Update 8 via
with K ~ Pois(A A Apt, HMM
(O)
D Jump
V Σ
(-p- Ur + w||St 5+113)
9:
if (i - 1) mod M = then
9: until converged
10:
B Sn, + B [n Sp p2 C] + 0z + + + G >
Restart
else
12:
In St + In Se + 0s +
+B++B B.+
13:
return (St,
where 0n,6 = In + (Up. >p, - /2)6 + nvp, and b² = +2.5-22 This approach
eliminates the need for numerical simulation to compute the integrals Eq. (TO) and has been widely adopted for jump process modeling [66] [74].
As for the conditional expectation, we have
Pr
E
[S|[c]
So exp
Σ My +t (t-ta- Pt + 1)Hp.).
(13)
Derivation details are shown in App. B Further, we jointly consider the likelihood of all future data:
Ty
SInnS1.-. ... In ST, = II P(In S,I(In = II
P(In S,
r=1
r=1
where we use the Markov property and the fact that In(-) is bijective. Therefore, the MLE objective is given by:
In P(In S ..... In ST, |C) = Σ In PIn (I S7-1, C).
(14)
Eq. 12
The training algorithm of our neural non-stationary MJD model is shown in Alg. I In computing the term In P(In S, of Eq. (4), instead of doing teacher forcing, we replace the ground truth S,-1 with the conditional mean prediction E[S,-1 C] from Eq. [33. This design mitigates the dis- crepancy between training and inference while redueing error accumulation in sequential predictions, especially for non-stationary data. As shown in the ablation study in Sec. 5.3 this approach improves performance effectively. To further improve accuracy, we introduce an additional regularization term that encourages the conditional mean to remain close to the ground truth. Additionally, the for loop in Alg. can executed in parallel, as the conditional mean computation does not depend on sequential steps, significantly improving efficiency. Notably, our model imposes no restrictions on the neural network architecture, and the specific design details are provided in App. D.I
Truncation error of likelihood function. Exact computation of P(In S, S) in Eq. 14 requires evaluating an infinite series, which is infeasible in practice. To address this, we truncate the series at order K E N+, i.e., limiting the maximum number of jumps between consecutive time steps. We establish the following theoretical result to characterize the decay rate of the truncation error:
Theorem 4.1. Let the likelihood approximation error in Eq. (12. truncated to at most jumps, be
(x( Ex P (AN n) P (In S.48 S,,C,N n).
Then, (t, 8) decays at least super-exponentially as - co, with convergence rate of O(
The proof provided App. CTT The truncation error is dominated by is, with other time-dependent parameters absorbed into the big- O notation. We set to 5 to achieve better empirical performance.

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Inference without
Uncerteinlyy Prediction
Uncertainty Prediction
2
3
Ty
3
Tj
So
Expected Meen la S,C
Sa.
S24.
82
Restart time
Sa.-5 Stea
82-5.
Inferencing results
Figure Comparison of numerical simulations with and without restart strategy during inference.
4.3 Inference based on Euler Scheme
Once trained, our Neural MJD model enables simulations following Eq. 7 by computing the non- stationary SDE parameters with single neural function evaluation (NFE) of fe. Unlike models limited to point-wise predictions, Neural MJD supports continuous-time simulation across the entire future horizon. Although the training data consists of finite set of time steps, Si.T,- the learned model can generate full trajectories from = 0 to = Ty at arbitrary resolutions.
The standard Euler-Maruyama (EM) method provides a general- purpose approach for simulating SDEs with simple implementation and proven convergence [75] 76). However, MJD SDEs exhibit analytically derived variance that grows over time (see App. B and the simulated trajectories produced using the vanilla EM, assuming sufficiently small error, reflect this growth as well. Notably, the resulting high-variance simulations can undermine the empirical reliability of future forecasts.
In our MJD model, it is possible compute closed-form expressions for statistical properties such as the mean and variance at any point in time [65]; for instance, the analytical mean can be derived from Eq. (3).. Building on this insight, propose hybrid analytic-simulation solver, the Euler- Maruyama with restart method, which periodically injects the exact analytical mean to improve accuracy and enhance stability, as shown in Alg. 2 and Fig. 2 Specifically,
discretize time using ...
uniform step size 1/M for simulation and set the restart points as the target times {I, .1, J. The solver follows the standard EM method for Eq. 7 whenever a restart unnecessary. Otherwise, resets the state using the conditional expectation from Eq. (13).
Further, prove that this restart strategy has tighter weak-convergencee error, particularly helpful for empirical forecasting tasks where the mean estimation is critical. Let e, = [ [S)) be the standard weak convergence error 775. where S, is the ground truth state, S, is the estimated one using certain sampling scheme and g is a K-Lipschitz continuous function. We denote the weak
convergence errors of our restarted solver and the standard EM solver by e, and c,- respectively.
Let 1/M be the step size. Both standard EM and our solver exhibit weak convergence rate of OLLMM). Specifically, the vanilla EM has weak error of of < K exp(Lt)/M for some constant L > O, while ours achieves tighter weak error of c < K <exp(L(t [t)))M
Proposition 4.2.
The proof and details are left to App. Our sampler is in the same spirit the Parareal simulation algorithms [7] it first obtains estimates at discrete steps and then runs fine-grained simulations for each interval. By resetting the state the true conditional mean the start of each interval, sampler reduces mean estimation error and prevents error accumulation over time. Notably, the SDE simulation requires no additional NFEs and adds negligible computational overhead compared to neural-nee neural-network inference, since it involves only simple arithmetic operations that can be executed efficiently on CPUs. For reference, also present the standard EM solver in App. D.5
5 Experiments
In this section, we extensively examine Neural MJD's performance on synthetic and real-world time-series datasets, highlighting its applicability in business analytics and stock price prediction.
7

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Table 1: Quantitative results the synthetic dataset.
Pat Sequence
Menn
Witnes taics-ill
Probubilistic
ATIAA
Mixell
MAE]
R-t
musR"t
μ-MAEL
p
NPP
NIA
0.02
0.21
0.08
0.18
0.15
0.20
0.09
30
XGBoooot
0.18
0.37
NIA
NA
MI.F
0.14
0.21
NIA
N/A
NJ ODC
0.15
0.20
NIA
NA
Figure 3: Qualitative result on the synthetic dataset.
Neural RS
0.15
0.25
0.10
0.35
0.14
0.29
Neural MJD (ouns)
0.09
0.32
0.07
0.39
0.09
0.34
Baselines. We evaluate Neural MJD against a wide range of competitors, including statistical methods such as ARIMA (8, the BS model, and the MJD model. Additionally, we compare against learning- based approaches, including supervised learning models such as XGBoost [82]. MLPs, GCNs 83 as well as denoising diffusion models like DDPM 45) EDM 84, and flow matching (FM) 47 We include recent neural ODE or SDE based learning methods such as NeuralCDE [33
and LatentSDE [59] for comparisons. NJ-ODE [85] was further included on the S&P 500 dataset. We also design a baseline model, Neural BS, which shares the same architecture as Neural MJD but omits the jump component. For DDPM, EDM, FM, Neural BS, and Neural MJD, we share the same transformer-based backbone to ensure fair comparison. Since some datasets contain graph-structured data as seen in the following section, we incorporate additional graph encoding steps based on Graphormer [86] to capture spatial features, which also justifies the inclusion of GCN as baseline. Further details are provided in App.D
Evaluation metrics. We employ Mean Absolute Error (MAE), Mean Squared Error (MSE), and the R-squared (R²) score as the primary evaluation metries. To account for stochastic predictions, we run each stochastic models 10 times and report results across three types of metrics: 1) Mean Metries: used for deterministic models or to average the results of stochastic models; 2) Alller-Takes-All Metrics: we select the best prediction from ten stochastic samples to compute minMAF, minMSE, and maxR?; 3) Probabilistic Metrics: these metrics assess the likelihood of stochastic predictions and select the most probable outcome to calculate p-MAE, p-MSE, and p-R3. We mark N/A for inapplicable metrics for certain methods. Please refer to App. for more details.
5.1 Synthetic Data
Data generation. We evaluate our algorithm scalar Merton jump diffusion model. The dataset consists of N 10, 000 paths, generated using the standard EM scheme with 100 time steps. Using sliding window with stride 1, we predict the next 10 frames from the past 10. The data is split into 60% training. 20% validation, and 20% testing. Refer to App. D for details.
Results. Tab. D reports quantitative results on the jump-driven synthetic dataset. Learning- Jearning-based methods outperform traditional statistical models (ARIMA, BS, and MJD), and our Neural MJD tops all three evaluation protocols, surpassing Neural BS thanks to its explicit jump modeling objective. Qualitatively, as shown Fig. 3 our Neural MJD generates larger, realistic jumps, while the baselines produce smoother but less accurate trajectories.
5.2 World Business and Financial Data
Business analytics dataset. The SafeGraph& Advan business analytics dataset combines proprietary data from Advan [87] and SafeGraph [88 to capture daily customer spending at points of interest (POIs) in Texas, USA. It includes time-series features (e.g., visits, spending) and static features (e.g.. parking availability) for each POI, along with ego graphs linking each POI to its 10 nearest neighbors. Using a sliding window of 14 input days to predict the next 7, the dataset spans Jan.-Dec. 2023 for training. Jan. 2024 for validation, and Feb.Apr. 2024 for testing.
Stock price dataset. The S&P 500 dataset [89] is public dataset containing historical daily prices for 500 major US stocks. It comprises time-series data without additional contextual information. We construct a simple fully connected graph among all listed companies. Similarly to the business analytics dataset, we employ sliding window approach with stride of 1, using the past 14 days as input to predict the next days. The dataset divided into training (Jan.-Dec. 2016), validation (Jan. 2017), and testing (Feb.-Apr. 2017) sets. Refer to App. D for further details about the datasets.
8

