Continuous-time Stochastic Models Research Articles

Resource planning strategies for healthcare systems during a pandemic

We study resource planning strategies, including the integrated healthcare resources’ allocation and sharing as well as patients’ transfer, to improve the response of health systems to massive increases in demand during epidemics and pandemics. Our study considers various types of patients and resources to provide access to patient care with minimum capacity extension. Adding new resources takes time that most patients don't have during pandemics. The number of patients requiring scarce healthcare resources is uncertain and dependent on the speed of the pandemic's transmission through a region. We develop a multi-stage stochastic program to optimize various strategies for planning limited and necessary healthcare resources. We simulate uncertain parameters by deploying an agent-based continuous-time stochastic model, and then capture the uncertainty by a forward scenario tree construction approach. Finally, we propose a data-driven rolling horizon procedure to facilitate decision-making in real-time, which mitigates some critical limitations of stochastic programming approaches and makes the resulting strategies implementable in practice. We use two different case studies related to COVID-19 to examine our optimization and simulation tools by extensive computational results. The results highlight these strategies can significantly improve patient access to care during pandemics; their significance will vary under different situations. Our methodology is not limited to the presented setting and can be employed in other service industries where urgent access matters.

Large and moderate deviations for a discrete-time marked Hawkes process

Hawkes process is a continuous-time stochastic model that captures temporal stochastic self-exciting phenomena. In particular, the linear Hawkes process has been well studied and widely used in practice because of its mathematical tractability. However, in some contexts, a Hawkes model is not directly applicable because data is recorded in a discrete-time scheme or an aggregated way. Thus, a discrete-time Hawkes model is appealing for applications. In this paper, we study large and moderate deviations for a discrete-time marked Hawkes process first proposed in Xu, Zhu, and Wang (2020).

Closed-form portfolio optimization under GARCH models

This paper develops an approximate closed-form optimal portfolio allocation formula for a spot asset whose variance follows a GARCH(1,1) process. We consider an investor with constant relative risk aversion (CRRA) utility who wants to maximize the expected utility from terminal wealth under a Heston and Nandi (2000) GARCH (HN-GARCH) model. Based on an approximation of the log returns from Campbell and Viceira (1999), we obtain closed formulas for the optimal investment strategy, the value function and the optimal terminal wealth. We find the optimal strategy is independent of the development of the risky asset, and the solution converges to that of a continuous-time Heston stochastic volatility model (Kraft, 2005), albeit under additional conditions. For a daily trading scenario, the optimal solutions are quite robust to variations in the parameters, while the numerical wealth equivalent loss (WEL) analysis shows good performance of the Heston solution, with a quite inferior performance of the Merton solution.The solution is extended to two dimensions under the multivariate affine GARCH in Escobar-Anel et al. (2020).

Portfolio optimization and intergenerational risk sharing for a collective defined contribution pension plan

Abstract In this paper, we use a continuous time stochastic model to study a collective defined contribution pension plan when the interest rate is stochastic, and where the benefit levels are adjusted depending on the performance of the plan, and with risk sharing between different generations. The nominal interest rate is characterized by the Vasicek model, and the pension fund is invested in a financial market consisting of three assets: one risk-free asset, one bond and one risky asset. The participants of the pension plan are the risk bearers, and the plan seeks optimal investment and risk-sharing arrangements for plan sponsors and participants that maximize the expected accumulated discount utility of intermediate benefit adjustments and terminal wealth. Closed-form solutions are derived via the stochastic optimal control approach under constant relative risk aversion utility function. Numerical results show the effects of financial market parameters on the optimal investment strategy and how the optimal benefit changes with respect to different risk aversions and wage increase rates.

