Continuous-time Model Research Articles

A Non-Standard Finite Difference Discretization Scheme Applied to a Malaria Model

In this research work, a dynamically consistent non-standard finite difference (NSFD) scheme is developed to solve a continuous-time model of malaria transmission with herbal medicine as control strategy. We compared results from NSFD scheme with the standard finite difference methods (4th order Runge-kutta and forward Euler methods). The numerical investigation showed that the proposed NSFD method remains consistent, preserves the positivity of solutions and converges to true equilibrium points of the continuous model independent of the step size h.

Modeling Survey Time Series Data with Flow-Observed CARMA Processes

Published survey data often are delivered as estimates computed over an epoch of time. Customers may desire to obtain survey estimates corresponding to epochs, or time points, that differ from the published estimates. This “change of support” problem can be addressed through the use of continuous-time models of the underlying population process, while taking into account the sampling error that survey data is subject to. The application of a Continuous AutoRegressive Moving Average (CARMA) model is investigated as a tool to provide change of support applications, thereby allowing interpolation for published survey estimates. A simulation study provides comparisons of competing estimation methods, and a synthetically constructed data set is developed in order to elucidate real data applications. The proposed method can be successful for change of support problems, despite modeling challenges with the CARMA framework.

Analysis of the dynamics of transition from non-colonization to colonization and Staphylococcus aureus bacteremia in hemodialysis patients using Markov models.

Background Hemodialysis patients are frequently colonized by Staphylococcus aureus, leading to severe infections with high mortality rates. However, little is known about transition from non-colonization to colonization or bacteremia over time. The aim was to analyze the behavior of S. aureus colonization, identifying the probability of transition from non-colonized to colonized state or bacteremia, and the influence of specific covariates. Methods The study was conducted in a dialysis unit associated with a tertiary care hospital in Medellín between October 2017 and October 2019. An initial measurement was taken to evaluate S. aureus colonization, and follow-up measurements were performed 2 and 6 months later. Bacteremia evolution was monitored for 12 months. A two-state recurrent continuous-time Markov model was constructed to model transition dynamics from non-colonization to S. aureus colonization in hemodialysis patients. Subsequently, the model was applied to a third state of bacteremia. Results Of 178 patients on hemodialysis, 30.3% were colonized by S. aureus. Transition intensity from non-colonization to colonization was three times higher (0.21; CI: 0.14-0.29) than from colonization to non-colonization (0.07; CI: 0.05-0.11). The colonization risk increased in patients with previous infections (HR: 2.28; CI: 0.78-6.68), hospitalization (HR: 1.29; CI: 0.56-2.99) and antibiotics consumption (HR: 1.17; CI: 0.53-2.58). Mean non-colonized state duration was 10.9 months, while in the colonized state was 5.2 months. In the 3-state model, it was found that patients colonized were more likely to develop S. aureus infection (13.9%). Conclusion A more likely transition from non-colonization to colonization was found, which increases with factors such as previous infection. In addition, the development of bacteremia was more likely in colonized than in non-colonized patients. These results underline the importance of surveillance and proper management of S. aureus colonization to prevent serious complications, such as bacteremia, and improve prognosis in this vulnerable population.

Exact results on the dynamics of the stochastic Floquet-East model

Abstract We introduce a stochastic generalisation of the classical deterministic Floquet-East model, a discrete circuit with the same kinetic constraint as the East model of glasses. We prove exactly that, in the limit of long time and large size, this model has a large deviation phase transition between active and inactive dynamical phases. We also compute the finite time and size scaling of general space-time fluctuations, which for the case of inactive regions gives rise to dynamical hydrophobicity. We also discuss how, through the Trotter limit, these exact results also hold for the continuous-time East model, thus proving long-standing observations in kinetically constrained models. Our results here illustrate the applicability of exact tensor network methods for solving problems in many-body stochastic systems.

