Continuous-time Stochastic Models Research Articles

From fine-scale foraging to home ranges: a semivariance approach to identifying movement modes across spatiotemporal scales.

Understanding animal movement is a key challenge in ecology and conservation biology. Relocation data often represent a complex mixture of different movement behaviors, and reliably decomposing this mix into its component parts is an unresolved problem in movement ecology. Traditional approaches, such as composite random walk models, require that the timescales characterizing the movement are all similar to the usually arbitrary data-sampling rate. Movement behaviors such as long-distance searching and fine-scale foraging, however, are often intermixed but operate on vastly different spatial and temporal scales. An approach that integrates the full sweep of movement behaviors across scales is currently lacking. Here we show how the semivariance function (SVF) of a stochastic movement process can both identify multiple movement modes and solve the sampling rate problem. We express a broad range of continuous-space, continuous-time stochastic movement models in terms of their SVFs, connect them to relocation data via variogram regression, and compare them using standard model selection techniques. We illustrate our approach using Mongolian gazelle relocation data and show that gazelle movement is characterized by ballistic foraging movements on a 6-h timescale, fast diffusive searching with a 10-week timescale, and asymptotic diffusion over longer timescales.

A numerical study for a mining project using real options valuation under commodity price uncertainty

Commodity price is an important factor for mining companies, as price volatility is a key parameter for mining project evaluation and investment decision making. The conventional discounted cash flow (DCF) methods are broadly used for mining project valuations, however, based on commodity price uncertainty and operational flexibilities, it is difficult and often inappropriate to determine mining project values through traditional DCF methods alone. In order to more accurately evaluate the economic viability of a mining project, the commodity price and its inherent volatility should be modelled appropriately and incorporated into the evaluation process. As a consequence, researchers and practitioners continue to develop and introduce real options valuation (ROV) methods for mining project evaluations under commodity price uncertainty, incorporating continuous time stochastic models. Although the concept of ROV arose a few decades ago, most of the models that have been developed to-date are generally limited to theoretical research and academia and consequently, the application of ROV methods remains poorly understood and often not used in mining project valuations. Analytical and numerical solutions derived through the application of ROV methods are rarely found in practice due to the complexity associated with solving the partial differential equations (PDE), which are dependent on several conditions and parameters. As a consequence, it may not generally be applicable to evaluate mining projects under all project-specific circumstances. Therefore, the greatest challenge to ROV modelling is in finding numerically explicit project values. This paper contributes towards the further development of known theoretical work and enhances an approach to approximating explicit numerical project values. Based on this work, it is possible to formulate more complex PDEs under additional uncertainties attached to the project and to approximate its numerical value or value ranges. To ensure the project is profitable and to reduce commodity price uncertainty, delta hedging and futures contracts have been used as options for deriving the PDE. Moreover, a new parameter for taxes has been incorporated within the PDE. This new PDE has been utilised to approximate the numerical values of a mining project considering a hypothetical gold mine as a case study. The explicit finite difference method (FDM) and MatLab software have been used and implemented to solve this PDE and to determine the numerical project values considering the available options associated with a mining project. In addition, commodity price volatility has been determined from historical data, and has again revealed price volatility as having a significant impact on mining project values.

When to Sell Apple and the Nasdaq? Trading Bubbleswith a Stochastic Disorder Model

In this paper, the authors apply a continuous-time stochastic process model developed by Shiryaev and Zhutlukhin for optimally stopping random price processes that appear to be bubbles, defined as price increases that are largely based on the expectation of higher and higher future prices. Futures traders, such as George Soros, attempt to trade such markets, trying to exit near the peak from a starting long position. The model applies equally well to the question of when to enter and exit a short position. In this article, the authors test the model in two technology markets. These include the price of Apple computer stock from various times in 2009–2012 after the local low of March 6, 2009, plus a market in which the generally very successful bubble trader George Soros lost money by shorting the NASDAQ-100 stock index too soon in 2000. The model provides good exit points in both situations; these would have been profitable to speculators who employed the model.

Research on congestible government production and welfare expenditure in stochastic endogenous growth model

Based on the endogenous growth model, a continuous-time stochastic endogenous growth model with congestible government production and welfare expenditure is proposed. The impact of model parameters on the equilibrium economic growth and welfare is discussed and some optimal policy parameters are obtained. Furthermore, production effect of congestible government expenditure is investigated from two cases whether congestion has production utility or not. Finally, macroeconomic effects of government expenditure and its volatility on economic growth are analysed and demonstrated.

Implications of a lowered damage trajectory for mitigation in a continuous-time stochastic model

We provide counterexamples to the idea that mitigation of greenhouse gas emissions, and adaptation to climate change, are always substitutes. We consider optimal mitigation policy when climate damages follow a geometric Brownian motion process with positive drift and mitigation is lumpy. Climate damages can be affected by adaptation in two main ways: 1) reduced proportionately for given climate impact; or 2) their growth path down-shifted. In either case expectation and variance of the climate damage are both reduced by adaptation. In case 1, the variance effect (which leads to more rapid mitigation as the option value of waiting is reduced) may dominate over the expectation effect (which reduces mitigation), thus on balance increasing mitigation when damages are reduced. Mitigation and adaptation are then complements. A family of functions relating climate damage to adaptation cost in this way includes the Cobb–Douglas specification. In case 2, mitigation and adaptation are always substitutes.