Efficient estimation of bounded gradient-drift diffusion models for affect on CPU and GPU

Computational modeling plays an important role in a gamut of research fields. In affect research, continuous-time stochastic models are becoming increasingly popular. Recently, a non-linear, continuous-time, stochastic model has been introduced for affect dynamics, called the Affective Ising Model (AIM). The drawback of non-linear models like the AIM is that they generally come with serious computational challenges for parameter estimation and related statistical analyses. The likelihood function of the AIM does not have a closed form expression. Consequently, simulation based or numerical methods have to be considered in order to evaluate the likelihood function. Additionally, the likelihood function can have multiple local minima. Consequently, a global optimization heuristic is required and such heuristics generally require a large number of likelihood function evaluations. In this paper, a Julia software package is introduced that is dedicated to fitting the AIM. The package includes an implementation of a numeric algorithm for fast computations of the likelihood function, which can be run both on graphics processing units (GPU) and central processing units (CPU). The numerical method introduced in this paper is compared to the more traditional Euler-Maruyama method for solving stochastic differential equations. Furthermore, the estimation software is tested by means of a recovery study and estimation times are reported for benchmarks that were run on several computing devices (two different GPUs and three different CPUs). According to these results, a single parameter estimation can be obtained in less than thirty seconds using a mainstream NVIDIA GPU.

Economic heat control of mixing loop for residential buildings supplied by low-temperature district heating

Increasing the penetration of Renewable Energy Sources (RES), the heat pumps are an alternative solution to facilitate the integration of RES into district heating. To hedge against the intermittency of RES, the flexibility potentials of the thermal inertia of buildings are unlocked and integrated into power systems. This paper suggests a novel structure to control the mixing loop of residential heating systems supplied by electrically operated district heating. The Economic Model Predictive Control (EMPC) is proposed to optimize the heat demand of the buildings in response to electricity market price. The EMPC controls the temperature and flow rate of forward/return water in the mixing loop to minimize the energy consumption cost of households. The thermal dynamics of the buildings are modeled by 3-state differential equations to integrate the flexibility potentials of thermal inertia into the district heating. The approach optimizes the heat consumption of the building with multi-temperature zones. The Continuous-Time Stochastic Model (CTSM) is addressed to determine the thermal dynamics of the building using sensor data. Finally, a 150 m2 Danish test house with four temperature zones is simulated. The simulation results show that the suggested approach not only minimizes the energy consumption cost but also provides flexibility for the Danish Electricity Market.

Stochastic energy optimization of residential heat pumps in uncertain electricity markets

The penetration of renewable energy sources is increasing in power systems all over the world. Due to the intermittency and volatility of renewable energies, demand-side management is a practical solution to overcome the problem. This paper proposes a stochastic model predictive control for heat pumps to supply space heating and domestic hot water consumption of residential buildings. Continuous-Time Stochastic Model is coded in R language to address the model identification approach. The approach uses the sensor data of households to extract the thermal dynamics of the building. The controller participates in three floors of power markets with high renewable power penetration. The three-stage stochastic programming is suggested to unlock power-to-heat flexibility in the day-ahead, intraday, and balancing markets on long, mid, and short advance notices, respectively. Regarding the close correlation between renewable power availability and electricity price, the price data is modeled as probabilistic scenarios through Auto-Regressive Integrated Moving Average. The ambient temperature, as well as the domestic hot water consumption, are addressed as envelope-bounds with upper and lower thresholds. Finally, the operational strategies of the controller are examined on a 150 m2 test house under uncertain electricity prices, weather variables, and occupancy patterns.

The high frequency risk attitude implied by the volatility risk premium

Risk attitude measures risk preference, determines micro subjects’ investment portfolios and ultimately affects the macro financial market structure, and is commonly measured by risk aversion coefficient. However, constant or lower frequency risk attitude data makes it difficult to match financial variables in frequency. The risk aversion coefficient characterizes the risk premium of unit risk, so there should be a risk attitude behind each volatility and its risk premium. Based on this principle, this paper introduces risk aversion into the continuous-time stochastic volatility model and establishes a state-space model to separate the implied risk attitude time series from the high-frequency volatility and its risk premium time series, which solves the problem of mismatch between the low-frequency risk aversion and the high-frequency market data in the literature.

