Continuous-time Setting Research Articles

Investment–consumption optimization with transaction cost and learning about return predictability

In this paper, we investigate an investment–consumption optimization problem in continuous-time settings, where the expected rate of return from a risky asset is predictable with an observable factor and an unobservable factor. Based on observable information, a decision-maker learns about the unobservable factor while making investment–consumption decisions. Both factors are supposed to follow a mean-reverting process. Also, we relax the assumption for perfect liquidity of the risky asset through incorporating proportional transaction costs that are incurred in trading the risky asset. In such way, a form of friction posing liquidity risk to the investor is examined. Dynamic programming principle coupled with an Hamilton–Jacobi–Bellman (HJB) equation are adopted to discuss the problem. Applying an asymptotic method with small transaction costs being taken as a perturbation parameter, we determine the frictional value function by solving the first and second corrector equations. For the numerical implementation of the proposed approach, a Monte-Carlo-simulation-based approximation algorithm is adopted to solve the second corrector equation. Finally, numerical examples and their economic interpretations are discussed.

On dynamical system modeling of learned primal-dual with a linear operator K : stability and convergence properties

Learned Primal-Dual (LPD) is a deep learning based method for composite optimization problems that is based on unrolling/unfolding the primal-dual hybrid gradient algorithm. While achieving great successes in applications, the mathematical interpretation of LPD as a truncated iterative scheme is not necessarily sufficient to fully understand its properties. In this paper, we study the LPD with a general linear operator. We model the forward propagation of LPD as a system of difference equations and a system of differential equations in discrete- and continuous-time settings (for primal and dual variables/trajectories), which are named discrete-time LPD and continuous-time LPD, respectively. Forward analyses such as stabilities and the convergence of the state variables of the discrete-time LPD to the solution of continuous-time LPD are given. Moreover, we analyze the learning problems with/without regularization terms of both discrete-time and continuous-time LPD from the optimal control viewpoint. We prove convergence results of their optimal solutions with respect to the network state initialization and training data, showing in some sense the topological stability of the learning problems. We also establish convergence from the solution of the discrete-time LPD learning problem to that of the continuous-time LPD learning problem through a piecewise linear extension, under some appropriate assumptions on the space of learnable parameters. This study demonstrates theoretically the robustness of the LPD structure and the associated training process, and can induce some future research and applications.

Multiple-Population Discrete-Time Mean Field Games with Discounted and Total Payoffs: The Existence of Equilibria

In the paper we present a model of discrete-time mean-field game with several populations of players. Mean-field games with multiple populations of the players have only been studied in the literature in the continuous-time setting. The main results of this article are the first stationary and Markov mean-field equilibrium existence theorems for discrete-time mean-field games of this type. We consider two payoff criteria: β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-discounted payoff and total payoff. The results are provided under some rather general assumptions on one-step reward functions and individual transition kernels of the players. In addition, the results for total payoff case, when applied to a single population, extend the theory of mean-field games also by relaxing some strong assumptions used in the existing literature.

Discretization of prescribed-time observers in the presence of noises and perturbations

An implicit Euler discretization scheme of the prescribed-time converging observer from Holloway and Krstic, (2019) is proposed, which preserves all main properties of the continuous-time counterpart, and it is investigated how to apply recursively this discretization on any interval of time. In addition, it is demonstrated that the estimation error is robustly stable with respect to the measurement noise with a linear gain (the property that does not hold in the continuous-time setting). The efficiency of the suggested discrete-time observer is illustrated through numeric experiments.

Non-fragile output-feedback control for time-delay neural networks with persistent dwell time switching: A system mode and time scheduler dual-dependent design

This paper is concerned with non-fragile output-feedback control for time-delay neural networks with persistent dwell time (PDT) switching in a continuous-time setting. The main purpose is to design an output-feedback controller subject to gain fluctuations, guaranteeing both asymptotic stability and L2-gain of the closed-loop control system. To achieve reduced conservatism, the controller is formulated to depend not only on the system mode but also on a time scheduler constructed based on the PDT switching rule and minimum time span. A criterion for the asymptotic stability and L2-gain analysis is established through the application of the Gronwall–Bellman inequality and mathematical induction. Then, a numerically tractable design approach for the desired controller is proposed, utilizing a four-section piecewise time-dependent Lyapunov–Krasovskii functional and several nonlinearity decoupling techniques. For comparative purposes, a simple case, independent of the time scheduler, is also investigated, and the corresponding controller design approach is presented. Finally, a simulation example is given to illustrate the effectiveness and superiority of the proposed system mode and time scheduler dual-dependent controller design approach.

The kind of silence: managing a reputation for voluntary disclosure in financial markets

We create a continuous-time setting in which to investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies) in the period between two consecutive mandatory disclosure dates: one with full and transparent disclosure termed candid, the other, termed sparing, under which values only above a dynamic threshold are disclosed. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and the optimal times for management to switch between the two. The model presented develops a number of insights, based on a very simple ordinary differential equation characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black–Scholes model and Poisson arrival of signals of firm value. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about). Significantly, with constant pay-for-performance parameters, at most one switching can occur.