Foundations of Market Power in Monetary Economies

We study the foundations of firms' market power in a continuous-time model where agents are price-makers who interact explicitly with each other. Market power arises from the existence of rents, the size of which depends on consumers' outside options, and firms' ability to appropriate these rents through rent seeking. We study how measures of market power (e.g., markups, concentration) are affected by search frictions, monetary policy, and self-fulfilling beliefs. An increase in the nominal interest rate affects market power by changing consumers' payment capacity and the value of their outside options, and by altering firms' rent-seeking behavior.

A general valuation framework for rough stochastic local volatility models and applications

Rough volatility models are a new class of stochastic volatility models that have been shown to provide a consistently good fit to implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with the corresponding Hurst parameter less than a half. Albeit the empirical success, the valuation of derivative securities under rough volatility models is challenging. The reason is that it is neither a semi-martingale nor a Markov process. This paper proposes a novel valuation framework for rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semi-martingale approximation for the RSLV model and establish its existence, uniqueness, Markovian representation and convergence. Then we propose a fast continuous-time Markov chain (CTMC) approximation algorithm to the PSLV model and establish its convergence. Numerical experiments demonstrate the convergence of our approximation method to the true prices, and also the remarkable accuracy and efficiency of the method in pricing European, barrier and American options. Comparing with existing literature, a significant reduction in the CPU time to arrive at the same level of accuracy is observed.

Insider trading at a random deadline with correlation between dynamic asset and stochastic liquidity

We propose a generalized continuous-time insider trading model, building upon the frameworks of Caldentey and Stacchetti (2010) and Collin-Dufresne and Fos (2016), with a correlation between the value of a risky asset following an Ornstein-Uhlenbeck-type process and the noise trading volume with volatility characterized by a general stochastic process. And a closed form of the market equilibrium is established, consisting of the insider's trading strategy and the market makers' pricing rule. It shows that at the equilibrium: (i) all of the insider's private information is released at the end of the transaction; (ii) market depth and market liquidity evolve as semi-martingales, respectively; and (iii) the equilibrium price is driven by a bridge process that solves an Ornstein-Uhlenbeck-type SDE. Numerical simulations show that as the correlation coefficient increases, the equilibrium price becomes more informative, leading to a decrease in both the trading intensity and the expected payoff for the insider.

Design of LQ regulators with prescribed degree of stability for dual-rate systems based on reinforcement learning

This paper considers dual-rate systems, where the output is measured at a relatively slow rate while the control signal is adjusted at a faster rate. The output sampling time is an integer multiple of the input sampling time. The paper examines dual-rate inferential control systems, which consist of a fast model, a slow model, and a switch. Missing output samples are estimated using the fast single-rate model. The single-rate control algorithm is then implemented at the fast-sampling rate. The fast-sampling discrete-time model is derived from the plant’s continuous-time model using the first-order hold (FOH) element. A discrete LQ regulator is proposed for this plant model, with a prescribed degree of stability (all closed-loop eigenvalues are within the range 0 < λ < 1 in magnitude). The matrix gain is calculated offline, and an online method for calculating the regulator gain is provided. The regulator gain is calculated using policy iteration, specifically Hewer’s algorithm. Finally, it is demonstrated that the presented inferential control system remains effective in the presence of multiplicative unmodeled dynamics. The main contributions of the paper are: (i) Designing the LQ regulator with a prescribed degree of stability using reinforcement learning (RL) (generalized policy iteration); and (ii) Considering the robust stability of the inferential control system in the presence of multiplicative unmodeled dynamics using the lifting technique.

Estimating Time-Varying Exposure Effects Through Continuous-Time Modelling in Mendelian Randomization.

Mendelian randomization is an instrumental variable method that utilizes genetic information to investigate the causal effect of a modifiable exposure on an outcome. In most cases, the exposure changes over time. Understanding the time-varying causal effect of the exposure can yield detailed insights into mechanistic effects and the potential impact of public health interventions. Recently, a growing number of Mendelian randomization studies have attempted to explore time-varying causal effects. However, the proposed approaches oversimplify temporal information and rely on overly restrictive structural assumptions, limiting their reliability in addressing time-varying causal problems. This article considers a novel approach to estimate time-varying effects through continuous-time modelling by combining functional principal component analysis and weak-instrument-robust techniques. Our method effectively utilizes available data without making strong structural assumptions and can be applied in general settings where the exposure measurements occur at different timepoints for different individuals. We demonstrate through simulations that our proposed method performs well in estimating time-varying effects and provides reliable inference when the time-varying effect form is correctly specified. The method could theoretically be used to estimate arbitrarily complex time-varying effects. However, there is a trade-off between model complexity and instrument strength. Estimating complex time-varying effects requires instruments that are unrealistically strong. We illustrate the application of this method in a case study examining the time-varying effects of systolic blood pressure on urea levels.

