Continuous-Time Maximum Likelihood Estimation

• Continuous-time maximum likelihood estimation is a method for inferring parameters by optimizing likelihood functions defined on continuous-time stochastic processes.
• It encompasses exact, approximate, and quasi-likelihood approaches tailored to various observation schemes including fully observed, discretely sampled, or hidden states.
• Practical applications include estimating drift, diffusion, and interaction parameters in models such as diffusion, jump-diffusion, and state-space systems.

Continuous-time maximum likelihood estimation denotes a family of likelihood-based procedures in which the statistical model is specified in continuous time and the estimator is obtained by maximizing an exact, approximate, or quasi-likelihood derived from that model. In the cited literature, this includes exact pathwise likelihoods under continuous observation, likelihoods built from finite-time transition laws for discretely or irregularly sampled processes, hidden-Markov reformulations of continuous-time state-space models, and transformed likelihoods for non-Markovian or function-valued dynamics (Mai, 2014, Mews et al., 2020, Kirkby et al., 2021, Nag, 2021). The common objective is to infer structural parameters—drift, diffusion, generator, mean-function, or interaction parameters—from observations that are indexed by continuous time even when the data themselves are sampled discretely, irregularly, or through a latent layer.

1. Observation schemes and the scope of the estimator

The literature treats several observation regimes as instances of continuous-time likelihood inference. At one extreme is full path observation on a time interval $[0,T]$, as in continuously observed diffusion, jump-diffusion, threshold diffusion, Ornstein–Uhlenbeck, CIR, Volterra Ornstein–Uhlenbeck, and mixed fractional Vasicek models (Mai, 2014, 1711.02140, Lejay et al., 2018, Alaya et al., 2024, Cai et al., 2020). At another extreme are continuous-time models observed only at irregular or low-frequency time points, such as $0=t_0<t_1<\cdots<t_T$ for general state-space models or $Y_k^{(h)}=Y(kh)$ for cointegrated continuous-time state-space models (Mews et al., 2020, Fasen-Hartmann et al., 2017). Between these cases lie partially observed diffusions, continuous-time models with latent states, and continuous-time processes approximated by continuous-time Markov chains or hidden Markov models (Surace et al., 2016, Kirkby et al., 2021, McGibbon et al., 2015).

This breadth matters because “continuous-time MLE” does not refer to a single formula. In some models the likelihood is a Radon–Nikodym derivative on path space. In others it is a product of transition probabilities $p_\Delta(x_t\mid x_s)$, a matrix-product HMM likelihood, or a pseudo-Gaussian contrast built from pseudo-innovations. The cited papers therefore use the same inferential principle—maximize a likelihood-type criterion—under markedly different stochastic structures (Mews et al., 2020, Fasen-Hartmann et al., 2017, Ströh, 2021).

A recurring distinction is between models whose continuous-time transition law is available in closed form and those for which it is not. When $p_\Delta(x_t\mid x_s)$ is explicit, irregular sampling can be handled directly through the elapsed-time gaps $\Delta_\tau=t_\tau-t_{\tau-1}$, as in the general continuous-time state-space framework and in several diffusion examples (Mews et al., 2020). When the transition density is unavailable, the literature replaces it by state-space discretisation, CTMC approximation, functional Fokker–Planck transition densities, or quasi-likelihood constructions (Kirkby et al., 2021, Nag, 2021, Fasen-Hartmann et al., 2017).

2. Exact continuous-observation likelihoods

For continuously observed semimartingale models, the central construction is a likelihood ratio between path measures, typically derived by Girsanov-type arguments. In the Lévy-driven Ornstein–Uhlenbeck model

$dX_t=-aX_t\,dt+dL_t,$

the likelihood relative to $P^0$ on $\mathcal F_T$ is

$$\frac{dP_T^a}{dP_T^0} = \exp\!\left( -\frac{a}{\sigma^2}\int_0^T X_s\,dX_s^c -\frac{a^2}{2\sigma^2}\int_0^T X_s^2\,ds \right),$$

which yields the explicit continuous-time MLE

$0=t_0<t_1<\cdots<t_T$0

The estimator is strongly consistent, asymptotically normal, and asymptotically efficient in the Hájek–Le Cam sense when the stated conditions hold (Mai, 2014).

A similar linear-quadratic likelihood appears in continuously observed CIR-type models. For the $0=t_0<t_1<\cdots<t_T$1-stable CIR process

$0=t_0<t_1<\cdots<t_T$2

the paper derives the explicit MLE

$0=t_0<t_1<\cdots<t_T$3

together with the identity

$0=t_0<t_1<\cdots<t_T$4

This identity becomes the basis for regime-dependent asymptotics in the subcritical, critical, and supercritical cases (1711.02140). The jump-type CIR model driven by a subordinator has the analogous explicit estimator

$0=t_0<t_1<\cdots<t_T$5

again with a martingale-ratio representation for the estimation error (Barczy et al., 2016).