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Table 2: Quantitative results the SafeGraph&Addye Advan business analytics dataset.
Metries
Mean
Winner- takee--takes-al
Probabilistic
Model
MAEL
MSEJ
R²
minMAE4
minMSE
maxR"1
P-MAEL
p-MSEL
p-R
ARIMA
152.6
1.66e05
-0.183
N/A
N/A
BS
135.5
1.05c05
0.102
112.8
9.01c04
0.159
121.5
9.87c04
0.138
MJD
131.8
9.98c04
0.127
109.6
8.48c04
0.169
117.6
9.02c04
0.144
XGBoost
124.0
9.76004
0.303
N/A
N/A
MLP
109.5
8.18c04
0.416
N/A
GCN
95.2
7.12c04
0.432
N/A
N/A
DDPM
68.5
4.75c04
0.501
58.9
4.48c04
N/A
EDM
57.6
4.35c04
0.525
49.4
3.76c04
0.556
N/A
FM
54.5
4.32e04
0.540
47.8
3.58e04
0.552
N/A
NeuralCDE
94.6
7.09e04
0.425
N/A
N/A
LatentSDE
75.7
5.26c04
0.487
66.5
4.58c04
0.498
N/A
Neural BS
56.4
4.17e04
0,539
45.6
3,45e04
0,561
55.9
4.16c04
0.538
Neural MJD (ours)
54.1
4.18c04
0.549
42.3
3.19e04
0,565
53.0
4.10el4
0,550
Table 3: Quantitative results on the S&P 500 stock dataset.
Metries
Mean
Winner-takes-all
Probabilistic
Model
MAEI
MSEJ
R²t
minMAE!
minMSEJ
maxR21
P-MAEL
p-MSEJ
P-R²1
ARIMA
3,67e04
-0.863
N/A
N/A
BS
65.1
4.01c04
0.052
44.6
1.79004
0.145
52.8
2,03c04
0.105
MJD
64.3
3.58c04
0.092
40.7
1.22c04
0.235
49.7
1.67e04
0.112
XGBoost
44.3
1.64004
0.170
N/A
N/A
MIPP
44.4
1,57004
0.205
N/A
N/A
GCN
44.7
1,53e04
0.224
N/A
N/A
NJ-ODE
1.69004
0.208
N/A
N/A
DDPM
42.2
L88804
0.235
8,42603
0.470
N/A
EDM
37.1
1.68044
0.249
27.6
5.01c03
0.542
N/A
FM
34.9
8.47e03
0.368
19.8
3.59e03
0.625
N/A
NeuralCDE
42.8
1, 1,46004
0.201
N/A
N/A
LatentSDE
39.8
1. 1.44004
0.212
20.8
3,49e03
0.617
N/A
Neural BS
31.6
4.32e03
0.781
12.6
8.04e02
0.959
22.3
2. 19t93
0.889
Neural MJD (ours)
15.4
1.36603
0.953
4.3
1.46e02
0.995
13.6
1.08e03
0.963
Results. Tab. Preports results on the SafeGraph&Advan dataset covering POIs revenue prediction, which is measured in dollars. Denoising generative models (e.g., DDPM, EDM, FM) show strong performance, significantly outperforming simple supervised baselines like GCN. Neural MJD further improves upon the strong FM baseline, especially winner-takes-al metrics, indicating better diver- sity and accuracy in generating plausible outcomes through simulated jumps. While denoising models support likelihood evaluation, their high computational cost-requiring hundreds NFEs-maker them unsuitable for large datasets. In contrast, Neural MJD enables fast likelihood evaluation without such overhead, which enables the computation probabilistic metrics.
Tab. 3 shows similar results on the S&P 500 dataset. FM again outperforms conventional baselines, including ODE based NJ-ODE and NeuralCDE, and Neural MJD achieves the best overall perfor- mance, effectively capturing volatility and discontinuities in stock time-series data. For completeness, we also report results for additional deterministic time-series baselines in Appendix D.3
Table Ablation study on the effect of teacher foreing (TF) and Euler Marlyamm (EM).
Table 5: Runtime comparison.
Model
Train (ms)
1-run (ms)
10-run (ms)
Meun
Winner takes
Probabilistic
MI.P
65.2
52.2
N/A
Model
MAE]
R²
minMAE]
maxRt
P-MAEL
p-
GCN
271.3
250.7
N/A
Ours
66.7
0.495
$7.4
0,511
64.5
0.499
FM
184.6
275.4
2696.3
w.
101.5
0.325
99.8
0.324
Ours
183.5
166.8
179.2
EM
85.6
0.397
79.4
0.423
84.4
0.405

---

5.3 Ablation Study
We perform ablation studies to evaluate (i) the training algorithm described in Alg. I and (ii) the Euler-Maruyand with restart solver introduced in Sec. 4.3 For the ablations, we use 10% of the SafeGraph& Advan business analyties training for traminn and evaluate on the full validation set.
The results are presented Tab. Our training algorithm computes the MLE loss using the model predictions instead of ground truth, unlike teacher forcing. This improves training stability and reduces the generalization gap. Additionally, we empirically show that the vanilla EM solver results in higher variance and worse performance compared to our solver.
Additionally, we compare the runtime of our method against various baselines in Tab. 5 Thanks to the efficient numerical simulation-base forecasting framework that does not increase NFEs, models are particularly well-suited for efficient multi-run stochastic predictions.
Conclusion
We introduced Neural MJD, neural non-stationary Merton jump diffusion model for time series forecasting. By integrating time-inhomoggnnouus Itō diffusion and a time-inhomogeneous compound Poisson process, our approach captures non-stationary time series with abrupt jumps. We further proposed likelihood truncation mechanism and an improved solver for efficient training and inference respectively. Experiments demonstrate that Neural MJD outperforms state-of-the-ar approaches. Future work includes extending to more challenging data types like videos.
Acknowledgments and Disclosure of Funding
This work was funded, in part, by the NSERC DG Grant (No. RGPIN-2022-04636)) the Vector Institute for AI, Canada CIFAR AI Chair, Google Gift Fund, and the CIFAR Pan-Canadian AI Strategy through a Catalyst award. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through the Digital Research Alliance of Canada alliance. can. ca, and companies sponsoring the Vector Institute www. wwwtorinstitttt.. ai/ #partners| and Advanced Research Computing at the University of British Columbia. Additional hardware support was provided by John R. Evans Leaders Fund CFI grant. Y.L. and Y.G. are Q.Y. Y. is supported by UBC Four Year Doctoral Fellowship.
supported by the NSF grant IIS-21 113468.
10

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Appendix
TABLE OF CONTENTS
Derivations of the Stationary Merton Jump Diffusion Model
17
A.I MJD and Lévy Process
17
Explicit Solution MJD
17
Likelihood Function of MJD
18
Derivations of the Non-stationary Merton Jump Diffusion Model
19
B.1
Non-stationary MJD and Additive Process
19
B Explicit Solution to Non-stationary MJD
19
B.3 Likelihood Function of Non-stationary MJD
20
Proofs of Theorem and Proposition
22
Proof of Theorem
22
Proof of Proposition 4.2
24
Experiment Details
25
D.I
Baseline, Model Architecture, and Experiment Settings
25
Datasets Details
25
D.3 Additional Deterministic Series Baselines (Third Party Implementations
27
D Limitations
27
D.5
Vanilla Fuler Solver
27
Impact Statement
28
16

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A Derivations of the Stationary Merton Jump Diffusion Model
In this section, we briefly review the mathematical derivations from classical textbooks ensure the paper is self-contained. Our primary focus is on the case where the state variable S is scalar, as is common many studies. However, in Sec. 4 we extend our analysis to the more general IR setting. Notably, in our framework, we do not account for correlations among higher-ding dimeer-aimesional variables. For instance, the covariance matrix of the Brownian motion is assumed be isotropic, meaning all components have the same variance. To maintain clarity and consistency with standard textbook conventions, we adopt scalar notations throughout this section for simplicity.
A.1 MJD and Lévy Process
Definition A.1. Lévy process
Definition 3.1] A càdlàg (right-continuous with left limits) stochastic process (20 on (S, with values in IR such that Xo 0 is called Lévy process it possesses the following properties:
655
1. Independent increments: For every increasing sequence of times to f1...., bu the random - Xx... are independent.
variables X X,, - -.o...., X,,.
2. Stationary increments: The law of X1th - X, does not depend on f.
3. Stochastic continuity: For all E > O, lim,- P[[ - X²l 2 =.
A Lévy process (20) is a stochastic process that generalizes jump-diffusion dynamics, incorporat- ing both continuous Brownian motion and discontinuous jumps. The Merton Jump Diffusion (MJD) model given by,
ds. S.((H Ak)dkt + odW, + dQ.).
(15)
is a specific example of a Lévy process, as it comprises both a continuous diffusion component and a jump component. According to the Lévy-Itô decomposition [65] Proposition 3.7], any Lévy process can be expressed as the sum of a deterministic drift term, Brownian motion component, and a pure jump process, which is represented as a stochastic integral with respect a Poisson random measure.
A.2 Explicit Solution to MJD
To derive the solution to MJD in Eq. to the SDE:
based on 665 Proposition 8.14], we first apply Itô's formula
df(S,,t) Of(S,,()
aS(S,. 1)
OS(S.. 1) as,
as, dt + + (f(()) - f(S:))),
dt b.
dt + wi
dw,
Øt
2 asp
(16)
where b, = (" (m- Ak)S,, Wt S,, and S._ represents the value of S before the jump at time By setting the function [(S., 1) = Sr, the formula can rearranged as:
d S, Sm
01 dt (μ Ak)S, Im 025t 0² Sn as,
In dW.
dt +
dt oS
2 0s
0S,
[In(S.) - In(S,-)]
(n 1; 1
025?
dt + oSe( 1
{ JaW, + [In(S)) In(S,-)]
dt +
5,
2
5,
²² dt + odW + [In(S.) In(S
(p Aw)
(17)
2
From the definition of the Compound Poisson process, we have that S, Y,S, such that In(S,) In(S; ) = In Y. Here, Y; is the magnitude the multiplicative jump. Therefore, integrating both sides of Eq. (17 we get the final explicit solution for MJD model:
N
In S₁ - In Sg (μ Ak
2 oW, + Σ Iy,
(18)
2
im
17

---

We can reorganize the explicit solution as:
S. = So exp (m-N t
-
Nr In Y
+
(19)
since the drift term, diffusion term and jump term are independent, derive the mean of S conditional on So:
exp (-- (M- -k 2
0²)
E Su]
=S,.E
ow +
In Yi
N₂
)
=5 cxp
11- - Ak
exp (oW
E
exp
Σ i==l
In
Yi
((-
((
02)
2 1) ))
=So exp
|1 Ak
exp
exp (Akt)
=So exp (1e)
(20)
A.3 Likelihood Function of M.JD
For the log- log-likelhhood derivation, given the conditional probability in Eq. the MJD model can be expressed as:
4 the log- log-likelihood of
20
log P (In Sc = log
P (NN = n) P (In Sc N₂ = n)
n=
exp (-X0) (at) 1 exp
(In Si St an)
log
2b²
(At)
(In S: - an))
log
exp
-Xt + log
log
n!!
v /
20
log b²
(In St ap. )2
log
exp
-Xt + n log (At)
log n!! 2
26
(21)
where a, In So + (e - Ak Ak- 2- t and b2 o 2 In maximum likelihood estimation
o²
(MLE), the initial asset price Sn is assumed to be constant (non-lcarnable) and can therefore be ex- cluded from optimization. The objective of MLE estimate the parameter set θ {1,, a, X, 7, v. by maximizing the likelihood of the observed data under the estimated parameters. For the MJD model, the MLE objective is to determine the optimal parameters Ô. By omitting constant terms and expanding 8n and 0n, the final expression of the MLE objective can be simplified as:
θ arg max log P(I SI
log (02t + ny2)
arg max log
exp 1==
At nog (At)
2
(In S, - In So
o Ak) t nw/²
t-n
(22)
2 2 + 2
According to [90]. the Fourier transform can be applied to the Merton Jump Diffusion log-return density function. The characteristic function is then given by:
(.(w) = exp(iwx))(( P(z)dx
xc
iwv + 7Pw2 - -1 + ii ((- 0²
exp At exp
2 -Ake
1k)
t
(23)
2
2
18