Opinions dynamics for a generalized nonlinear model with state-dependent susceptibility in signed networks on time scales

This article studies the opinions dynamics for a generalized nonlinear model with state-dependent susceptibility in signed networks on time scales. Unlike continuous-time or discrete-time models, time-scale systems can integrate continuous and discrete systems within a unified framework. Nevertheless, time-scale induced discontinuity properties can make it difficult to study the well-posedness and dynamics analysis of the opinions dynamics. In this paper, we first give a time-scale related condition to ensure the well-posedness of the nonlinear model. Then, based on time-scale theory, the comparison principle and the generalized Gronwall inequality, we focus on analyzing the dynamic behavior of the nonlinear model with three distinct susceptibility functions, which represent the behavior pattern of stubborn positives, stubborn extremists, and stubborn neutrals respectively. For the behavior pattern of stubborn positives and stubborn extremists, some sufficient conditions are given such that all agents’ viewpoints will converge exponentially towards a specific equilibrium point or achieve consensus under certain initial conditions. For the behavior pattern of stubborn neutrals, all agents’ opinions will finally reach neutrality.

Robust dynamic contracts with multiple agents

We develop a continuous-time dynamic multi-agent contracting model in which the principal is unsure about the distributions of the project's terminal payoffs and worries about model misspecification. With model uncertainty, workers' wages depend on the outputs of other unrelated projects and the optimal contracts exhibit overdetermination. We demonstrate an inverse U-shaped relationship between the extent of overdetermination and group size. Moreover, model uncertainty induces wage compression, especially in small firms as the empirical evidence demonstrates. Finally, expanding the group size increases the average project value by mitigating the negative impacts of ambiguity.

Momentum mechanisms under heterogeneous beliefs

We establish a continuous-time heterogeneous beliefs model to discuss the mechanisms of Momentum and Reversal. Price learning, information transmission and extrapolative expectation are incorporated into a unified framework for the Momentum and Reversal. The calibration results from SP500 show that the presence of Extrapolators and Information-driven Traders are important influences of Momentum and Reversal in all phases. We also find that momentum and reversal become significantly stronger as belief weights approach true belief weights.

Contracting with cost synergies: Continuous-time double-sided moral hazard

We study optimal effort and compensation in two continuous-time double-sided models with cost synergies. In the co-work synergy model, cost synergies exist between two agents with ongoing effort: each agent's effort reduces his colleague's marginal cost of effort. The agents completely divide the project's cash flow. In the optimal contract, the agent with higher productivity and a bigger cost-reduction influence claims a larger fraction of the cash flow. In the chain synergy model, one agent exerts initial effort to start the project, and her colleague exerts ongoing effort to manage it. The upfront effort reduces the marginal cost of her colleague's ongoing effort. The timing of optimal payments reflects cost synergies across agents and the timing of efforts: Upfront effort corresponds to early payments. We show that the introduction of cost synergies not only alters the allocation of the cash flow but also improves the expected social surplus. This study suggests that cost synergies increase efficiency for a broad set of contracting problems involving teams.

