Stochastic Differential Equation Approach Research Articles

DynGMA: A robust approach for learning stochastic differential equations from data

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Robust asset-liability management games for n players under multivariate stochastic covariance models

This paper investigates a non-zero-sum stochastic differential game among n competitive CARA asset-liability managers, who are concerned about the potential model ambiguity and aim to seek the robust investment strategies. The ambiguity-averse managers are subject to uncontrollable and idiosyncratic random liabilities driven by generalized drifted Brownian motions and have access to an incomplete financial market consisting of a risk-free asset, a market index and a stock under a multivariate stochastic covariance model. The market dynamics permit not only stochastic correlation between the risky assets but also path-dependent and time-varying risk premium and volatility, depending on two affine-diffusion factor processes. The objective of each manager is to maximize the expected exponential utility of his terminal surplus relative to the average among his competitors under the worst-case scenario of the alternative measures. We manage to solve this robust non-Markovian stochastic differential game by using a backward stochastic differential equation approach. Explicit expressions for the robust Nash equilibrium investment policies, the density generator processes under the well-defined worst-case probability measures and the corresponding value functions are derived. Conditions for the admissibility of the robust equilibrium strategies are provided. Finally, we perform some numerical examples to illustrate the influence of model parameters on the equilibrium investment strategies and draw some economic interpretations from these results.

Exploring the effects of temperature on demersal fish communities in the Central Mediterranean Sea using INLA-SPDE modeling approach

Climate change significantly impacts marine ecosystems worldwide, leading to alterations in the composition and structure of marine communities. In this study, we aim to explore the effects of temperature on demersal fish communities in the Central Mediterranean Sea, using data collected from a standardized monitoring program over 23 years. Computationally efficient Bayesian inference is performed using the integrated nested Laplace approximation and the stochastic partial differential equation approach to model the spatial and temporal dynamics of the fish communities. We focused on the mean temperature of the catch (MTC) as an indicator of the response of fish communities to changes in temperature. Our results showed that MTC decreased significantly with increasing depth, indicating that deeper fish communities may be composed of colder affinity species, more vulnerable to future warming. We also found that MTC had a step-wise rather than linear increase with increasing water temperature, suggesting that fish communities may be able to adapt to gradual changes in temperature up to a certain threshold before undergoing abrupt changes. Our findings highlight the importance of considering the non-linear dynamics of fish communities when assessing the impacts of temperature on marine ecosystems and provide important insights into the potential impacts of climate change on demersal fish communities in the Central Mediterranean Sea.

Comparing spatiotemporal species distribution models: A case study of a Scotian Shelf sea cucumber (Cucumaria frondosa)

AbstractNumerous spatiotemporal species distribution modeling frameworks are now available to the ecological practitioner. This study compared three such frameworks accessible in the R programming language: generalized additive models with spatiotemporal smooths as implemented by mgcv, spatiotemporal generalized linear mixed models based on nearest neighbor Gaussian processes as implemented by starve, and spatiotemporal generalized linear mixed models based on the stochastic partial differential equations approach as implemented by sdmTMB. The primary focus was to compare the inferences obtained from applying these frameworks to the case study of the orange‐footed sea cucumber, Cucumaria frondosa, on the Scotian Shelf off Nova Scotia, Canada. Each model was fit to catch data (2000–2019) from Fisheries and Oceans Canada's annual Research Vessel and Snow Crab surveys. Environmental covariates were sourced from high‐resolution data layers, including physical oceanographic, bathymetric, and seafloor morphometric datasets. The three models captured variability in sea cucumber distribution that would have been overlooked without a spatiotemporal approach. Although their predictions were similar, including within C. frondosa spatial reserves, the models provided different inferences regarding covariate effects. This suggests that while practitioners primarily interested in mapping species distributions need only apply the most familiar framework, those most concerned with identifying predictive environmental covariates may benefit from comparing the output from multiple approaches. Employing multiple approaches can also serve as a validation technique.

Optimal investment strategy for asset-liability management with a defaultable bond under stochastic default intensity

This paper studies a continuous-time asset-liability management problem with a defaultable bond under stochastic interest rates and stochastic default intensity. Specifically, an asset-liability manager is allowed to invest in a cash, a treasury bond, a defaultable bond and a stock. Using the stochastic control scheme and partial differential equation approach, we derive closed-form expressions for optimal investment strategies and corresponding value functions under the power and exponential utility functions. In addition, we also provide numerical experiments to illustrate the impact of relevant parameters in the default process and recovery rate on the optimal asset allocation strategy.