The effect of dispersal on asymptotic total population size in discrete- and continuous-time two-patch models

Many populations occupy spatially fragmented landscapes. How dispersal affects the asymptotic total population size is a key question for conservation management and the design of ecological corridors. Here, we provide a comprehensive overview of two-patch models with symmetric dispersal and two standard density-dependent population growth functions, one in discrete and one in continuous time. A complete analysis of the discrete-time model reveals four response scenarios of the asymptotic total population size to increasing dispersal rate: (1) monotonically beneficial, (2) unimodally beneficial, (3) beneficial turning detrimental, and (4) monotonically detrimental. The same response scenarios exist for the continuous-time model, and we show that the parameter conditions are analogous between the discrete- and continuous-time setting. A detailed biological interpretation offers insight into the mechanisms underlying the response scenarios that thus improve our general understanding how potential conservation efforts affect population size.

Finansinių aktyvų dinamikos skaitmeninis modelis

In the article is proposed the algorithm how to model the dynamics of asset prices by Markov process with continuous time and countable set of states.

Analysis of Theoretical and Numerical Properties of Sequential Convex Programming for Continuous-Time Optimal Control

Sequential Convex Programming (SCP) has recently gained significant popularity as an effective method for solving optimal control problems and has been successfully applied in several different domains. However, the theoretical analysis of SCP has received comparatively limited attention, and it is often restricted to discrete-time formulations. In this paper, we present a unifying theoretical analysis of a fairly general class of SCP procedures for continuous-time optimal control problems. In addition to the derivation of convergence guarantees in a continuous-time setting, our analysis reveals two new numerical and practical insights. First, we show how one can more easily account for manifold-type constraints, which are a defining feature of optimal control of mechanical systems. Second, we show how our theoretical analysis can be leveraged to accelerate SCP-based optimal control methods by infusing techniques from indirect optimal control.

Estimating Treatment Effects from Irregular Time Series Observations with Hidden Confounders

Causal analysis for time series data, in particular estimating individualized treatment effect (ITE), is a key task in many real world applications, such as finance, retail, healthcare, etc. Real world time series, i.e., large-scale irregular or sparse and intermittent time series, raise significant challenges to existing work attempting to estimate treatment effects. Specifically, the existence of hidden confounders can lead to biased treatment estimates and complicate the causal inference process. In particular, anomaly hidden confounders which exceed the typical range can lead to high variance estimates. Moreover, in continuous time settings with irregular samples, it is challenging to directly handle the dynamics of causality. In this paper, we leverage recent advances in Lipschitz regularization and neural controlled differential equations (CDE) to develop an effective and scalable solution, namely LipCDE, to address the above challenges. LipCDE can directly model the dynamic causal relationships between historical data and outcomes with irregular samples by considering the boundary of hidden confounders given by Lipschitz constrained neural networks. Furthermore, we conduct extensive experiments on both synthetic and real world datasets to demonstrate the effectiveness and scalability of LipCDE.

Continuous-time locally stationary time series models

AbstractWe adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Asymptotic results for a class of Markovian self-exciting processes

Hawkes process is a class of self-exciting point processes with clustering effect whose jump rate relies on their entire past history. This process is usually defined as a continuous-time setting and has been widely applied in several fields, including insurance, finance, queueing theory, and statistics. The Hawkes model is generally non-Markovian because the future development of a self-exciting point process is determined by the timing of past events. However, it can be Markovian in special cases such as when the exciting function is an exponential function or a sum of exponential functions. Difficulty arises when the exciting function is not an exponential function or a sum of exponentials, in which case the process can be non-Markovian. The inverse Markovian case for Hawkes processes was introduced by Seol (Stat. Probab. Lett. 155:108580, 2019) who studied some asymptotic behaviors. An extended version of the inverse Markovian Hawkes process was also studied by Seol (J. Korean Math. Soc. 58(4):819–833, 2021). In the current work, we propose a class of Markovian self-exciting processes that interpolates between the Hawkes process and the inverse Hawkes process. We derived limit theorems for the newly considered class of Markovian self-exciting processes. In particular, we established both the law of large numbers (LLN) and central limit theorems (CLT) with some key results.