A low-complexity solution for LMIs of hardly constrained nonlinear MIMO tracking control systems

This research is dedicated to the discrete-time tracking control of nonlinear multi-input multi-output (MIMO) Newtonian mechanics systems with an equal number of inputs and outputs, in the presence of input saturation and high-order dynamics of the actuation system. Firstly, the continuous-time model of the plant and actuation system has been transformed into the discrete-time domain using Adams–Bashforth and Adams–Moulton discretization techniques. In the next step, considering saturation boundaries, the modified position and velocity references are derived using the discrete-time reaching law-based control approach. These modified references are obtained through an optimization algorithm that calculates the closest achievable position and velocity trajectories with respect to the desired reference trajectory. Modifying desired trajectories in MIMO systems results in the formation of a system of linear matrix inequalities (LMIs). Solving such systems of inequalities requires the use of complex methods, and this difficulty increases with the increase in the system’s degrees of freedom (DOFs). The proposed strategy offers a more straightforward approach to solving these inequalities. Once the modified references are calculated, the corresponding position and velocity control signals can be obtained using the mentioned reaching law strategies. The final step of the proposed algorithm is to combine the resulting position and velocity control signals using a Gaussian-weighted averaging method to form the main control signal. A desirable performance for the control system can be achieved by appropriately adjusting the standard deviations of the Gaussian weighing functions. To evaluate the performance characteristics of the proposed method, it has been applied in the tracking control procedure of a nonlinear 3-DOF mechanical system. The numerical simulation results demonstrate the effectiveness of the method.

Childhood and adult socioeconomic status influence on late-life healthy longevity: evidence from the Chinese longitudinal healthy longevity survey.

Older people in low- and middle-income countries are more susceptible to the impact of childhood experiences. This study comprehensively examines how childhood socioeconomic status (SES) and adult SES collectively influence late-life healthy longevity from a life course perspective, providing insights for shaping health-related policies. This study analyzed data from the Chinese Longitudinal Healthy Longevity Survey (1998-2018) with 37,264 individuals aged 65 and above. Using R software, we applied continuous-time multi-state models incorporating the Rockwood frailty index with 38 indicators to assess participants' health. Childhood SES or life course SES trajectories were core explanatory variables, while age and gender were controlled. Multinomial regression estimated annual transition probabilities between different states, and the multi-state life table method calculated total and frailty-specific life expectancy (LE). (1) Social mobility among older people in China showed an upward trend from childhood to adulthood. (2) Transition probabilities for robust-frailty, robust-dead, and frailty-dead increased with age, while frailty-robust decreased. Transition probabilities and LE varied across different childhood SES (low, medium, high) or life-course SES trajectory categories (low-low, low-medium, low-high, medium-low, medium-medium, medium-high, high-low, high-medium, high-high), with probabilities of robust-frailty, robust-dead, and frailty-dead decreasing sequentially across different categories, and frailty-robust increasing sequentially across different categories. Total LE, robust LE, and robust LE proportion increased sequentially across different categories, while frailty LE decreased sequentially across different categories. (3) Women had higher total LE and frailty incidence, but lower recovery rate, mortality risk, robust LE, and robust LE proportion compared to men. Favorable childhood SES and lifelong accumulation of SES advantages protect against frailty morbidity, improve recovery rate, reduce mortality risk, and increase total LE, robust LE, and robust LE proportion. High childhood SES has a stronger protective effect than high adult SES, indicating the lasting impact of childhood conditions on healthy longevity. Systematic interventions in education, food supply, and medical accessibility for children from impoverished families are crucial.

Asymptotics for Finite-Time Ruin Probabilities of a Dependent Bidimensional Risk Model with Stochastic Return and Subexponential Claims

The paper considers a bidimensional continuous-time risk model with subexponential claims and Brownian perturbations, in which the price processes of the investment portfolio of the two lines of business are two geometric Lévy processes and the two lines of business share a common claim-number process, which is a renewal counting process. The paper mainly considers the claims of each line of business having a dependence structure. When the claims have subexponential distributions, the asymptotics of the finite-time ruin probabilities ψand(x1,x2;T) and ψsim(x1,x2;T) have been obtained. When the distributions of claims belong to the intersection of long-tailed and dominatedly varying-tailed distribution classes, the asymptotics of the finite-time ruin probability ψor(x1,x2;T) is given.

Profiling the sleep architecture of ageing adults using a seven-state continuous-time Markov model.