A unifying nonlinear probabilistic epidemic model in space and time

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher–Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher–Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher–Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

Estimating energy storage size for Nuclear-Renewable hybrid energy systems using data-driven stochastic emulators

The increased electricity production from renewable energy sources, such as wind, in recent years has contributed to an increase in grid variability. This creates the need for clean, reliable, and flexible back-up electricity generation to complement the renewables and match the grid demand. Nuclear power is a clean, reliable energy source for back-up energy generation but cannot be operated in a manner flexible enough to compensate for the high uncertainties in renewable energy production. To maintain the grid stability with clean energy production and limited operational flexibility of nuclear power, appropriately sized energy storage solutions are needed. Typically, such integrated energy systems are modeled using deterministic methods which are not suitable to accommodate fluctuating energy production and safety constraints on nuclear power plants. This work developed a continuous-time stochastic model for integrating wind and nuclear power with energy storage. The model was then demonstrated for an isolated microgrid using the yearly demand and wind generation data, with nuclear and wind energy as the only power generation sources. Multiple scenarios were analyzed with varying wind penetrations and nuclear flexibility limitations, and the energy storage sizes and their utilization fractions were calculated for three different energy storage types. Without the implementation of energy storage, the required nuclear power capacity should be above the average grid load, and most of the wind power is wasted. By implementing energy storage and using nuclear as a constant base-load, the wind power can account for a larger fraction of the grid, however the storage size requirements are large and utilization fractions are small. If the nuclear power is allowed to operate with greater flexibility than current ramp rate limitations, the storage requirement decreases and storage utilization increases. Of the three storage types analyzed, thermal storage is found to be lowest in size requirement with the highest utilization factors for all scenarios.

A two-phase dynamic contagion model for COVID-19

In this paper, we propose a continuous-time stochastic intensity model, namely, two-phase dynamic contagion process (2P-DCP), for modelling the epidemic contagion of COVID-19 and investigating the lockdown effect based on the dynamic contagion model introduced by Dassios and Zhao [24]. It allows randomness to the infectivity of individuals rather than a constant reproduction number as assumed by standard models. Key epidemiological quantities, such as the distribution of final epidemic size and expected epidemic duration, are derived and estimated based on real data for various regions and countries. The associated time lag of the effect of intervention in each country or region is estimated. Our results are consistent with the incubation time of COVID-19 found by recent medical study. We demonstrate that our model could potentially be a valuable tool in the modeling of COVID-19. More importantly, the proposed model of 2P-DCP could also be used as an important tool in epidemiological modelling as this type of contagion models with very simple structures is adequate to describe the evolution of regional epidemic and worldwide pandemic.

A review of multistate modelling approaches in monitoring disease progression: Bayesian estimation using the Kolmogorov-Chapman forward equations.

There are numerous fields of science in which multistate models are used, including biomedical research and health economics. In biomedical studies, these stochastic continuous-time models are used to describe the time-to-event life history of an individual through a flexible framework for longitudinal data. The multistate framework can describe more than one possible time-to-event outcome for a single individual. The standard estimation quantities in multistate models are transition probabilities and transition rates which can be mapped through the Kolmogorov-Chapman forward equations from the Bayesian estimation perspective. Most multistate models assume the Markov property and time homogeneity; however, if these assumptions are violated, an extension to non-Markovian and time-varying transition rates is possible. This manuscript extends reviews in various types of multistate models, assumptions, methods of estimation and data features compatible with fitting multistate models. We highlight the contrast between the frequentist (maximum likelihood estimation) and the Bayesian estimation approaches in the multistate modeling framework and point out where the latter is advantageous. A partially observed and aggregated dataset from the Zimbabwe national ART program was used to illustrate the use of Kolmogorov-Chapman forward equations. The transition rates from a three-stage reversible multistate model based on viral load measurements in WinBUGS were reported.