---

where r = In S.
90
With simplification (.(w) = exp [tg(w)], we can find the characteristic exponent, namely, the cumulant generating function (CGF):
iwv + 7Pw²)
o²
o²w² (24)
g(w)
exp
2 -1 1 + iw (1- 2
Ak }
2
where & = exp (u 12) 1.
The series expansion of CFG is:
w2
w k3
g(w) = iwk1
(25)
2
6
According to 165 Proposition 3.13], the cumulants of the Lévy distribution increase linearly with t.
Therefore, the first cumulant kn is the mean of the standard MJD:
In St
(t) = E
= (μ Al - 0²/2 Avit
(26)
Sp
The second cumulant l is variance of the standard MJD, which is:
S.
k2(() Var
(o² 1(72 + x²))
(27)
S
The corresponding higher moments can also be calculated as:
Skewness = ks(() =A(3> )t
(28)
Excess Kurtosis ka(() =A(3>² + 6u29² + v²)
(29)
B Derivations of the Non-stationary Merton Jump Diffusion Model
B.1 Non-stationary MJD and Additive Process
Definition B.1. Additive process 665 Definition 14.1] A stochastic process (Xe) on Rd is called
an additive process if is cådlåg, satisfies Xo 0, and has the following properties:
1. Independent increments: For every increasing sequence of times to, t1...., tn. the random variables X X1, - X,o...., Xt., - -i.... are independent.
P(|X+++ 2. Stochastic continuity: For all & > 0, Pi - Xel 2 2) =
In the non-stationary MJD model, given by,
ds.- Si((p de o, / (y- dy)).
(30)
the parameters governing drift, volatility, and jump intensity evolve over time, resulting in non- stationary increments. This violates the key stationarity property required for Lévy processes, discussed in App. A Consequently, the non-stationary MJD no longer falls within the Lévy process framework. Instead, according to the definition above, a stochastic process with independent increments that follow non-stationary distribution is classified as an additive process. Similar to the relationship between the stationary MJD and the Lévy process, the non-stationary MJD can be viewed specific instance of additive process. Thus, we apply corresponding mathematical tools for additive processes to study the non-stationary MJD.
B.2 Explicit Solution Non-stationary MJJD
To derive the explicit solution to the non-stationary MJD, according to (65 Proposition 8.19], we have the Itô formula for semi-marting mamiimmattggals::
f(t, X.) - f(0, //' Of(s X.)
f Xf.s dX, 2
//d
o
fs.
ds +
2
x]x
Os
dr
Ox
AX, 0f(x,X..
Σ
[f(s, -(8,
(31)
Ox
05
19

---

According to 665 Remark 8.3], for a function independent of time (i.e., (tt = f(xe)), when we have finite number of jumps, we can rewrite the above equation as:
f(x,) f(xo) - ((x 1 L (XX
E
I (X,,)))
where X is the continuous part of Xx, and [X X| is the continuous quadratic variation of over the interval s].
In our case, let Xx St, and define S(1,, X.) In St, the corresponding derivatives are 8f(.
Bln 85:
S. and 0² s(e_xi) and
8X? - The dynamics of non-stationary MJD is defined by:
ds. = St (ndt Ackett + o,dW. + (u IdN dy)
3d
The continuous part of the quadratic variation of S, is d[S, S); - of A jump at time 8 is mod- eled as multiplicative change S, yS._. Thus, the jump contribution is x f(X,-)] = Eog.st.Ax, In(yS,-) In(SS-)] EosIgrAx, In y). Since the jump process is driven by Poisson random measure .V (dtt dy) on [0, t] x R we can rewrite the sum over all jump times as an integral with respect to this measure. When there are finitely many jumps on [0, t).
we have Eos [In = So f& In (ds,
Based on 665 Ch. 14), even when the parameters (drift, volatility, jump intensity, etc.) are time- dependent, the -ta-stationary MJD remains a semi-martingale. Therefore, we can simplify the equation follows for time
(" S. (us Ackis) do
1 (/" S00 (' a,dW, 2 5
In S. - In So
S
+ L [o
In Nny,
(32)
Therefore, the explicit solution is:
In - (v. roka - [ o,dW, S'' S In (ds, dy).
St So
(33)
< The only assumption needed for the derivation is the finite variation condition: Ii Jg Wel dy) 0o. Based the explicit solution for Sc, easily compute the conditional expectations as,
E [In(S:/S)) = S (mo Aohi
0² (d)ds + 1 / Inydefr dydds.
(34)
2
and
[| = So Sx (f p,ds),
(35)
The variance can also be calculated as,
[ od of / (In X y)dyds.
Var [In(S:/So)]
(36)
Given the results for the general time- tite-inhhoogeneouus system, can directly substitute the coeffi- cients into the discrete formulation implemented in Sec. 4.2 to obtain the corresponding results.
B.3 Likelihood Function Non-stationary MJD
Let X, In Ss/ In So, 2 0 be the log-return of the asset price S,. Under the non-stationary MJD settings, X, an additive process, therefore by the general property of additive process (65] Ch 14), the law of X, is infinitely divisible and its characteristic function is given by the Lévy-Khintchine formula:
E Elexp(iu X.)] exp v/ (u),
20

---

where
(b(u)) - u A + iu Te+ + / 7(dy) (c-==1-- iu -t)xx))
-
(37)
where we have the integrated volatility term A, = So a,ds, the integrated drift term I' (1= Adk,)ds, and the Lévy measure ryddy)= Av (4, y).
Since the jumps follow time-inhomogeneous Poisson random measure and the process is additive, we can denote the integrated intensity of jumps by A(t) = J" A,ds, then the number of jumps N₃ in the time range [0, t] a Poisson distribution with this integrated jump intensity At).. When conditioning on No, will have:
P(N. =n) exp(eepppp ((t))A(t)
(38)
n!!
We now derive the conditional density P(In S. n. So). and here can start with the case of one jump. When there exactly one jump [0, I]. the jump time 81 random. Given jump time 81, in time-inhonogeneous setting, the instantaneous probability of a jump at time 81 is proportional to dey. According to the dynamics of non non-stationary MJD, the continuous part of the log-return
leads a normal distribution with mean being 01 Sq + So² (Ho Agh, 2 ds + Vs, and variance being 6f = Jo ds + 2 Thus, the conditional density of In S. given one jump at time $1 is 6(In Sy: 01, b7). where o(; (1;;1.,)) denotes the Gaussian density with mean 01 and variance b7. Since the jump could have occurred at any time in [0, f), we must integrate over the possible jump time 81. Therefore, the conditional density given N, is:
1 P(In N, = 1, So) = A(t) / >a oln bf) dsi.
(39)
where Nay normalizes the density.
When generalizing the case of N n. the conditional density P(In S, nI So) defined via an integration over the n jump times, with the jump times denoted by 0 S $1, S t.
Because the process is time-in inhomogeneous, the probability density that a jump occurs specific time s; is given by the instantaneous rate Xa,. therefore for a given set jump times, the joint density for the jumps proportional to I 1 The conditional density be written as:
1
S.|N, nI, So)
1, Hln Sti On b dsi .an
(40)
A(L)
[0.4]
Here @IIn Se; an 62,) is the density of a normal distribution with mean a, and variance bū, which are defined by:
On = In + /-' (wo -
o² ds + Σ
[o
doke
2
b2 [ de +
1s:
For convenience, we may write the mixture term as
2 / S/1n
[o.1] 11 1i on b2) dsi ...
(41)
Therefore, for the varying SDEs, the conditional probability of In is given by.
exp(-A(t)) T...
P S,(S)) =
(42)
22 n!
21

---

C Proofs of Theorem and Proposition
C.1 Proof of Theorem
Theorem 4.1. Let the likelihood approxiaation error in Eq.
12
truncated to at most jumps, be
P = n) P S44 SN,CAN==M
Then, (t, 8) decays at least super-exponentially - cc, with convergence rate of OR
Before diving into the proof, we first introduce two important lemmas.
Lemma C.1 (Theorem 2 in (2). Let ~ Pois(m) a Poisson-distribure random variable with mean Its distribution function is defined as P(Y S k) == exp(-m) Σ m² J. with integer support & E {0.1...., )0). For is = O and k = oc, one has P(Y - 0) exp(-m), S 00)=
1. For every other k E 21, ... the following inequalities hold:
(sign(k- (agu(k-m)y 24 (m,,)) k) P(Y S k)
(sign(k +1 m) 2H 2H (m, k (m, 1)).
where Hm,, k) is the Kullback- Leibler (KL) divergence between two Poisson-distrrbuteed random variables with respective means and k:
H (m,, k) Dxr. (Pois(m) ((oos(()) = k: kln
And (z)) is the cumulative distribution function (CDF) of the standard normal distribution and sign(.) is the signum function..
Lemma C.I is particularly helpful in our proof below. We also acknowledge its foundation in an earlier work 91. which provides many insights and a profound amount of valuable knowledge on its own
Lemma C.2 (Bounds the Standard Normal CDF). The following upper bound for o(-) holds when x < 0:
((x)
+ (xx)
where (bxx) expl-x²/2) is the probability density function of the standard normal distribution.
V2*
Proof. By the Mills' ratio inequality for the Gaussian distribution 992 we have 1 - (xx) < d(x)
Vr 0. Using the identity ((-x) ((x) for we immediately obtain: o <
d( Va 0. For < O, substituting -r into the previous bound and noting that @(-x) o(x).
we obtain (xx) ((r)) Var < 0.
Proof of Theorem
Proof. The original likelihood objective in Eq. 112 is as follows:
P (In S St, Σ ( P AN = n)
n=
exp (-)p,0) 0)
o (In K2.5)
n!
(43)
0p,
(In S an,8) )2
exp (-)
exp
n!
v 2
2172 11,8
where 8 is small time change that Pt 0% 22) + nt,, and s 0 8 +2 22.n.
1 S t pe, an,5 = In (Mpr Aprkes
22