An approximate expectation maximisation algorithm for estimating parameters in nonlinear dynamic models with process disturbances

Stochastic terms are included in fundamental dynamic models of chemical processes to account for disturbances, input uncertainties and model mismatch. The resulting equations are called stochastic differential equations (SDEs). An approximate expectation maximisation (AEM) algorithm using B‐splines is developed for estimating parameters in SDE models when the magnitude of the disturbances and model mismatch is unknown. The AEM method is evaluated using a two‐state nonlinear continuous stirred tank reactor (CSTR) model. The proposed algorithm is compared with two other maximum‐likelihood‐based methods (continuous time stochastic modelling (CTSM) [Kristensen and Madsen, Continuous Time Stochastic Modelling: CTSM 2.3 User's Guide, 2003; Kristensen et al., Automatica 2004; 40: 225] and extended approximate maximum likelihood estimation (AMLE) [Varziri et al., Can. J. Chem. Eng. 2008; 86: 828]). For the CSTR examples studied, the AEM algorithm provides more accurate estimates of model parameters, unknown initial conditions and disturbance intensities. SDE models and associated parameter estimates obtained using AEM will be helpful to engineers who subsequently implement on‐line state estimation and process monitoring schemes because the two types of uncertainties that are considered (i.e. measurement noise and stochastic process disturbances) are consistent with the error structure used in extended Kalman filters.

A causal continuous-time stochastic model for the turbulent energy cascade in a helium jet flow

We discuss continuous cascade models and their potential for modelling the energy dissipation in a turbulent flow. Continuous cascade processes, expressed in terms of stochastic integrals with respect to Lévy bases, are examples of ambit processes. These models are known to reproduce experimentally observed properties of turbulence: the scaling and self-scaling of the correlators of the energy dissipation and of the moments of the coarse-grained energy dissipation. We compare three models: a normal model, a normal inverse Gaussian model, and a stable model. We show that the normal inverse Gaussian model is superior to both, the normal and the stable models, in terms of reproducing the distribution of the energy dissipation; and that the normal inverse Gaussian model is superior to the normal model and competitive with the stable model in terms of reproducing the self-scaling exponents. Furthermore, we show that the presented analysis is parsimonious in the sense that the self-scaling exponents are predicted from the one-point distribution of the energy dissipation, and that the shape of these distributions is independent of the Reynolds number.

Consumption, investment and life insurance strategies with heterogeneous discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent’s preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.

A State Space Approach to Estimating the Integrated Variance under the Existence of Market Microstructure Noise

This article develops a state space method for estimating the integrated variance under the existence of market microstructure noise (MMN). Our method is based on a state space representation of the noise-contaminated RV (NCRV), namely, the realized variance (RV) calculated with observed prices contaminated by MMNs. The main idea of our method is to filter out the bias component, which we call the microstructure noise (MN) component, from the NCRV using the Kalman filter. We apply the proposed method to yen/dollar exchange rate data. We find that about half of the variation in NCRV is because of the MN component. The proposed method can serve as a convenient way to estimate a general class of continuous-time stochastic volatility models under the existence of MMN.

A consumption–investment problem with heterogeneous discounting

We analyze a stochastic continuous time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different discount rates of time preference. Within this framework we can model problems in which, when the time t approaches to the final time, the valuation of the final function increases compared with previous valuations. We study a consumption and portfolio rules problem for CRRA and CARA utility functions for time-consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of random terminal time is also discussed. Differences with both the mathematical treatment and agent’s behavior in the case of hyperbolic discounting are stressed.

A specification test of stochastic diffusion models

In this paper, we propose a hypothesis testing approach to checking model mis-specification in continuous-time stochastic diffusion model. The key idea behind the development of our test statistic is rooted in the generalized information equality in the context of martingale estimating equations. We propose a bootstrap resampling method to implement numerically the proposed diagnostic procedure. Through intensive simulation studies, we show that our approach is well performed in the aspects of type I error control, power improvement as well as computational efficiency.

Sequential Bayesian Inference for Detection and Response to Seasonal Epidemics

We study sequential Bayesian inference in continuous-time stochastic compartmental models with latent factors. A motivating application of our methods is to modeling of seasonal infectious disease outbreaks, notably influenza. Assuming continuous observation of all the epidemiological transitions, our focus is on joint inference of the unknown transition rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. We demonstrate inference in such models with several numerical illustrations and also discuss predictive analysis of epidemic countermeasures using sequential Bayes estimates.