Formal Verification of Unknown Discrete- and Continuous-Time Systems: A Data-Driven Approach

This paper is concerned with a formal verification scheme for both discrete- and continuous-time deterministic systems with unknown mathematical models. The main target is to verify the safety of unknown systems based on the construction of barrier certificates via a set of data collected from trajectories of systems while providing an a-priori guaranteed confidence on the safety. In our proposed framework, we first cast the original safety problem as a robust convex program (RCP). Solving the proposed RCP is not tractable in general since the unknown model appears in one of the constraints. Instead, we collect finite numbers of data from trajectories of the system and provide a scenario convex program (SCP) corresponding to the original RCP. We then establish a probabilistic closeness between the optimal value of SCP and that of RCP, and as a result, we formally quantify the safety guarantee of unknown systems based on the number of data points and a required level of confidence. We propose our framework in both discrete-time and continuous-time settings. We illustrate the effectiveness of our proposed results by first applying them to an unknown continuous-time room temperature system. We verify that the temperature of the room maintains in a comfort zone with some desirable confidence by collecting data from trajectories of the system. To show the applicability of our techniques to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">higher dimensional</i> systems with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> dynamics, we then apply our results to a continuous-time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nonlinear</i> jet engine compressor and a discrete-time DC motor.

Large Deviations for Hawkes Processes with Randomized Baseline Intensity

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics.

Robust optimal investment strategy for a DC pension plan in the market with mispricing and constant elasticity of variance

This paper considers an optimal asset allocation problem for a defined contribution pension fund in a continuous time setting, and the manager is concerned about potential model misspecification. We suppose that the financial market consists of one risk-free asset, one market index fund whose price satisfies the constant elasticity of variance model, and a pair of risky assets with mispricing. Under the objective of maximizing the expected utility of the pension fund wealth at the retirement, the closed form expressions of the robust optimal investment strategy and the corresponding value function are obtained by using the stochastic control theory. Finally, some numerical examples are given to compare the difference between the ambiguity averse and the ambiguity neutral strategies, and to investigate the effect of relevant parameters on the strategy. We find that the optimal investment strategy is not a long-short symmetric strategy when the mispricing assets have different mispricing correct capability, and it will lead to significant utility loss for the manager if the model uncertainty risk is ignored.

An exit contract optimization problem

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents’ exit times. Under a technical monotonicity condition, and by using Bank-El Karoui’s representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimization problem over the class of all Markovian and/or continuous exit contracts.

Stabilization using shifted equilibria for saturated discrete-time linear systems⋆

We discuss controller designs for asymmetric saturated discrete-time linear systems. Under the assumption that a locally stabilizing controller of the origin is known, we augment the original controller with an additional term that vanishes in a neighborhood of the origin. The augmented controller outperforms the original controller in terms of the estimate of the region of attraction. The paper translates the results discussed in Braun et al. (2022a,b), from the continuous-time setting to the discrete-time setting, and numerically verifies that the results derived for continuous-time systems are recovered if the discrete-time system is obtained through an Euler discretization of a continuous-time system with a sufficiently small sampling rate.

Active causal structure learning in continuous time

Research on causal cognition has largely focused on learning and reasoning about contingency data aggregated across discrete observations or experiments. However, this setting represents only the tip of the causal cognition iceberg. A more general problem lurking beneath is that of learning the latent causal structure that connects events and actions as they unfold in continuous time. In this paper, we examine how people actively learn about causal structure in a continuous-time setting, focusing on when and where they intervene and how this shapes their learning. Across two experiments, we find that participants’ accuracy depends on both the informativeness and evidential complexity of the data they generate. Moreover, participants’ intervention choices strike a balance between maximizing expected information and minimizing inferential complexity. People time and target their interventions to create simple yet informative causal dynamics. We discuss how the continuous-time setting challenges existing computational accounts of active causal learning, and argue that metacognitive awareness of one’s inferential limitations plays a critical role for successful learning in the wild.

Asset-liability management with state-dependent utility in the regime-switching market

This paper considers a state-dependent optimal asset-liability management problem in continuous-time settings. The investor maximizes the expected state-dependent utility of the terminal asset-liability ratio in a regime-switching market to better describe insurance companies’ needs for asset-liability matching and regulation with background risk. We apply the stochastic dynamic programming method to get closed-form results. We find that the optimal strategy in the state-dependent utility case is the same, and the optimal value function differs from the state-independent utility case. Finally, we study how the parameters impact the optimal investment strategy and corresponding value functions through numerical examples.

Rejoinder: A formal causal interpretation of the case-crossover design.

We thank the discussants for their insightful commentary and questions. In our rejoinder, we extend our analysis to additional settings, control strategies, and sources of bias relevant to how case-crossover is often used in practice, as suggested by multiple discussants. In particular, we consider: control exposures that follow occurrence of events, settings with shared exposure trajectories (which are common in case-crossover studies of effects of air pollution), bias due to non-transient treatment effects, removing bias due to time trends in treatment using control subjects, and extending our results to the continuous time setting. We also take the opportunity to clarify an easily misinterpreted comment we made about collapsibility, which we thank Andersen and Martinussen for highlighting. Throughout, in the spirit of the discussants. contributions, we iterate between general bias formulas and simulations and numerical studies from illustrative simplified scenarios to build insight.