---

We define the truncation error with a threshold xt as:
20
(x((t = Σ
P(AN - n) P Sen S. C.C.AN AN = =)).
(44)
BERTI
The second term P PANN- = n) (In S St, C.C.AN = n) is Gaussian density function and upper bounded by Venk
so the truncation error V (t, 8) is bounded by:
v. (t,
Σ
P(AN= =))
(Gaussian density bound)
22 n,
m=*+1
P(AN=1 =))
(bx,5 increases as k goes up)
2Tb2 x11 n=R+1
1
P P(AN=1 = n))
(property of Poisson CDF)
2m22
N416
n==
1
1 D(sinnK Xo, 5) 2D (Pois ())((ois())))
(Lemma
2xxb
n+16
1
1 (sign(AA6 - (P)
(Gaussian CDF)
2mbb
k+1.6
As stated above, the KL divergence between two Poisson distributions follows
DRL. (Pois(a)]Poois()) b + blnl )
Therefore,
1
(sign(App. - x) o)
V (1, <
+
/ / x+1,6
1
2( sign(Ap, 2 x
**
1 2t 2 8 12, (n 1))
1
Ap, sign(
-- Apo )) - -- 1) -
$ 1 2 8 7b. (x 1))
Intuitively, the truncation error decreases to zero as k approaches infinity. Below, we analyze the convergence rate. When is sufficiently large, the term sign 6 -1) is negative. Consequently. the upper bound becomes:
Ap, 2 - --
(l, <
1 2T 2T 8 1B. (K 1))
K
22(2 K In( dee
(Lemma
C.2
2n (02 8 + 22 (K + 1))
2( K - K kln Am1
In( Amr
exp Ap, 8 + k
2x / 2(02 8 22 (K ()p 5 - K is In )
As k - oo, the numerator dominated by exp(- -x x In R:), which decays super-exponentially (faster than any polynomial or exponential decay). The denominator consists of two components:
23

---

The first term, Vo2.0 0 7,2,( + 1), scales asymptotically as 7ps VR.
The second term, - -- Am=")
scales as Valak.
Combining all terms, the upper bound scales as:
exp(=xinn)
K
2 / 2
RyIn
RVIn
The term K =* decays super-exponentially, while the denominator grows algebraically (as syIn n.). The rapid decay of =-* dominates the polynomial growth in the denominator. The overall convergence rate is super-exponentiall fast, the rate of O(exp(- - In x)) or equivalently (x***).
Since the upper bound of y (t, 8) decays at the rate of O(x=*) and v (t, o) is strictly positive, this implies that the original quantity (t 6) must decay at least as fast as the upper bound. This completes the proof.
C.2 Proof of Proposition 4.2
Proposition C.3. Let 1/M be the step size. Both standard EM and our solver exhibit weak convergence rate of (1/M). Specifically, the vanilla EM has a weak error of of S exp(Lt)/M for some constant L O. while achieves tighter weak error of cl K exp(L(t L
Proof. Here, we prove that this restart strategy has tighter weak-convergente error than the standard EM solver. Recall that we let e, - be the standard weak convergence error (75). where S₁ is the ground truth state, St is the estimated one using certain sampling scheme and is K-Lipschitz continuous function. We denote the weak convergence errors of our restarted solver and the standard EM solver by ee and c., respectively.
Step 1: Standard EM Results Time- Homogennm MJD SDEs
Early works on jump-diffusion SDE simulations explored the weak error bounds, which we summarize as follows. For time-homogeneous MJD SDEs, the error term ef of the standard EM method is dominated by ef < K / This is supported by the following: (a) Theorem 2.2 in [76] establishes the (11M)) rate; (b) Sec. 4-5 of [66 and Theorem 2.1 of [3] shows that cf grows exponentially regarding time with big-O factor (t)). In particular, the time-dependent term in the error bound (t(()) used in the proof of (76) rooted in their Lemma 4.1, which can be proven in a more general setting in [93]; e.g., Eq. (6) in 993 discusses concrete forms of (t))), which can be absorbed into O(e*tt for some constant L > O. Lastly, the K-Lipschitz condition of the function provides the coefficient K the bound. For detailed proof-which is more involved and not central ww
to the design and uniqueness of our algorithm-aa we refer the reader to [75, [5]. When combining the above existing results from the literature, we can derive the error bound of c S K exp(I.)/M.
Step 2: Standard EM Results on Time-Inhomogeneous MJD SDEs
Our paper considers imm--inhomogeneuus MJD SDEs, with parameters fixed within each interval [T 1,7) (T E N. T 2 1). This happens to align with the Euler-Peano scheme for general time- inhomogeneous SDEs approximation. As a specific case of time-varying Lévy processes, our MJD SDEs retain the same big-O O bounds as the time- teme-homogeneuus case. Namely, the standard EM solver has the same weak convergence error cf < K exp( Lt) /MM f as in the homogennuuu MJD SDEs. This can be justified by extending Section of [76] that originally proves the EM's 's weak convergence for time-homogeneous Lévy processes. Specifically, the core technique lies the Lemma 4.1 of (76), which, based on [X3] is applicable to both tine-homogeneous and Buler-Peano-ee
inhomogeneous settings (see Remark 3.3.3 (93). Therefore, equivalent weak convergence bounds could be attained by extending Lemma 4.1 of [66 with proofs from [93] thanks to the Euler- Peaano formulation.
24

---

Training Loss
Contextual Information
MLE
Reg term
C
MJD parameters
P1
1/1
S1
Neural MID
Pa
1/2
S2
PT
yry
St
S S_
S.
S
S
ST
Past data
Ground truth
Figure 4: Neural MJD training pipeline. {Mr, F, Ar, (r) in model.
The symbol p represents the MJD parameters
Step Our Restarted EM Solver Error Bound
We now discuss the error bound for the restarted EM solver, <R. Thanks explicit solutions for future states ( }. we can analytically compute their mean [[S, 2 1, based on Eq. 13 which greatl simplifies the analysis. Using the restart mechanism in line 10 of Alg. 2 we ensure that [[])|C] from our restarted EM solver closely approximates the true E[S, [C] at restarting times. ef is significantly reduced when restart happens (when / is integer in context for simplicity), then it grows again at the same rate as the standard EM method until the next restart timestep. This explains the ( difference in the error bounds of cf and c where [t] is the last restart time. Note that could make the restart timing more flexible potentially achieve tighter bound in terms of weak convergence. However, this may affect the diversity of the simulation results, as the fidelity of path stochasticity could be impacted.
D Experiment Details
D.1 Baseline, Model Architecture, and Experiment Settings
For the statistical BS and MJD baselines, we assume a stationary process and estimate the parameters using numerical MIE objective based past sequences. For the other deep learning baselines, including DDPM, EDM, FM, Neural BS, and Neural MJD, we implement our network using the standard Transformer architecture 25 All baseline methods released by their authors, with minor modifications to adapt to our datasets. Note that the technical term diffusion in the context of SDE modeling (e.g.. Merton jump diffusion) should not be conflated with diffusion-based generative models [45]. While both involve SDE-based representations of data, their problem formulations and learning objectives differ significantly.
based on the open-source code
We illustrate the training loss computation pipeline for Neural MJD in Fig. Notably, the loss computation can be processed in parallel across the future time-step horizon, climinating the need for recursive steps during training. We normalize the raw data into the range of j0, for stability and regularization weight 1.0 during training. All experiments were NVIDIA A40 and A100 GPUs (48 GB and 80 GB VRAM, respectively).
D.2
Datasets Details
For all datasets, we normalize the input data using statistics computed from the training set. For non-denoising models, normalization maps the data to the range [0. 1]. contrast, for denoising models (DDPM, EDM, FM), we scale the data to [-1, to align with standard settings used in image generation. Importantly, normalization coefficients are derived solely from the training set statistics. Further details on this process are provided below.
25

---

Connectons Festau Other Fating
General Nirchantose Stonea
Geeoline Stations
Figure 5: Visualization of Ego Graph Dataset Construction in Austin, Texas
Synthetic Data. We generate synthetic data using scalar Merton Jump Diffusion model. The dataset consists of N 10, 000 paths over the interval [0, 1]. simulated using the Euler scheme with 100 time steps. To facilitate time-series forecasting. we employ a sliding window approach with a stride of 1, where the model predicts the next 10 frames based on the previous 10. The dataset is divided into 60% training, 20% validation, and 20% testing. For each simulation, model parameters are randomly sampled from uniform distributions: ja ~ U (0.1, 0.5). a ~ (O.1, 0.5), U(3, 10), ~ (-1.1 0.1), and y ~ (0.5, 1.0). These parameter choices ensure the presence of jumps, capturing the stochastic nature of the process.
SafeGraph& Advan Business Analytics Data. The Adde business analytics dataset a proprietary dataset created by integrating data from Advan [87 and SafeGraph [8] to forecast daily customer spending at points of interest (POIs) across Texas, USA. Both datasets are licensed through Dewey Data Partners under their proprietary commercial terms, and we comply fully with the terms. For each POI, the dataset includes time-series data with dynamic features and static attributes. Additionally, ego graphs are constructed based on geodesic distances, where each POI serves as a central node connected to its 10 nearest neighbors. An visualization is shown in Fig. 5 Specifically, we use POI area, brand name, city name, top and subcategories (based on commercial behavior), and parking lot availability static features. The dynamic features include spending data, visiting data, weekday, opening hours, and closing hours. These features are constructed for both ego and neighboring nodes. Based on the top category, we determine the maximum spending in the training data and use it to normalize the input data for both training and evaluation, ensuring a regularized numerical range. For training stability, we clip the minimum spending value to 0.01 instead of 0 to enhance numerical stability for certain methods.
We adopt sliding window approach with stride of 1, using the past 14 days as input to predict spending for the next 7 days. The dataset spans multiple time periods: the training set covers January-December 2023, the validation set corresponds to January 2024, and the test set includes February-April 2024. This large-scale dataset consists of approximately 3.9 million sequences for training. 0.33 million for validation, and 0.96 million for testing.
S&P 500 Stock Price Data. The S&P 500 dataset [89] is a publicly available dataset from Kaggle that provides historical daily stock prices for 500 of the largest publicly traded companies in the U.S (CCO 1.0 Universal license). It primarily consists of time-series data with date information and lacks additional contextual attributes. We include all listed companies and construct a simple fully connected graph among them. Therefore, for models capable of handling graph data, such as GCN, our implemented denoising models, and Neural MJD, we make predictions for all companies (represented as nodes) simultaneously. This differs from the ego-graph processing used in the SafeGraph& Advan dataset, where predictions are made only for the central node, while neighbor nodes serve purely as contextual information. To normalize the data, we determine the maximum stock price for each company in the training data, ensuring that input values fall within the [0, 1] range during training.
Following the approach used for the business analytics dataset, we apply sliding window method with a stride of 1, using the past 14 days as input predict stock prices for the next days. The dataset is split into training (Jen.-Dec. 2016), validation (Jan. 2017), and testing (Feb.-Apr. 2017)
26