When to Sell Apple and the NASDAQ? Trading Bubbles with a Stochastic Disorder Model

In this paper, the authors apply a continuous time stochastic process model developed by Shiryaev and Zhutlukhin for optimal stopping of random price processes that appear to be bubbles. By a bubble we mean the rising price is largely based on the expectation of higher and higher future prices. Futures traders such as George Soros attempt to trade such markets. The idea is to exit near the peak from a starting long position. The model applies equally well on the short side, that is when to enter and exit a short position. In this paper we test the model in two technology markets. These include the price of Apple computer stock AAPL from various times in 2009{2012 after the local low of March 6, 2009; plus a market where it is known that the generally very successful bubble trader George Soros lost money by shorting the NASDAQ-100 stock index too soon in 2000. The Shiryaev-Zhitlukhin model provides good exit points in both situations that would have been pro table to speculators following the model. who employed the model.

Functional Relationships Between Price and Volatility Jumps and Their Consequences for Discretely Observed Data

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

Functional Relationships Between Price and Volatility Jumps and Their Consequences for Discretely Observed Data

Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

Estimation and Forecasting of Continuous Time Stochastic Volatility Model

This paper improves the estimation of continuous time stochastic model that treats volatility as a latent variable and compares the forecasting performance of the Kalman filter procedure with Exponential model of Autoregressive Conditional Heteroscedastisity. Our empirical study examines the stock indice TUNINDEX by using the daily close price data over the period December 31, 1997, its creation date, to December 31, 2009. The results suggest the significant existence of leverage effect between TUNINDEX returns and its volatility. Indeed, an unanticipated increase in Tunindex return leads to increased uncertainty that is greater than that induced by an unanticipated drop in return. Thus, the volatility forecasts based on Kalman filter model may outperform those of EGARCH model.

Information ratio test for model misspecification on parametric structures in stochastic diffusion models

We develop a hypothesis testing approach to checking model misspecification on parametric structures in continuous-time stochastic diffusion models. The key idea behind the development of our test statistic is rooted in a ratio of two types of information matrices, the negative sensitivity matrix and the variability matrix, in the context of martingale estimating equations. We propose a bootstrap resampling method to implement numerically the proposed diagnostic procedure. Through intensive simulation studies, we compare the proposed approach with several currently popular methods and show that our approach is advantageous in the aspects of type I error control, power improvement as well as computational efficiency. Two real-world data examples are included to illustrate the practical use of our proposed testing procedure.

Calibration and Filtering for Multi Factor Commodity Models with Seasonality: Incorporating Panel Data from Futures Contracts

We construct a general multi-factor model for estimation and calibration of commodity spot prices and futures valuation. This extends the multi-factor long-short model in Schwartz and Smith (Manag Sci 893–911, 2000) and Yan (Review of Derivatives Research 5(3):251–271, 2002) in two important aspects: firstly we allow for both the long and short term dynamic factors to be mean reverting incorporating stochastic volatility factors and secondly we develop an additive structural seasonality model. In developing this non-linear continuous time stochastic model we maintain desirable model properties such as being arbitrage free and exponentially affine, thereby allowing us to derive closed form futures prices. In addition the models provide an improved capability to capture dynamics of the futures curve calibration in different commodities market conditions such as backwardation and contango. A Milstein scheme is used to provide an accurate discretized representation of the s.d.e. model. This results in a challenging non-linear non-Gaussian state-space model. To carry out inference, we develop an adaptive particle Markov chain Monte Carlo method. This methodology allows us to jointly calibrate and filter the latent processes for the long-short and volatility dynamics. This methodology is general and can be applied to the estimation and calibration of many of the other multi-factor stochastic commodity models proposed in the literature. We demonstrate the performance of our model and algorithm on both synthetic data and real data for futures contracts on crude oil.

Stochastic volatility and option pricing with long-memory in discrete and continuous time

It is commonly accepted that certain financial data exhibit long-range dependence. We consider a continuous-time stochastic volatility model in which the stock price is Geometric Brownian Motion with volatility described by a fractional Ornstein–Uhlenbeck process. We also study two discrete-time models: a discretization of the continuous model via a Euler scheme and a discrete model in which the returns are a zero mean i.i.d. sequence where the volatility is a fractional ARIMA process. We implement a particle filtering algorithm to estimate the empirical distribution of the unobserved volatility, which we then use in the construction of a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long-memory parameter we compute an implied value by calibrating the model with real data. We compare the performance of the three models using simulated data and we price options on the S&P 500 index.

Consumption, Investment and Life Insurance Strategies with Heterogeneous Discounting

In this paper we analyze how the optimal consumption, investment and life insurance rules are modified by the introduction of a class of time-inconsistent preferences. In particular, we account for the fact that an agent's preferences evolve along the planning horizon according to her increasing concern about the bequest left to her descendants and about her welfare at retirement. To this end, we consider a stochastic continuous time model with random terminal time for an agent with a known distribution of lifetime under heterogeneous discounting. In order to obtain the time-consistent solution, we solve a non-standard dynamic programming equation. For the case of CRRA and CARA utility functions we compare the explicit solutions for the time-inconsistent and the time-consistent agent. The results are illustrated numerically.