On the Two Approaches to Incorporate Wave‐Particle Resonant Effects Into Global Test Particle Simulations

AbstractEnergetic electron dynamics in the Earth's radiation belts and near‐Earth plasma sheet are controlled by multiple processes operating on very different time scales: from storm‐time magnetic field reconfiguration on a timescale of hours to individual resonant wave‐particle interactions on a timescale of milliseconds. The most advanced models for such dynamics either include test particle simulations in electromagnetic fields from global magnetospheric models, or those that solve the Fokker‐Plank equation for long‐term effects of wave‐particle resonant interactions. The most prospective method, however, would be to combine these two classes of models, to allow the inclusion of resonant electron scattering into simulations of electron motion in global magnetospheric fields. However, there are still significant outstanding challenges that remain regarding how to incorporate the long term effects of wave‐particle interactions in test‐particle simulations. In this paper, we describe in details two approaches that incorporate electron scattering in test particle simulations: stochastic differential equation (SDE) approach and the mapping technique. Both approaches assume that wave‐particle interactions can be described as a probabilistic process that changes electron energy, pitch‐angle, and thus modifies the test particle dynamics. To compare these approaches, we model electron resonant interactions with field‐aligned whistler‐mode waves in dipole magnetic fields. This comparison shows advantages of the mapping technique in simulating the nonlinear resonant effects, but also underlines that more significant computational resources are needed for this technique in comparison with the SDE approach. We further discuss applications of both approaches in improving existing models of energetic electron dynamics.

Robust optimal asset-liability management with mispricing and stochastic factor market dynamics

This paper investigates a robust optimal asset-liability management problem under an expected utility maximization criterion. More specifically, the manager is concerned about the potential model uncertainty and aims to seek the robust optimal investment strategies. We incorporate an uncontrollable random liability described by a generalized drifted Brownian motion. Also, the manager has access to an incomplete financial market consisting of a risk-free asset, a market index with potentially path-dependent, time-varying risk premium and volatility, and a pair of mispriced stocks. The market dynamics are assumed to rely on an affine-form, square-root factor process and the price error is modeled by a co-integrated system. We adopt a backward stochastic differential equation approach hinging on the martingale optimality principle to solve this non-Markovian robust control problem. Closed-form expressions for the robust optimal investment strategies, the probability perturbation process under the well-defined worst-case scenario and the corresponding value function are derived. The admissibility of the robust optimal controls is verified under some technical conditions. Finally, we perform some numerical examples to illustrate the effects of model parameters on the robust investment strategies and draw some economic interpretations from these results.

Monitoring regional soil organic matter content using a spatiotemporal model with time-series synthetic Landsat images

Accurately monitoring soil organic matter (SOM) content is crucial for food and soil security. Current methods of monitoring are expensive and existing sparse data cannot provide detailed spatial information about SOM content changes in an area. This study proposes using a spatiotemporal model with time-series synthetic Landsat images to monitor SOM content dynamics at the regional scale. The approach was implemented in Google Earth Engine (GEE) platform and tested in Jiangsu province, China, using the soil survey data from 2006 to 2007 and synthetic Landsat images from 1986 to 2007. The model generated SOM maps every three years between 1986 and 2007 and was evaluated using another soil survey from 2000 in southern Jiangsu. The model associated with 20 covariates derived from the synthetic Landsat image explained 70% of the variation in SOM content with root mean squared error (RMSE) and Lin's concordance correlation coefficient (CCC) of 5.17 g/kg and 0.57, respectively. The results showed that the model could reveal regional spatial and temporal differences in SOM content distribution. The SOM content in Jiangsu increased in the north, while decreasing in the central and southern areas. Temporally, the mean SOM contents increased from 1986 to 1992 by 0.17 g/kg, decreased in 1995, and increased again from 1998 to 2000 before decreasing from 2004 to 2007 by 0.14 g/kg. The validation based on the soil data in 2000 showed that the approach generated an RMSE of 5.97 g/kg, accounting for 22.77% of the average SOM content of the data. The study concluded that this approach could be used for monitoring SOM content and other soil properties. This approach had a relatively better accuracy than a previous study using the Integrated Nested Laplace Approximation with the Stochastic Partial Differential Equation approach with the same soil data but 4 times more samples.