Optimal parking management of connected autonomous vehicles: A control-theoretic approach

In this paper we develop a continuous-time stochastic dynamic model for the optimal parking management of connected autonomous vehicles (CAVs) in the presence of multiple parking lots within a given area. Inspired by the well-known Lotka-Volterra equations, a mathematical model is developed to explicitly incorporate the interactions among those parking garages under consideration. Time-dependent parking space availability is considered as the system state, while the dynamic price of parking is naturally used as the control input which can be properly chosen by the parking garage operators from the admissible set. By regulating parking rates, the total demand for parking can be distributed among the set of parking lots under question. Further, we formulate an optimal control problem (called Bolza problem) with the objective of maintaining the availability (managing the demand) of each parking garage at a desired level, which could potentially reduce traffic congestion as well as fuel consumption of CAVs. Based on the necessary conditions of optimality given by Pontryagin’s minimum principle (PMP), we develop a computational algorithm to address the nonlinear optimization problem and formally prove its convergence. A series of Monte Carlo simulations is conducted under various scenarios and the corresponding optimization problems are solved determining the optimal pricing policy for each parking lot. Since the stochastic dynamic model is general and the control inputs, i.e., parking rates, are easy to implement, it is believed that the procedures presented here will shed light on the parking management of CAVs in the near future.

Representation of simulation errors in single step methods using state dependent noise

The local error of single step methods is modelled as a function of the state derivative multiplied by bias and zero-mean white noise terms. The deterministic Taylor series expansion of the local error depends on the state derivative meaning that the local error magnitude is zero in steady state and grows with the rate of change of the state vector. The stochastic model of the local error may include a constant, “catch-all” noise term. A continuous time extension of the local error model is developed and this allows the original continuous time state differential equation to be represented by a combination of the simulation method and a stochastic term. This continuous time stochastic differential equation model can be used to study the propagation of the simulation error in Monte Carlo experiments, for step size control, or for propagating the mean and variance. This simulation error model can be embedded into continuous-discrete state estimation algorithms. Two illustrative examples are included to highlight the application of the approach.

Semimartingale and continuous-time Markov chain approximation for rough stochastic local volatility models

Rough volatility models have recently been empirically shown to provide a good fit to historical volatility time series and implied volatility smiles of SPX options. They are continuous-time stochastic volatility models, whose volatility process is driven by a fractional Brownian motion with Hurst parameter less than half. Due to the challenge that it is neither a semimartingale nor a Markov process, there is no unified method that not only applies to all rough volatility models, but also is computationally efficient. This paper proposes a semimartingale and continuous-time Markov chain (CTMC) approximation approach for the general class of rough stochastic local volatility (RSLV) models. In particular, we introduce the perturbed stochastic local volatility (PSLV) model as the semimartingale approximation for the RSLV model and establish its existence , uniqueness and Markovian representation. We propose a fast CTMC algorithm and prove its weak convergence. Numerical experiments demonstrate the accuracy and high 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.

An Envelope Method for Solving Continuous-Time Stochastic Models with Occasionally Binding Constraints

An Envelope Method for Solving Continuous-Time Stochastic Models with Occasionally Binding Constraints