---

sets. In total, it contains approximately 62K sequences for training, 5K for validation, and 15K for testing. To better distinguish the effects of different methods on the S&P 500 dataset, we use adjusted R² score R² = 1 (1 R n-1 of explanatory variables p to be (k 1)(n 1)/k, where k 70.0.
n-p-I- where n is the sample size and we set the number
D.3 Additional Deterministic Time-Series Baselines (Third-Party Implementations)
For completeness, we also report results from third-party implementations of Autoformer [95]. TiDE (96). and N-HiTS [97]. provided by the NeuralForecast library (NIXTLA). Results in the table were produced with the publicly available NeuralForecast package on the same train/validationntest splits and identical data input (e.g. exogenous stock ticker information) as our main experiments, using the package's 's default training settings without modification.
Table 6: Quantitative results from NeuralForecast (NIXTLA) implementations on the S&P 500 stock dataset.
Model
MAE
MSE
R² 1
Autoformer
81.0
2.73e04
0.061
TiDE
27.7
7.28e03
0.750
N-HiTS
15.5
1.86e03
0.936
D.4 Limitations
Our approach explicitly models discontinuities (jumps) in the time series. Consequently, if the underlying data lack such jump behaviors- i.c., if they are extremely smooth and exhibit no abrupt shifts-ou our jump component may inaccurately estimated effectively unused. In these scenarios, the model can underperform compared to simpler or purely continuous alternatives that do not rely on capturing sudden changes. For applications where jumps are absent or extremely rare, users should first verify the presence (or likelihood) of discontinuities in their dataset before adopting our framework. Additionally, one potential extension is to design an adaptive mechanism that can automatically deactivate or regularize the jump component when the data do not exhibit significant jump behavior, thereby reducing unnecessary complexity and improving general performance on smooth series.
D.5 Vanilla Euler Solver
Algorithm Vanilla Maruyaaa Method
Require: Total solver steps M
1: C Drest, with C = S T o, C7
2: {uT, Tr, Xr, Ur, ) + Co
3: A +- Ti
Solver time-step
4: for = =0,..., ....,M--. do
5: t; + iA,t+11 (i 1)A, Pt. |t:| +1
6: a; & (Hpp - Axp. kpp σ² /2)
> Drift
7:
B. + Op, VA21. with 21 ~ (0,1)
b Diffusion
8:
Ci + KVp, + /K7 22
with K ~ Pois pr, A), ~ ~(
> Jump
9: In S -- In Se, 03 +
return 111 JM
We present the standard Marur-Maruyama solver in Alg. which is used in the ablation study for comparison with our restarted Euler solver.
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E Impact Statement
This paper introduces Neural MJD, a learning-based time series modeling framework that integrates principled copp-diffuiion-base SDE techniques. Our approach effectively captures volatile dynam- ies, particularly sudden discontinuous jumps that govern temporal data, making it broadly applicable to business analytics, financial modeling, network analysis, and climate simulation. While highly useful for forecasting, we acknowledge potential ethical concerns, including fairness and unintended biases in data or applications. We emphasize responsible deployment and continuous evaluation to mitigate inequalities and risks.
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Reviews:
[{"review_id": "YqCMCr4y6g", "reviewer": "Reviewer_PuBS", "summary": "This paper presents the Neural MJD, a transposition of ideas from neural ODEs into a specific family of differential equations optimized to handle data with jumps. A collection of efficiency-improving suggestions are made. These are then benchmarked on a small-scale synthetic experiment, and two financial datasets. The method appears to perform well.", "questions": "1. What is the “MLP” baseline.  Would like to see how best-in-class forecasting algorithms such as N-BEATS perform (even though they can’t handle irregularly spaced data).\n2. Are the time series used univariate or multivariate?  I can’t quite tell from the description in 5.2. Then how do you use some baselines like ARIMA that don’t handle vector valued data? (Or are you actually use VARIMA models?)\n3. (cf. line 208) Something I don’t quite understand is how the model produces inferences for any t with a single evaluation?  This means that it is basically a feed-forward network that outputs the effective Euler derivative of the state (multiplied by the time interval and added to the previous state results in the new estimate).  \n    - Similarly, this is quite a departure from the neural differential equations way of operating, right?  To the point that it isn’t really a neural ODE that _requires_ integrating; or am I missing something…?\n4. (cf. Section 4.3) Trying to square this with Algorithm 2, why do you need to do the integration steps at all?  Why not just evaluate the exact target value and their variances at the times of interest?  I don’t love the introduction of these restart points where the prediction radically changes.  I can see that it improves performance, but it feels messy and unfinished. I would be more interested if you then fitted a GP or did smoothing over it to at least get a smooth estimate of the mean and variance which is constrained by these “gold-standard” evaluations at the restart points.", "limitations": "No societal impacts.\n\nMain technical limitations are in reach and strength of empirical validation.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Summary\nI think this paper is an interesting and very detailed assessment of extending themes from neural differential equations into very specialized model families.  It appears well motivated from a historical and finance perspective, then tying in nicely with some modern themes. I have some reservations really striking at the generality and strength of evaluation. As there is not enough differentiating material across a broad base, I find myself being _indifferent_ to the paper (it seems to work fine under a narrow definition), and find it difficult to campaign ardently for its inclusion (or indeed exclusion).  Since I can see this paper having some impact and is a nice combination of very domain-specific models with very general concepts, I err towards acceptance. I would like to see a revision to this paper that tackles more broad use-cases, compares to more appropriate baselines, and attempts to leverage the underlying structure of the model to unlock new insights.  Good luck.\n\n## Strengths\n1.\tModeling data with jumps is a practical tool, since real-world data is often very messy with lots of jumps.\n2.\tA lot of the paper is well written and clearly expressed despite some pretty intense material (modulo some points of confusion I do have; see Questions).\n3.\tI think the restarts are an interesting idea.\n4.\tI like some of the simplifications and modeling assumptions, enabling easier inference.\n5.\tSeems to perform admirably.\n6.\tAppreciate the inclusion of runtime comparison.\n7.\tThe authors are very precise in their exposition and should be commended as such.  I did lose the thread a little bit around 4.3, I have detailed some questions below.\n\n## Weaknesses (roughly in order of significance)\n1. My main weakness with this paper is that it is presented in a very domain-specific way.  the model is effectively only benchmarked on two univariate real-world prediction tasks under fairly restricted settings, which does temper my enthusiasm for the generality and potential impact of the method. I would love to see it benchmarked on some non-financial timeseries, because I think there could be some real performance benefits in other domains (e.g. vitals monitoring in healthcare).  Similarly, it is a transplantation (if a very thorough and precise transplantation!) of ideas from neural differential equations into another, more specialized family of differential equations.  This again limits the impact and reach to me, without seeing applications against domain-specific baselines in a wider range of tasks.  I think this is a bit of a missed opportunity.  I don’t think this is terminal in my opinion, the paper still clearly has merit, but it does reduce my enthusiasm. \n2. I think the baselines are sort of set up to fail. Looking through the baselines in Table 2: ARIMA models don’t capture jumps, and BS and MJD are “static” versions of your models. XGBoost is good but limited, see question below on MLP, and I don’t know much about the GCN or other diffusion baselines, but I doubt they are very well configured for jumps.  Your model basically has heavier tails than those models, and so I would expect it to do fairly well on data with jumps. I would be much more interested in comparisons with state-of-the-art forecasting methods like N-BEATS and N-HITS, or TCN or TiDE.  I think these are much more representative baselines for this sort of data (and should be relatively easy to compare to, packages like DARTS implement them and have stochastic decoding methods).  Until seeing these more even-handed evaluations, my enthusiasm is still tempered.\n3. I do not agree with some of the parlance like “inability to explicitly model underlying stochastic processes often limit their generalization to non-stationary data”. Not only is this unsubstantiated as a claim and biases the reader towards your method unfairly, I don’t think it is correct. Maybe a neural network doesn’t explicitly parameterize an SDE, but I don’t see why that would limit generalization (arguably it would increase generalization!).  Then no model can truly expand to non-stationary timeseries, because the timeseries might enter a dynamics space it hasn’t seen before and so is just out-of-distribution and its randomly guessing.  I don’t see why your model is any more or less robust to this. These claims or hints are scattered throughout the paper. I think choosing to model an SDE directly is interesting, but since you do not discuss mechanistic interpretability, causality, or parameter efficiency (which you also lose because of the neural networks), I think the parameterization with an SDE is incidental and doesn’t directly contribute to this – I think the more powerful framework of defining a neural differential equation contributes to this, and so this language should (to my taste) be stripped out.\n4. (this is a comparatively minor point.) I would like to see some of the weight of the exposition stripped out. It is quite a “heavy” paper, for what is a relatively simple idea (at least once it has been worked through). I would love to see a revision that added some more examples and exploration of the results (successes and failures!) and demoted some of the derivations to the supplement. I also find Algorithm 1 and 2 quite unhelpful.  I would prefer if they were spaced out and properly commented to explain what each line does, because a lot of the equations don’t actually appear in the main text (e.g. $\\zeta$ randomly appears and is not referenced in the rest of the paper?).\n5. I’m not quite sure how you’d improve this, but the $N(dt, dy)$ notation I find confusing.  Maybe you could explicitly separate it into two integrals, and also denote that $N$ is actually a function of $t$? (cf. line 150).  This suppression makes it quite hard to keep a track of the dependencies.", "was_revised": true, "final_justification": "Thank you to the authors for their response.  As far as I am concerned: the original paper seemed correct and it seemed to work well within the applied domain.  In the rebuttal phase, the authors have made some attempt to expand the domains and to test more baselines.  Based on this, I find no reasonable cause from my own review to not upgrade my score (although I do note not being able to see the updated manuscript/results/baselines/justification tempers my confidence in this somewhat).\n\nI am also waiting to see how the discussion with qPpS plays out.  I don't fully understand the core critique: it seems like the discussion is about the ability of the model to overfit, which is valid, but is also a relatively omnipresent criticism that can be resolved (regularization etc).  If the model performs well in the domains it is tested on, then I find it difficult to add credence to the reviewers line of questioning.  