Threshold diffusions provide a distinct exact-likelihood example in which discontinuity at zero enters through occupation times and local time. For drifted Oscillating Brownian motion with piecewise constant drift $0=t_0<t_1<\cdots<t_T$6 and volatility $0=t_0<t_1<\cdots<t_T$7, the continuous-time likelihood ratio under the driftless model is

$0=t_0<t_1<\cdots<t_T$8

and the MLEs are

$0=t_0<t_1<\cdots<t_T$9

Using Itô–Tanaka, these can be written in terms of the terminal value and the symmetric local time $Y_k^{(h)}=Y(kh)$0, which makes the threshold contribution explicit (Lejay et al., 2018).

Exact continuous-time likelihoods also occur in higher-order linear systems. For the second-order Gaussian autoregression

$Y_k^{(h)}=Y(kh)$1

the MLE is obtained from the continuous-time Gaussian likelihood and can be written as a matrix inverse involving

$Y_k^{(h)}=Y(kh)$2

The paper proves strong consistency for every $Y_k^{(h)}=Y(kh)$3 and a full regime classification of asymptotic behavior based on the roots of $Y_k^{(h)}=Y(kh)$4 (Lin et al., 2012).

3. Approximate and quasi-likelihood formulations for sampled data

When the underlying model is continuous time but the data are sampled discretely or irregularly, the likelihood is often unavailable in closed form. One response is numerical integration over the latent state path. In the general continuous-time state-space model with irregular observations $Y_k^{(h)}=Y(kh)$5, the exact likelihood is a multiple integral over latent states: $Y_k^{(h)}=Y(kh)$6 The paper approximates this by discretising the state space into bins $Y_k^{(h)}=Y(kh)$7, which turns the model into a continuous-time HMM with structured, time-inhomogeneous transitions and yields the matrix-product approximation

$Y_k^{(h)}=Y(kh)$8

The forward recursion reduces computation to order $Y_k^{(h)}=Y(kh)$9 (Mews et al., 2020).

A different route is state-space discretisation without time discretisation. For a univariate diffusion

$p_\Delta(x_t\mid x_s)$0

the CTMC approximation constructs a finite-state generator $p_\Delta(x_t\mid x_s)$1 on a spatial grid and uses

$p_\Delta(x_t\mid x_s)$2

as the exact transition matrix of the approximating CTMC. With transition counts $p_\Delta(x_t\mid x_s)$3, the log-likelihood becomes

$p_\Delta(x_t\mid x_s)$4

The paper emphasizes that this introduces no time-discretization error during parameter estimation and proves $p_\Delta(x_t\mid x_s)$5 as $p_\Delta(x_t\mid x_s)$6 under its smoothness assumptions (Kirkby et al., 2021).

For finite-state continuous-time Markov jump processes observed every lag time $p_\Delta(x_t\mid x_s)$7, the likelihood may be written directly in terms of the generator $p_\Delta(x_t\mid x_s)$8 through

$p_\Delta(x_t\mid x_s)$9

The resulting MLE estimates $p_\Delta(x_t\mid x_s)$0 itself rather than a lag-dependent discrete-time transition matrix. The same paper develops a reversible parameterization based on a symmetric matrix $p_\Delta(x_t\mid x_s)$1 and stationary distribution $p_\Delta(x_t\mid x_s)$2, which enforces detailed balance exactly during optimization (McGibbon et al., 2015).

Quasi-likelihood constructions arise when exact innovations are unavailable. In cointegrated continuous-time linear state-space models observed at low frequencies, the estimator is based on pseudo-innovations from the Kalman filter and the pseudo-Gaussian criterion

$p_\Delta(x_t\mid x_s)$3

In locally stationary continuous-time models, the analogous object is a kernel-localized $p_\Delta(x_t\mid x_s)$4-contrast, and in the state-space case it becomes a localized quasi maximum likelihood criterion (Fasen-Hartmann et al., 2017, Ströh, 2021).