Robert C. Merton and the Science of Finance

Starting with his 1970 doctoral dissertation and continuing to today, Robert C. Merton has revolutionized the theory and practice of finance. In 1997, Merton shared a Nobel Prize in Economics “for a new method to determine the value of derivatives.” His contributions to the science of finance, however, go far beyond that. In this article I describe Merton's main contributions. They include the following: 1. The introduction of continuous-time stochastic models (the Ito calculus) to the theory of household consumption and investment decisions. Merton's technique of dynamic hedging in continuous time provided a bridge between the theoretical complete-markets equilibrium model of Kenneth Arrow and the real world of personal financial planning and management. 2. The derivation of the multifactor Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM generalizes the single-factor CAPM and explains why that model might fail to properly account for observed market excess returns. It also provides a theory to identify potential forward-looking risk premia for use in factor-based investment strategies. It is therefore both a positive and normative theory. 3. The invention of Contingent Claims Analysis (CCA) as a generalization of option pricing theory. CCA applies the technique of dynamic replication to the valuation and risk management of a wide range of corporate and government liabilities. Merton's CCA model for the valuation and analysis of risky debt is known among scholars and practitioners alike as the Merton Model. 4. The development of financial engineering, which employs CCA to design and produce new financial products. Merton was the first to apply CCA to analyze government guaranty programs such as deposit insurance, and to suggest improvements in the way those programs are managed. He and his students have applied his insights at both the micro and macro policy levels. 5. And finally, the development of a theory of financial intermediation that explains and predicts how financial systems differ across countries and change over time. Merton has applied that theory, called functional and structural finance, to guide the design and regulation of financial systems at the levels of the firm, the industry, and the nation. He has also used it to propose reforms in pensions, sovereign wealth funds, and macrostabilization policy. This article is one of a pair of articles published in this volume about Robert C. Merton's contributions to the science of financial economics. This article was originally published in Volume 11 of the Annual Review of Financial Economics. The other article in this pair is “ Robert C. Merton: The First Financial Engineer ” by Andrew W. Lo.

The effect of graph connectivity on metastability in a stochastic system of spiking neurons

We consider a continuous-time stochastic model of spiking neurons originally introduced by Ferrari et al. in Ferrari et al. (2018). In this model, we have a finite or countable number of neurons which are vertices in some graph G where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction G that is chosen. In this model the spiking activity of any given neuron is represented by a point process, whose rate fluctuates between 1 and 0 over time depending on whether the membrane potential is positive or null. The membrane potential of each neuron evolves in time by integrating all the spikes of its adjacent neurons up to the last spike of the said neuron, so that when a neuron spikes, its membrane potential is reset to 0 while the membrane potential of each of its adjacent neurons is increased by one unit. Moreover, each neuron is exposed to a leakage effect, modeled as an abrupt loss of membrane potential which occurs at random times driven by a Poisson process of some fixed rate γ. It was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to the parameter γ. It was also proven that, when γ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result does not hold anymore if γ is large enough, and in fact we prove that for γ>1 the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if G is the complete graph, the result of metastability holds for any positive γ.

Volatility and Liquidity on High-Frequency Electricity Futures Markets: Empirical Analysis and Stochastic Modeling

This paper investigates the relationship between volatility and liquidity on the German electricity futures market based on high-frequency intraday prices. We estimate volatility by the time-weighted realized variance acknowledging that empirical intraday prices are not equally spaced in time. Empirical evidence suggests that volatility of electricity futures decreases as time approaches maturity, while coincidently liquidity increases. Established continuous-time stochastic models for electricity futures prices involve a growing volatility function in time and are thus not able to capture our empirical findings a priori. We demonstrate that incorporating increasing liquidity into the established models is key to model the decreasing volatility evolution.

VOLATILITY AND LIQUIDITY ON HIGH-FREQUENCY ELECTRICITY FUTURES MARKETS: EMPIRICAL ANALYSIS AND STOCHASTIC MODELING

This paper investigates the relationship between volatility and liquidity on the German electricity futures market based on high-frequency intraday prices. We estimate volatility by the time-weighted realized variance acknowledging that empirical intraday prices are not equally spaced in time. Empirical evidence suggests that volatility of electricity futures decreases as time approaches maturity, while coincidently liquidity increases. Established continuous-time stochastic models for electricity futures prices involve a growing volatility function in time and are thus not able to capture our empirical findings a priori. In Monte Carlo simulations, we demonstrate that incorporating increasing liquidity into the established models is key to model the decreasing volatility evolution.