Mapping the timeliness of routine childhood vaccination in The Gambia: A spatial modelling study

IntroductionTimeliness of routine vaccination shapes childhood infection risk and thus is an important public health metric. Estimates of indicators of the timeliness of vaccination are usually produced at the national or regional level, which may conceal epidemiologically relevant local heterogeneities and makeitdifficultto identify pockets of vulnerabilities that could benefit from targeted interventions. Here, we demonstrate the utility of geospatial modelling techniques in generating high-resolution maps of the prevalence of delayed childhood vaccination in The Gambia. To guide local immunisation policy and prioritize key interventions, we also identified the districts with a combination of high estimated prevalence and a significant population of affected infants. MethodsWe used the birth dose of the hepatitis-B vaccine (HepB0), third-dose of the pentavalent vaccine (PENTA3), and the first dose of measles-containing vaccine (MCV1) as examples to map delayed vaccination nationally at a resolution of 1 × 1-km2 pixel. We utilized cluster-level childhood vaccination data from The Gambia 2019–20 Demographic and Health Survey. We adopted a fully Bayesian geostatistical model incorporating publicly available geospatial covariates to aid predictive accuracy. The model was implemented using the integrated nested Laplace approximation—stochastic partial differential equation (INLA-SPDE) approach. ResultsWe found significant subnational heterogeneity in delayed HepB0, PENTA3 and MCV1 vaccinations. Specificdistricts in the central and eastern regions of The Gambia consistentlyexhibited the highest prevalence of delayed vaccination, while the coastal districts showed alower prevalence forallthree vaccines. We also found that districts in the eastern, central, as well as in coastal parts of The Gambia had a combination of high estimated prevalence of delayed HepB0, PENTA3 and MCV1 and a significant population of affected infants. ConclusionsOur approach provides decision-makers with a valuable tool to better understand local patterns of untimely childhood vaccination and identify districts where strengthening vaccine delivery systems could have the greatest impact.

Covariance–Based Rational Approximations of Fractional SPDEs for Computationally Efficient Bayesian Inference

ABSTRACT The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field u on R d as the solution of an elliptic SPDE L β u = W where L is a second-order differential operator, 2 β ∈ N is a positive parameter that controls the smoothness of u and W is Gaussian white noise. A few approaches have been suggested in the literature to extend the approach to allow for any smoothness parameter satisfying β > d / 4 . Even though those approaches work well for simulating SPDEs with general smoothness, they are less suitable for Bayesian inference since they do not provide approximations which are Gaussian Markov random fields (GMRFs) as in the original SPDE approach. We address this issue by proposing a new method based on approximating the covariance operator L − 2 β of the Gaussian field u by a finite element method combined with a rational approximation of the fractional power. This results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference. A rigorous convergence analysis of the method is performed and the accuracy of the method is investigated with simulated data. Finally, we illustrate the approach and corresponding implementation in the R package rSPDE via an application to precipitation data which is analyzed by combining the rSPDE package with the R-INLA software for full Bayesian inference. Supplementary materials for this article are available online.

A continuous-time Markov chain and stochastic differential equations approach for modeling malaria propagation

Malaria is the world’s most fatal and challenging parasitic disease, caused by the Plasmodium parasite and transmitted to humans by the bites of infected female mosquitos. This study presents a stochastic process representing the evolution in time of an unexpected phenomenon. We use different probability techniques to assume the possible outcome in a Malaria model since state variables or parameters are random in a stochastic model. This study considers a deterministic vector-host malaria model and converts this model into the corresponding stochastic model with a Continuous-time Markov chain (CTMC) and Stochastic differential equation (SDE). We prove the positivity and boundedness of solutions and calculate the equilibrium points. The threshold values are evaluated using different approaches. Moreover, the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) procedures are used to analyze the sensitivity of each model parameter. Finally, a transient numerical simulation is discussed, and their outcomes strongly correlate with the actual scenario.

Stochastic Differential Equation-based Mixed Effects Model of the Fluid Volume in the Fasted Stomach in Healthy Adult Human.

The rate and extent of drug dissolution and absorption from a solid oral dosage form depend largely on the fluid volume along the gastrointestinal tract. Hence, a model built upon the gastric fluid volume profiles can help to predict drug dissolution and subsequent absorption. To capture the great inter- and intra-individual variability (IAV) of the gastric fluid volume in fasted human, a stochastic differential equation (SDE)-based mixed effects model was developed and compared with the ordinary differential equation (ODE)-based model. Twelve fasted healthy adult subjects were enrolled and had their gastric fluid volume measured before and after consumption of 240mL of water at pre-determined intervals for up to 2hours postingestion. The SDE- and ODE-based mixed effects models were implemented and compared using extended Kalman filter algorithm via NONMEM. The SDE approach greatly improved the goodness of fit compared with the ODE counterpart. The proportional and additive measurement error of the final SDE model decreased from 14.4 to 4.10% and from 17.6 to 4.74mL, respectively. The SDE-based mixed effects model successfully characterized the gastric volume profiles in the fasted healthy subjects, and provided a robust approximation of the physiological parameters in the very dynamic system. The remarkable IAV could be further separated into system dynamics terms and measurement error terms in the SDE model instead of only empirically attributing IAV to measurement errors in the traditional ODE method. The system dynamics were best captured by the random fluctuations of gastric emptying coefficient Kge.