I will track the discussion and will upgrade my final score if there is no substantial flaws found.", "rebuttal": "We sincerely thank the reviewer for their time and effort in evaluating our work. We appreciate their recognition of the novelty of our model formulation and the use of SDE modeling for time-series analysis. Below, we address the main points raised in the review.\n\n\n## Applicable domain\n> benchmarked on some non-financial timeseries\n\nTo further address this concern, beyond the economic-financial time-series data used in the paper, we conducted an additional experiment predicting **daily visit volumes** to business points of interest (POIs) using the SafeGraph and Advan datasets. Our method continues to outperform the baseline models in this setting as well.\n\n|Model|MAE $\\downarrow$|R² $\\uparrow$|\n|----------|------|--------|\n|ARIMA|58.2|-0.125|\n|XGBoost|33.4|0.189|\n|MLP|30.3|0.225|\n|GCN|31.4|0.219|\n|FM|27.9|0.273|\n|**Ours**|**27.5**|**0.268**|\n\nAs future work, we aim to apply our method to more diverse settings and datasets, such as healthcare datasets as suggested by the reviewer.\n\n> more specialized family of differential equations\n\nWe respectfully argue that our approach, inspired by Merton jump diffusion SDEs formulations, provides a broader and more flexible framework than simpler models like the Black-Scholes SDE. This allows our model to better capture jump behavior, which standard neuralized SDEs may struggle with. While fully neural formulations (e.g., NeuralCDE, LatentSDE) can theoretically represent a wider range of SDE processes, they do not outperform our approach as seen in the experiments. Empirical results demonstrate that the use of jump diffusion SDEs in our model design leads to improved performance.\n\n\n## Baseline fairness\n> I would be much more interested in comparisons with state-of-the-art forecasting methods\n\nWe conducted further experiments on the S&P 500 dataset and obtained preliminary results for TCN and N-BEATS:\n\n|Model|MAE $\\downarrow$|R² $\\uparrow$|\n|----------|------|--------|\n|TCN | 47.9 | 0.215 |\n|N-BEATS| 33.5 |0.751|\n|**Ours**|**16.8**|**0.938**|\n\nWe plan to include additional experiments comparing with more baselines in the revised version.\n\n> I think the baselines are sort of set up to fail.\n\nWe would like to clarify that our experiments were designed for a fair comparison. Some baselines do not explicitly model jumps, motivating our Neural MJD, and the performance comparison supports the proposed model.\nExcept for the synthetic data, we do not control jump behavior in real-world datasets. Our goal is to ensure a fair, practical comparison on real data without any jump labels and to highlight the methodological value of modeling potential jumps.\n\n> What is the “MLP” baseline.\n\nThe MLP model consists of a stack of linear layers and serves as a simple baseline for time-series forecasting.\n\n> Are the time series used univariate or multivariate?\n\nWe use univariate time-series datasets, predicting a scalar at each time step.\n\n\n## Generalization to non-stationary data\n> I do not agree with some of the parlance like “inability to explicitly model underlying stochastic processes often limit their generalization to non-stationary data”.\n\nWe thank the reviewer for their valuable feedback. Our intention was to emphasize that explicitly modeling jump diffusion SDEs offers a useful inductive bias for capturing sudden changes and non-stationary behaviors that may be more challenging to represent with standard neural SDEs. We believe this approach complements the neural differential equation framework by providing additional flexibility. We appreciate the suggestion and will revise the manuscript to ensure accuracy.\n\n## Neural ODE vs. ours\n\n> how the model produces inferences for any t with a single evaluation?\n\n> why do you need to do the integration steps at all?\n\nWe would like to clarify that our model is not a neural ODE and does not require step-by-step integration of a *neural* vector field. Instead, it predicts all SDE parameters—drift, diffusion, and jump intensity—in a single forward pass. The SDE trajectory $S_t$ is then simulated using these predicted coefficients via a numerical method, without further neural network evaluations.\n\nA neural ODE typically learns a vector field:\n$\\frac{dS(t)}{dt} = f(S(t), t; \\theta),$ and computes $S(T_f) = S(0) + \\int_0^{T_f} f(S(t), t; \\theta)\\,dt$ using an ODE solver, which repeatedly queries the neural network during integration.\n\nIn our method, we do not learn a neural vector field. Instead, we predict the full set of Merton jump-diffusion (MJD) parameters: $[\\mu_1, \\sigma_1, \\lambda_1, \\gamma_1, \\nu_1, \\dots] \\in \\mathbb{R}^{5 \\times T_f}.$ Due to the stochastic nature of SDEs, numerical simulation is required to generate actual forecasts. We compute each next state (informally) as:\n$S_{t+\\Delta t} = S_t + \\mu_t\\,\\Delta t + \\sigma_t\\,\\Delta W + \\text{jump}(\\lambda_t, \\gamma_t, \\nu_t),$\nwhich involves stochastic integration—but no further neural network inference, as all parameters are predicted once at the start.\n\nIn summary, our model generates predictions for the SDE parameters at time $t$, followed by numerical simulations to obtain SDE trajectories for forecasting outcomes.\n\n> Why not just evaluate the exact target value and their variances at the times of interest?\n\nWe would like to emphasize that we do not forecast outcomes in a single step. Instead, we sequentially sample each increment across the entire SDE simulation trajectory to obtain stochastic forecasts. Relying solely on the exact expectation and variance would oversimplify the model, ignoring the jump dynamics. While our formulation allows for the computation of mean and variance, these values may deviate from the true behavior, which includes jumps.\n\n> get a smooth estimate\n\nWe thank the reviewer for raising this point. In Tables 1–3, the \"Mean\" metrics represent averages over 10 runs of the SDE simulation, serving as a simple proxy for smoothing. We agree that more advanced approaches, such as GP fitting, are promising and plan to explore them in future work.\n\n## Writing clarity\n\nThank you for your helpful comments. We will clarify the derivations and notation in the main paper and present our method more clearly in the revision. We followed the classical literature for the Poisson random measure $N(\\mathrm{d}t, \\mathrm{d}y)$ and will also include explanations highlighting its meaning and interpretation in practical applications to enhance understanding.\n\n## Concluding remarks\nWe thank the reviewer again for their valuable feedback. We hope that our rebuttal addresses their questions and concerns, and we kindly ask the reviewer to consider a fresher evaluation of our paper if the reviewer is satisfied with our responses. We are also more than happy to answer any further questions that arise."}, {"review_id": "y0aZDUsTQJ", "reviewer": "Reviewer_J6Nv", "summary": "This work introduces Neural MJD, a non-stationary Merton Jump Diffusion model parameterized by a series of neural networks, which captures a wide variety of stochastic processes particularly in the world of quantitative finance. The authors additionally propose a restart inference method, based on the celebrated Euler-Maruyama method which achieves exponentially tighter weak error, and demonstrate the efficacy of their methods on a swath of synthetic and real-world bench-marks.", "questions": "* What are the implications of this work for modeling option prices and more generally volatility? \n* In the proof of Theorem 4.1, what are the precise assumptions necessary for this theorem to hold? \n* Does the method obtain improved results when the discretization is performed on smaller intervals?", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "> Strengths: \n>  * The exposition in the text is clear and the motivation is well defined.\n>  * The method proposed in the paper is novel, and subsumes the important Black-Scholes and Merton Jump Diffusion models to account for non-stationarity by way of encoding the parameters in a time-varying transformer architecture. \n>   * The model derivation in Section 3 as well as the deferred proofs in the appendix are well-written and provide clear motivation\n>  * The experiments are run on a swath of synthetic and real-world tests, including modeling the S & P index, demonstrating the efficacy of their novel method. \n>  * The authors give many helpful figures to explain certain phenomena. \n\n> Weaknesses: \n> * The S & P experiment is run on a limited set of data (Jan 2016 - Apr 2017), which had limited volatility (evidenced by the VIX index during this time), it is unclear whether this method still outperforms given the increased volatility of the past 5 years. \n> * Additional discussion on the relationship between this work and other works relating neural learning in non-stationary time series would be appreciated.", "was_revised": true, "final_justification": "This work introduces a novel model whose experimental results surpass the best known alternatives across a range of benchmarks relevant to finance, economics, and other domains. Towards tractable learning, the authors present a likelihood truncation mechanism and provide theoretical error bounds for this approximation. The effectiveness of their methods is demonstrated on a broad set of synthetic and real-world benchmarks.\n\nMy primary concern with the work was with the S & P data experiment, worrying that a year and a half of data with relatively low market volatility may not be general enough to prove the efficacy of their methods. However, the authors have since extended the time frame and shown that their methods remain effective under the more challenging conditions. Moreover, the authors also answered my questions, revealing insights on the nature of the discretization interval. They also included additional experiments addressing concerns raised by other reviewers.\n\nFor these reasons, I maintain my positive evaluation and recommend a score of 5 (Accept).", "rebuttal": "We would like to express our sincere gratitude to the reviewer for their time and effort in evaluating our work. We are grateful for their acknowledgment of the novelty in our model formulation and the application of MJD SDE modeling to time-series analysis. Below, we respond to the key points raised in the review.\n\n\n## Additional experiments\n>The S & P experiment is run on a limited set of data (Jan 2016 - Apr 2017), which had limited volatility (evidenced by the VIX index during this time), it is unclear whether this method still outperforms given the increased volatility of the past 5 years.\n\nThank you for your valuable feedback.\nWe have extended the S&P 500 dataset to cover the period from January 2014 to December 2017, using the data from 2017 for performance evaluation. The preliminary experimental results are as follows:\n\n|Model|MAE $\\downarrow$|R² $\\uparrow$|\n|----------|------|--------|\n|ARIMA | 65.2 | -0.892 |\n|XGBoost| 56.4 |0.150|\n|MLP| 51.2 | 0.161 |\n|**Ours**|**39.7**|**0.389**|\n\nIn the extended dataset, our method still outperforms the baselines.\nIn future work, we plan to extend our experiments to cover the past five years comprehensively, alongside additional baselines.\nWe will also analyze the empirical effects of long-term forecasting and data volatility on model performance.\n\n\n## Discussion on related work\n>Additional discussion on the relationship between this work and other works relating neural learning in non-stationary time series would be appreciated.\n\nWe thank the reviewer for the helpful suggestion to clarify how our work relates to prior research on non-stationary time series. We will include this clarification in our revision to prevent any ambiguity.\n\nIn previous studies, \"non-stationarity\" typically refers to distributional shifts in the data over time. These works focus on mitigating such shifts using techniques like input-level normalization (e.g., DAIN [1], ST-norm [2], RevIN [3]), domain adaptation (e.g., DDG-DA [4]), or de-stationary attention mechanisms (e.g., Non-stationary Transformers [5]).