Sleep is a complex biological process regulated by networks of neurons and environmental factors. As one falls asleep, neurotransmitters from sleep-wake regulating neurones work in synergy to control the switching of different sleep states throughout the night. As sleep disorders or underlying neuropathology can manifest as irregular switching, analysing these patterns is crucial in sleep medicine and neuroscience. While hypnograms represent the switching of sleep states well, current analyses of hypnograms often rely on oversimplified temporal descriptive statistics (TDS, e.g., total time spent in a sleep state), which miss the opportunity to study the sleep state switching by overlooking the complex structures of hypnograms. In this paper, we propose analysing sleep hypnograms using a seven-state continuous-time Markov model (CTMM). This proposed model leverages the CTMM to depict the time-varying sleep-state transitions, and probes three types of insomnia by distinguishing three types of wake states. Fitting the proposed model to data from 2056 ageing adults in the Multi-Ethnic Study of Atherosclerosis (MESA) Sleep study, we profiled sleep architectures in this population and identified the various associations between the sleep state transitions and demographic factors and subjective sleep questions. Ageing, sex, and race all show distinctive patterns of sleep state transitions. Furthermore, we also found that the perception of insomnia and restless sleep are significantly associated with critical transitions in the sleep architecture. By incorporating three wake states in a continuous-time Markov model, our proposed method reveals interesting insights into the relationships between objective hypnogram data and subjective sleep quality assessments.

Performing global sensitivity analysis on simulations of a continuous-time Markov chain model motivated by epidemiology

Performing global sensitivity analysis on simulations of a continuous-time Markov chain model motivated by epidemiology

Predictive value of mono-exponential and multiple mathematical models in locally advanced rectal cancer response to neoadjuvant chemoradiotherapy.

This prospective study aimed to assess the predictive value of mono-exponential and multiple mathematical diffusion-weighted imaging (DWI) models in determining the response to neoadjuvant chemoradiotherapy (nCRT) in patients with locally advanced rectal cancer (LARC). The study included 103 LARC patients scheduled for preoperative chemoradiotherapy between December 2021 and June 2023 Magnetic resonance imaging (MRI) scans were performed using a 3.0-T MR scanner, encompassing sagittal, axial, and oblique coronal T2-weighted images without fat saturation, along with DWI perpendicular to the rectum's long axis. Various DWI parameters, including apparent diffusion coefficient (ADC), stretched exponential model (SEM), continuous-time random-walk model (CTRW), and fractional-order calculus model (FROC), were measured. The pathologic complete response (pCR) rate and tumor downstaging (T-downstage) rate were determined. After nCRT, SEM-α, SEM-DDC, CTRW-α, CTRW-β, CTRW-D, FROC-β, and ADC values were significantly higher in the pCR group compared to the non-pCR group (all P < 0.05). SEM-DDC, CTRW-α, CTRW-D, FROC-β, FROC-µ, and ADC values were significantly higher in the T-downstage group (ypT0-1) than in the non-T-downstage group (ypT2-4) (P < 0.05). The combination of CTRW (α + β + D) exhibited the best diagnostic performance for assessing pCR after nCRT (AUC = 0.840, P < 0.001). Pre-nCRT CTRW (α + β) demonstrated a predictive AUC of 0.652 (95%CI: 0.552-0.743), 90.3% sensitivity, and 43.1% specificity for pCR. Regarding T-downstage assessment after nCRT, the combination of CTRW (α + D) yielded the best diagnostic performance (AUC = 0.877, P = 0.048). In LARC patients, imaging markers derived from CTRW show promise in predicting tumor response before nCRT and assessing pCR after nCRT.

Machine Learning for Continuous-Time Finance

Abstract We develop an algorithm for solving a large class of nonlinear high-dimensional continuous-time models in finance. We approximate value and policy functions using deep learning and show that a combination of automatic differentiation and Ito’s lemma allows for the computation of exact expectations, resulting in a negligible computational cost that is independent of the number of state variables. We illustrate the applicability of our method to problems in asset pricing, corporate finance, and portfolio choice and show that the ability to solve high-dimensional problems allows us to derive new economic insights.