4. Functional states, hidden structure, and transformed likelihoods

Several papers extend continuous-time MLE beyond finite-dimensional Markov diffusions by enlarging the state or transforming the process. In the continuous-time higher-order Markov framework, the recent history over $p_\Delta(x_t\mid x_s)$5 is encoded by a state function $p_\Delta(x_t\mid x_s)$6, and the observed scalar process satisfies

$p_\Delta(x_t\mid x_s)$7

The associated functional state evolves as

$p_\Delta(x_t\mid x_s)$8

After deriving a Chapman–Kolmogorov relation over curves and a functional Fokker–Planck equation,

$p_\Delta(x_t\mid x_s)$9

the likelihood of an observed trajectory is formed as a product of segment-to-segment transition densities $\Delta_\tau=t_\tau-t_{\tau-1}$0 and maximized over a finite-dimensional parameterization $\Delta_\tau=t_\tau-t_{\tau-1}$1 of the drift and diffusion (Nag, 2021).

For partially observed diffusions,

$\Delta_\tau=t_\tau-t_{\tau-1}$2

the incomplete-data log-likelihood is expressed on the observation filtration through the innovation process

$\Delta_\tau=t_\tau-t_{\tau-1}$3

leading to

$\Delta_\tau=t_\tau-t_{\tau-1}$4

The gradient involves the tangent filter, and the paper studies a continuous-time stochastic gradient ascent recursion for online maximum likelihood (Surace et al., 2016).

Non-Markovian models are handled by transformation to a semimartingale. In the ergodic Volterra Ornstein–Uhlenbeck process,

$\Delta_\tau=t_\tau-t_{\tau-1}$5

the transformed process

$\Delta_\tau=t_\tau-t_{\tau-1}$6

satisfies

$\Delta_\tau=t_\tau-t_{\tau-1}$7

which restores a Girsanov likelihood and produces explicit estimators for $\Delta_\tau=t_\tau-t_{\tau-1}$8 and $\Delta_\tau=t_\tau-t_{\tau-1}$9 (Alaya et al., 2024). The mixed fractional Vasicek model uses an analogous canonical representation. After defining

$dX_t=-aX_t\,dt+dL_t,$0

the transformed dynamics become

$dX_t=-aX_t\,dt+dL_t,$1

and the likelihood takes the Gaussian-shift form needed for explicit MLEs of $dX_t=-aX_t\,dt+dL_t,$2 and $dX_t=-aX_t\,dt+dL_t,$3 (Cai et al., 2020).

For Gaussian processes with continuous observations, the paper on mean-function estimation identifies the class

$dX_t=-aX_t\,dt+dL_t,$4

for which the likelihood can be written explicitly. The contrast is

$dX_t=-aX_t\,dt+dL_t,$5

and the parametric MLE is $dX_t=-aX_t\,dt+dL_t,$6 (Kobayashi et al., 8 Jul 2025).

5. Asymptotic theory, rates, and efficiency

The asymptotic behavior of continuous-time MLEs is highly model-dependent. In ergodic settings the standard pattern is strong consistency and a Gaussian limit with $dX_t=-aX_t\,dt+dL_t,$7 normalization. This occurs for the continuous-time Lévy-driven Ornstein–Uhlenbeck MLE, for the ergodic Volterra Ornstein–Uhlenbeck estimator, and in the subcritical regimes of several CIR-type models (Mai, 2014, Alaya et al., 2024, 1711.02140). In the Gaussian-process small-noise setting, the analogue of increasing information is $dX_t=-aX_t\,dt+dL_t,$8, the model is locally asymptotically normal with normalization $dX_t=-aX_t\,dt+dL_t,$9, and the MLE is asymptotically efficient with asymptotic covariance $P^0$0 (Kobayashi et al., 8 Jul 2025).

Outside the ergodic setting, the asymptotics need not be normal and may depend on the dynamical regime. For the $P^0$1-stable CIR process, the subcritical case yields strong consistency and asymptotic normality, the supercritical case yields strong consistency and asymptotic mixed normality, and in the critical case “the description of the asymptotic behavior of the MLE in question remains open” (1711.02140). For the jump-type CIR process driven by a subordinator, the subcritical case is asymptotically normal, the critical case has a non-standard limit expressed through a critical diffusion CIR process, and the supercritical case has a mixed normal limit involving the almost sure limit of an exponentially rescaled process (Barczy et al., 2016).

Heavy tails and recurrence structure alter both rate and limit law. In the heavy-tailed Lévy-driven Ornstein–Uhlenbeck model, the experiment is locally asymptotically mixed normal, the correct rate is $P^0$2, and the MLE limit is a Gaussian scale mixture

$P^0$3

with random information driven by an $P^0$4-stable variable (Gushchin et al., 2019). In threshold diffusion, the occupation times $P^0$5 determine whether the estimator is consistent and whether the limit is Gaussian, mixed normal, or nonstandard; the paper treats ergodic, transient, and null recurrent cases separately (Lejay et al., 2018). For the second-order Gaussian autoregression, the MLE is always strongly consistent, but the paper identifies nine distinct asymptotic regimes. One of its main conclusions is that when $P^0$6 and $P^0$7, the MLE convergence rate is governed by the smaller root while the normalized likelihood ratio is governed by the larger root (Lin et al., 2012).