\n\nIn contrast, our notion of non-stationarity centers on modeling a Merton jump diffusion (MJD) process with parameters that evolve over time. While our model is not explicitly designed to handle distributional shifts, it predicts time-varying MJD parameters through a closed-form solution, which naturally leads to a changing predictive distribution as parameters evolve.\n\n\n## Model implications\n> What are the implications of this work for modeling option prices and more generally volatility?\n\nThank you for your feedback. We would like to highlight the following advantages of our approach:\n\n1. Reduced calibration effort: Classical option-pricing methods require significant effort to fit a time-inhomogeneous Merton jump-diffusion model to market or historical data. In contrast, Neural MJD treats past returns as conditioning inputs rather than direct calibration targets and directly outputs the drift, diffusion, and jump-intensity parameters for the future time horizon. This approach eliminates the need of time-consuming calibration due to fast neural network inference.\n\n2. Handling exogenous information: Most classical modeling methods either ignore exogenous drivers or can only handle numerical features. Our method, however, is capable of incorporating diverse types of data as inputs or conditioning variables, such as static features (e.g., parking lot sizes) in the SafeGraph and Advan datasets used for business revenue prediction.\n\n\n3. Potential extension to other SDE models: A similar neural parameterization technique can be extended to other SDE dynamics for more domain-specific applications, allowing for better capture of volatility. The training and inference techniques introduced in the paper would be beneficial for future model development.\n\n\n## Theorem proof\n\n> In the proof of Theorem 4.1, what are the precise assumptions necessary for this theorem to hold?\n\nTheorem 4.1 (the truncation-error bound for the non-stationary MJD) is proven under the following assumptions:\n\n\n1. We choose $\\delta$ small enough that $t$ and $t+\\delta$ are under the same set of parameters. We have this assumption simply to ease the proof of the theorem.\n2. Other standard Poisson/MJD assumptions:\n    a. The jump sizes in Poisson process are i.i.d.\n    b. Conditional on $\\Delta N$, $\\ln S_{t+\\delta}$ is Gaussian.\n\nWe will clarify the assumptions used in the proof of the theorem, and we thank the reviewer for bringing this up.\n\n\n## Discretization interval\n> Does the method obtain improved results when the discretization is performed on smaller intervals?\n\nTheoretically, using a finer discretization reduces both the Euler integration error and the jump-truncation bias per interval. Empirically, we find that using an interval that is small enough (such as 0.1 or 0.2) leads to good performance.\nWe conduct the comparison on the S&P 500 dataset as follows.\nPlease note that the interval is normalized relative to the time difference between two times of interest (which is 1 in our notation).\n\n\n| Intervals | 0.05 | 0.1  | 0.2  | 0.5  | 1.0  |\n|-----------|-------|-------|-------|-------|-------|\n| Mean MAE $\\downarrow$       | 17.1  | **16.8**  | 17.1  | 35.2  | 43.2  |\n\n\n## Concluding Remarks\n\nWe sincerely thank the reviewer once again for their valuable feedback. We hope our responses have satisfactorily addressed their questions and concerns. We remain fully available and happy to answer any further questions that may arise.\n\n\n## Reference\n\n[1] Passalis, Nikolaos, et al. \"Deep adaptive input normalization for time series forecasting.\" IEEE transactions on neural networks and learning systems 31.9 (2019): 3760-3765.\n\n[2] Deng, Jinliang, et al. \"St-norm: Spatial and temporal normalization for multi-variate time series forecasting.\" Proceedings of the 27th ACM SIGKDD conference on knowledge discovery & data mining. 2021.\n\n[3] Kim, Taesung, et al. \"Reversible instance normalization for accurate time-series forecasting against distribution shift.\" International conference on learning representations. 2021.\n\n[4] Li, Wendi, et al. \"Ddg-da: Data distribution generation for predictable concept drift adaptation.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 36. No. 4. 2022.\n\n[5] Liu, Yong, et al. \"Non-stationary transformers: Exploring the stationarity in time series forecasting.\" Advances in neural information processing systems 35 (2022): 9881-9893."}, {"review_id": "OCqu5pKVF3", "reviewer": "Reviewer_scGv", "summary": "This paper introduces time-inhomogeneous Merton jump diffusion models for stochastic time-series prediction. The authors introduce a truncated likelihood that makes learning the model practical, and also a 'restarting' Euler-Maruyama solver that has a tighter error bound. The model is evaluated on a number of synthetic and real-world datasets, against a large number of baseline models and performs well.", "questions": "Q1: The first experiment is on synthetic data, i.e., you know the *true* parameters of the MJD process that generates the data. Can you please show how the estimated parameters using your method compare to the true parameters?\n\nQ2: You sample 10 times to express mean metrics. Can you please also provide confidence intervals on these metrics? E.g. using t-test statistics as is done in Bartosh SDE Matching (2025) and earlier works on latent SDEs.\n\nQ3: You restart the EM solver at each observation. Do you think this has a negative effect on learning long-term dependencies? I mean, a change of the parameters at some time t will not have an effect on the dynamics after the next S_\\tau is crossed?\n\nQ4: I'm a bit surprised you needed to add the 'regularization' term. It resembles a Gaussian likelihood on the measurements (L2 term). Can you provide me more intuition on why (or why not) your likelihood does not already contains a similar effect?\n\nQ5: The related work on jump processes seems to focus on the financial application domain. But inference of jump processes has been investigated in machine learning, how does your work relate to these 'older' works? e.g.\nhttps://academic.oup.com/jrsssb/article-abstract/67/3/395/7109469\nhttps://proceedings.neurips.cc/paper/2007/hash/735b90b4568125ed6c3f678819b6e058-Abstract.html\n\n\nMinor questions or remarks:\n- line 166: There seems to be an unnecessary space at the start of this line.\n- line 178, this means you always discretize with time-step equal to the time unit?\n- line 233 parallel", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n- The paper is very well written. The work is clearly embedded and motivated in the context of (primarily) financial modelling. Specific concepts are explained with high clarity, this is very much appreciated. As well the clear citations to e.g. specific book chapters allows the reader to grasp more context.\n- The method is well explained. While the model might be rather specific, and not e.g. a general SDE driven by Lévy, I would like to see this as a plausible and rigorous step forwards for the research community.\n- Modelling stochastic jump processes is a difficult problem, which is very relevant to the machine learning community.\n- The experiments are convincing and extensive.\n\nWeaknesses:\n- Experiments are done by sampling 10 times from the models. Results don't show confidence intervals, and sometimes other models (such as flow matching) are very close the the author's model, but not indicated in bold. This is a bit misleading. I would like to see an evaluation where, if the best models are performing similar (statistically significantly), they are both in bold.", "was_revised": true, "final_justification": "Ok, thanks for your answers and the extra work during rebuttal. I retain my score.", "rebuttal": "We sincerely thank the reviewer for their time and effort in evaluating our work. We greatly appreciate their recognition of our model formulation and the application of MJD SDE modeling to time-series analysis. Below, we address the key points raised in the review. \n\n## Confidence interval\n> I would like to see an evaluation where, if the best models are performing similar (statistically significantly), they are both in bold.\n\n\n> You sample 10 times to express mean metrics. Can you please also provide confidence intervals on these metrics? E.g. using t-test statistics as is done in Bartosh SDE Matching (2025) and earlier works on latent SDEs.\n\nThank you for your valuable suggestions. We provide additional experimental results on the SafeGraph & Advan business analytics dataset, where we compute the confidence interval by calculating the standard deviation across multiple runs. The preliminary experiments are as follows:\n\n\n|Model|MAE $\\downarrow$|R² $\\uparrow$|\n|----------|------|--------|\n|DDPM| 68.5 $\\pm$ 3.1 |0.501 $\\pm$ 0.023|\n|FM | 54.5 $\\pm$  2.4| 0.540 $\\pm$ 0.020|\n|**Ours**|54.1 $\\pm$ 1.8|0.549 $\\pm$ 0.018|\n\nDue to the time constraints of the rebuttal period, we will compute the t-test statistics as suggested in future work and include these metrics for all stochastic methods in additional experiments. \n\n\n## Synthetic dataset \n> The first experiment is on synthetic data, i.e., you know the true parameters of the MJD process that generates the data. Can you please show how the estimated parameters using your method compare to the true parameters?\n\n\nThank you for your feedback.\nWe show the estimation parameter differences for the $\\mu$ and $\\sigma$ parameters using our Neural BS and full Neural MJD models. The results are averaged over time steps to obtain aggregated MAE results.\n\n| Model      | $\\mu$ error | $\\sigma$ error |\n|------------|--------|----------|\n| Neural BS  | 0.15   | 0.21     |\n| Neural MJD | 0.11   | 0.19     |\n\nAs we can see, the Neural MJD model performs slightly better, likely due to its ability to model jumps during the learning process, whereas the Neural BS model only captures the SDE without jumps.\nFor the other jump-related parameters, we don’t observe very close estimations from the Neural MJD. We hypothesize that different combinations of jump parameters may yield similar outcomes in the likelihood loss training. \nWe plan to devote more effort to exploring this aspect further in order to enhance model interpretability. Thank you again for your insightful comment.\n\n\n## Restart mechanism\n> You restart the EM solver at each observation. Do you think this has a negative effect on learning long-term dependencies? I mean, a change of the parameters at some time t will not have an effect on the dynamics after the next S_\\tau is crossed?\n\nWe would like to respectfully point out that the restart mechanism is based on the conditional mean, which is given by:\n$$\n    \\mathbb{E}\\left[S_{t} \\vert \\mathcal{C} \\right] = S_0 \\exp(\\sum_{j=1}^{\\rho_t-1} \\mu_{j} + (t-\\rho_t+1) \\mu_{\\rho_t}).\n$$\n\nAs such, the SDE parameters $\\mu_t$ still implicitly capture the long-term dependencies learned by the neural network.\n\nWe agree that changes in other parameters have a lesser impact on the SDE dynamics once $S_\\tau$ is crossed. This observation aligns with our design motivation, which aims to reduce simulation variance in the long term to achieve improved empirical performance.\n\n## Regularization loss\n\n>I'm a bit surprised you needed to add the 'regularization' term. It resembles a Gaussian likelihood on the measurements (L2 term). Can you provide me more intuition on why (or why not) your likelihood does not already contains a similar effect?\n\nThanks for your comment.\nThe extra regularization loss is added to address the practical issues associated with the likelihood-based learning objective.\nThe intuition behind this is that a low likelihood loss does not necessarily guarantee that the conditional mean (as represented by the regularization loss) is well learned.\nDue to the jump components of the SDEs, we conjecture that there may be local minima that result in a relatively low likelihood loss, but where the conditional mean can still vary significantly.\nSince our restart solver also uses the conditional mean, we find that adding this term generally enhances performance by constraining the optimization space when using likelihood-based loss.\n\n\n## Related work\n> The related work on jump processes seems to focus on the financial application domain. But inference of jump processes has been investigated in machine learning, how does your work relate to these 'older' works?\n\nWe thank the reviewer for highlighting these foundational works on jump-process inference. It is an insightful suggestion to include a comparison with classical inference methods for jump processes.\n\nWhile these methods share the goal of estimating non-stationary parameters from discrete observations, our approach differs in three key respects:\n\n1. Inputs vs. Calibration: We do not fit SDE parameters directly to past data using Maximum Likelihood Estimation (MLE). Instead, we treat historical observations and additional information as inputs to the neural network. The likelihood serves as a training loss to optimize forecast accuracy, rather than as a per-series estimator of past states.\n\n2. Classical Methods: Classical Expectation-Maximization (EM) or Markov Chain Monte Carlo (MCMC)-based methods estimate parameters for each individual sequence. In contrast, our single network is trained once over all series, capturing shared dynamics and generalizing across assets or nodes.\n\n3. Closed-Form Requirement: Classical methods do not require a closed-form transition density, while our method does. This limitation provides a promising direction for future work, where the closed-form constraint could potentially be relaxed.\n\nThese \"older\" works provide a strong statistical foundation and could inspire hybrid methods. We will incorporate a discussion of their contributions and distinctions in the Related Work section of the revision.\n\nAdditionally, we find that these works offer valuable insights into the challenges and non-uniqueness of estimating non-stationary event rates from discrete observations (which relates to the previous question).\n\n\n\n## Writing\n> line 178, this means you always discretize with time-step equal to the time unit?\n\nIn practice, we hold the parameters fixed during the time unit (1 in this case), but during this time unit, you can choose any smaller $\\delta$ to reduce discretization error.\n\n\nWe thank the reviewer for pointing out the typos and formatting issues. We will address these in the revision.\n\n## Concluding Remarks\n\nWe sincerely thank the reviewer once again for their valuable feedback. We hope our responses have satisfactorily addressed their questions and concerns. We remain fully available and happy to answer any further questions that may arise."}, {"review_id": "48HHYBwu2E", "reviewer": "Reviewer_qPpS", "summary": "In this paper, non-stationary Merton jump diffusion is used to model and predict time series. The model parameters, as five functions of time, are ''predicted'' by a neural network. The practical issues in both training phase (evaluating likelihood) and inference phase (time discretization) are well addressed. Experiments on synthetic and real-word data show that the proposed method has promising performance (low prediction error).", "questions": "See ''Strengths And Weaknesses''.", "limitations": "See ''Strengths And Weaknesses''.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The paper is easy to follow and its structure is complete (baseline, proposed model, addressing implementation challenges, performance evaluation). My concern is mainly about the proposed model itself.\n\nAt the beginning of Section 4.1, when the non-stationary MJD is introduced, there is no structural assumption on the five model parameters $\\{\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t\\}$, so they can literally be any deterministic functions of $t$ (integrability is mentioned later to make the solution well-defined and continuity is provided by the neural network expression). With this nearly arbitrary flexibility, ''fitting data to non-stationary MJD'' does not seem to be a well-defined task. It is like ''fitting a sequence of arrivals to a non-stationary Poisson process with arbitrary $\\lambda_t$'', which leads to sheer overfitting.\n\nThe expression (8) $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t=f_\\theta(S_0,S_{-1},...,S_{-T_p},C,t)$ may also be problematic. The deterministic future parameters ($\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$) is expressed as a deterministic function ($f_\\theta$) of past sample path ($S_0,S_{-1},...,S_{-T_p}$), which is a sequence of random variables. This violates the internal logic of the SDE (unless those parameters are some adaptive processes, not mentioned in the paper), especially when $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$ are generated from $S_0,S_{-1},...,S_{-T_p}$ in a sliding-window manner as in Section 5.1. Defining instead predicting $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$ by (8) (in a sliding-window manner) makes more sense to me, but that will lead to a much more complicated SDE (7) + (8) than non-stationary MJD with deterministic $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$, so all  equations after (8) no longer holds.\n\n''Open access to data and code, Answer: Yes, Justification: We will release our code once the paper is accepted.'' Why?", "was_revised": true, "final_justification": "In my review, my main concern is about whether the learning task is well-defined in the sense of standard statistical inference.\nAfter discussion, the authors clarify that they are working in a different amortized setting. After making the difference clear in the revision, I think it will be a good paper.", "rebuttal": "We appreciate the reviewer’s thorough evaluation. We believe that the concerns stem from some technical misunderstandings of our approach, and we value the opportunity to address each point below.\n\n## Model discussion\n> At the beginning of Section 4.1, when the non-stationary MJD is introduced, there is no structural assumption on the five model parameters, so they can literally be any deterministic functions of $t$ (integrability is mentioned later to make the solution well-defined and continuity is provided by the neural network expression). With this nearly arbitrary flexibility, ''fitting data to non-stationary MJD'' does not seem to be a well-defined task. It is like ''fitting a sequence of arrivals to a non-stationary Poisson process with arbitrary $\\lambda_t$'', which leads to sheer overfitting.\n\n**Well-defined task concern**\n\nWe respectfully **disagree** with the statement that fitting a non-stationary MJD is not a well-defined task.\n\nPrevious models, such as Neural Jump Stochastic Differential Equations [1] and Neural Jump-Diffusion Temporal Point Processes (NJDTPP) [2], adopt highly similar processes to learn SDEs from data using neural networks.\nWe find the claim of the task not being \"well-defined\" to be a vague assessment that contradicts established findings in the research community.\n\nSpecifically, in our implementation, we enforce that all learned parameters satisfy the classical MJD requirements. For example, jump intensities and variances are constrained to remain non-negative. This effectively regularizes the learning of SDE parameters.\n\n\n**Overfitting concern**\n\nWe believe the reviewer’s comparison to \"fitting a sequence of arrivals to a non-stationary Poisson process with arbitrary $\\lambda_t$\" may stem from a misunderstanding. Rather than fitting each series independently (e.g., as done in ARIMA), we train a single neural network using all sequences in the training data. This approach encourages the model to learn shared non-stationary dynamics, preventing it from merely memorizing individual trajectories.\n\nIn practice, we use non-overlapping train/validation/test splits and select model checkpoints based solely on validation performance to avoid overfitting. The experimental results on the unseen testing set clearly show no sign of what is referred to as \"sheer overfitting.\"\n\n\n> The expression (8) $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t=f_\\theta(S_0,S_{-1},...,S_{-T_p},C,t)$ may also be problematic. The deterministic future parameters $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$ is expressed as a deterministic function ($f_\\theta$) of past sample path ($S_0,S_{-1},...,S_{-T_p}$), which is a sequence of random variables. This violates the internal logic of the SDE (unless those parameters are some adaptive processes, not mentioned in the paper), especially when $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$ are generated from $S_0,S_{-1},...,S_{-T_p}$ in a sliding-window manner as in Section 5.1. Defining instead predicting $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$ by (8) (in a sliding-window manner) makes more sense to me, but that will lead to a much more complicated SDE (7) + (8) than non-stationary MJD with deterministic $\\mu_t,\\sigma_t,\\lambda_t,\\gamma_t,\\nu_t$, so all equations after (8) no longer holds.\n\n**SDE logic concern**\n\nWe respectfully **disagree** that our method violates the internal logic of the SDE.\n\nTo avoid confusion, we would like to emphasize that we do not fit jump-diffusion parameters by directly calibrating them on historical data. Instead, past observations $S_{-1},...,S_{-T_p}$ serve only as inputs to a shared neural network. In other words, they are not part of the SDE state variables to be modeled. They can be incorporated into the context information, denoted by $\\mathcal{C}$, in our setup.\n\nTo be specific, in our model, the SDE process at time $t$ depends only on information available up to time $t$, with no \"peeking\" into the future. The model has no exposure to future data before making predictions, which is standard practice in deep learning. All the SDE parameters are predicted based on past observations using neural networks once and for all.\n\nWe noted some ambiguity in the comment. Our model was described as \"sliding-window,\" yet the reviewer also suggested that \"sliding-window makes more sense.\" We interpret this as a possible preference for an autoregressive approach (please kindly clarify if we have misunderstood the review).\n\nThat said, we would like to clarify that our model does not operate in a sliding-window manner. Instead, it adopts a feedforward design that (1) predicts the SDE parameters for the target horizon and then (2) performs numerical simulation to generate actual forecasts. This architecture allows for a single forward pass to predict SDE parameters across all future time steps under consideration, significantly improving inference efficiency by using neural network only once.\n\n## Open source\n>''Open access to data and code, Answer: Yes, Justification: We will release our code once the paper is accepted.'' Why?\n\nWe promise to release our full codebase and data upon acceptance since we would like researchers to be able to reproduce our work. We will release our full codebase and data upon acceptance, as we expect to include additional experiments and corresponding code during the rebuttal phase.\n\n\n## Concluding remarks\n\nWe thank the reviewer again for their valuable feedback. We hope that our rebuttal addresses their questions and concerns, and we kindly ask the reviewer to consider a fresher evaluation of our paper if the reviewer is satisfied with our responses. We are also more than happy to answer any further questions that arise.\n\n## Reference\n\n[1] Jia, Junteng, and Austin R. Benson. \"Neural jump stochastic differential equations.\" Advances in Neural Information Processing Systems 32 (2019).\n\n[2] Zhang, Shuai, et al. \"Neural jump-diffusion temporal point processes.\" Forty-first International Conference on Machine Learning. 2024."}]

Author rebuttal:

Meta-review: The reviewers agree that this is an interesting and useful paper, bringing ideas from modern ML to more classical time-series modeling. There were some concerns about clarity, evaluations, and somewhat narrow scope (at least in the presentation). None of these were unanimous, and the authors in their rebuttal suggested some remedies. Please take these into account (as well as the overall author-reviewer discussion when preparing the camera-ready version).
Decision: Accept (poster)