Efficiency results are likewise model-specific. The Lévy-driven Ornstein–Uhlenbeck drift estimator is asymptotically efficient in the Hájek–Le Cam sense (Mai, 2014). The heavy-tailed Lévy-driven Ornstein–Uhlenbeck MLE is asymptotically efficient in the convolution-theorem sense under the LAMN structure (Gushchin et al., 2019). The Gaussian-process mean-function MLE is asymptotically efficient under LAN, and the discrete-sample $P^0$8-estimator matches the continuous-time efficiency when

$P^0$9

with the concrete sufficient condition $\mathcal F_T$0 under the stated mesh and Lipschitz assumptions (Kobayashi et al., 8 Jul 2025).

6. Computation, constraints, and methodological issues

A persistent computational theme is the replacement of intractable infinite-dimensional or continuous-state likelihoods by structured finite-dimensional surrogates. In the continuous-time higher-order Markov model, the drift and diffusion are assumed smooth or piecewise smooth and approximated by polynomials or splines, precisely to avoid the discrete higher-order parameter explosion $\mathcal F_T$1 (Nag, 2021). In the general irregularly sampled continuous-time state-space model, the approximation can be made arbitrarily accurate in principle by increasing the number of bins $\mathcal F_T$2, but larger $\mathcal F_T$3 increases computation time; the paper reports that $\mathcal F_T$4 was a conservative and often effective choice in its examples (Mews et al., 2020).

For finite-state CTMC likelihoods, computational efficiency is tied to gradient evaluation. One paper derives an $\mathcal F_T$5 gradient computation through eigendecomposition and a reusable matrix $\mathcal F_T$6, contrasting this with earlier $\mathcal F_T$7 or $\mathcal F_T$8 approaches, and uses L-BFGS-B for optimization (McGibbon et al., 2015). The diffusion-to-CTMC approximation similarly exploits matrix exponentials and count compression; its stated complexity is

$\mathcal F_T$9

where the raw sample is summarized once into a count matrix (Kirkby et al., 2021).

The literature is also explicit about methodological limits. Maximum approximate likelihood for general continuous-time state-space models requires that the transition density be available in closed form, which excludes many diffusions without tractable transitions (Mews et al., 2020). The finite-state generator MLE is non-convex and may have local minima; embeddability problems also mean that directly taking a matrix logarithm of an empirical transition matrix can produce invalid generators (McGibbon et al., 2015). Quasi-likelihood methods are used in low-frequency cointegrated state-space models precisely because the sampled process is not in standard innovation form and the true innovations are not directly available from finitely many observations (Fasen-Hartmann et al., 2017).

A further methodological issue is correct likelihood specification. The highway-capacity paper argues that the Kaplan–Meier product-limit method is structurally mismatched to stochastic capacity estimation and that an earlier parametric likelihood was flawed because it used the capacity PDF where the CDF was required. The corrected likelihood is

$$\frac{dP_T^a}{dP_T^0} = \exp\!\left( -\frac{a}{\sigma^2}\int_0^T X_s\,dX_s^c -\frac{a^2}{2\sigma^2}\int_0^T X_s^2\,ds \right),$$0

with $$\frac{dP_T^a}{dP_T^0} = \exp\!\left( -\frac{a}{\sigma^2}\int_0^T X_s\,dX_s^c -\frac{a^2}{2\sigma^2}\int_0^T X_s^2\,ds \right),$$1 (Mikolášek, 1 Jul 2025). This example does not concern diffusion inference, but it illustrates a broader point already visible across the continuous-time MLE literature: the asymptotic theory and computational machinery are secondary to specifying the correct likelihood object.

Taken together, these results show that continuous-time maximum likelihood estimation is not a single methodology but a stratified class of exact, approximate, and quasi-likelihood procedures. Exact pathwise likelihoods remain the canonical benchmark when continuous observation and measure equivalence are available. Approximate transition-density, CTMC, HMM, and quasi-innovation constructions extend the framework to sampled, hidden, irregular, or non-Markovian settings. Transformations, state augmentation, and stationary approximations are the principal devices that make these extensions possible, while consistency, LAN or LAMN structure, mixed normality, and efficiency are determined by the geometry of the underlying continuous-time model rather than by a universal asymptotic template.