Mean–variance portfolio selection under no-shorting rules: A BSDE approach

This paper revisits the mean–variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To relax the strong condition in Li et al. (Li et al. 2002), the above issue is formulated as a stochastic recursive optimal linear–quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known “completion of squares” no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton–Jacobi–Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean–variance portfolio optimization problem.

Robust optimal asset-liability management under square-root factor processes and model ambiguity: a BSDE approach

This article studies robust optimal asset-liability management problems for an ambiguity-averse manager in a possibly non-Markovian environment with stochastic investment opportunities. The manager has access to one risk-free asset and one risky asset in a financial market. The market price of risk relies on a stochastic factor process satisfying an affine-form, square-root, Markovian model, whereas the risky asset’s return rate and volatility are potentially given by general non-Markovian, unbounded stochastic processes. This financial framework includes, but is not limited to, the constant elasticity of variance (CEV) model, the family of 4/2 stochastic volatility models, and some path-dependent non-Markovian models, as exceptional cases. As opposed to most of the papers using the Hamilton-Jacobi-Bellman-Issacs (HJBI) equation to deal with model ambiguity in the Markovian cases, we address the non-Markovian case by proposing a backward stochastic differential equation (BSDE) approach. By solving the associated BSDEs explicitly, we derive, in closed form, the robust optimal controls and robust optimal value functions for power and exponential utility, respectively. In addition, analytical solutions to some particular cases of our model are provided. Finally, the effects of model ambiguity and market parameters on the robust optimal investment strategies are illustrated under the CEV model and 4/2 model with numerical examples.

Modelling height growth of temperate mixedwood forests using an age-independent approach and multi-temporal airborne laser scanning data

Forest inventories provide information regarding the status of a range of attributes as well as enabling predictive applications. Growth and yield models are essential tools for sustainable forest management, importantly enabling projections of future forest conditions (such as height growth). To select the most appropriate growth trajectory, site index models are commonly used to quantify the productivity of a given site. However, applying these methods to more complex, multi-species, and multi-age forests can be challenging due to deviations from the assumptions made for even-aged stands. In this study, we provide a comprehensive indicator of site quality for more complex and irregular stand structures by developing age-independent height growth models for various forest types. We used multi-temporal airborne laser scanning (ALS) data from 2005, 2012, and 2018 in the Great Lakes–St. Lawrence forest region in southern Ontario, Canada. The stochastic differential equations approach was used to develop age-independent height models and a height growth rate index as a proxy of site quality from ALS-derived height metrics. We evaluated the sensitivity of the models using two different modelling approaches and found that the model that incorporated data from both periods (i.e., 2005–2012 and 2012–2018) generally provided the lower root mean square error (RMSE) value for most forest types. Overall, our results showed good agreement between the model predictions of top height and observed top height in 2018 from field plots for all forest types. We demonstrated the use of these models by creating a system of height growth curves for each forest type and producing a map of site quality for a mixedwood forest (∼10,000 ha) at a spatial resolution of 25 m. The approach developed herein leverages the accurate, spatially detailed characterization of canopy heights afforded by ALS data and is independent of stand age, which is challenging to measure accurately and is typically not available at a spatial resolution that is commensurate with the ALS data. Additionally, the demonstrated approach can be adapted to other data sources that accurately capture canopy heights (i.e., digital aerial photogrammetric or DAP), thereby increasing the possible geographic extent of height growth estimates.

A new hybrid approach for nonlinear stochastic differential equations driven by multifractional Gaussian noise

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems.

Learning Robust Point-to-Point Motions Adversarially: A Stochastic Differential Equation Approach

This letter proposes a robust stochastic differential equation approach for learning point-to-point motions in an adversarial way. The proposed stochastic dynamical model combines the advantages of the stochastic differential equation and the transformer-like function together to achieve both robustness and accuracy of the learning. The adversarial training method is proposed to simplify the way of updating the parameters of the model. The state of the proposed stochastic dynamical system is mathematically proved to converge asymptotically in the mean square sense, and it has been experimentally validated on the LASA dataset and by the trajectory-programming task of the Franka Emika robot. The experimental results show that: (1) the adversarial training method helps the model to achieve higher reproduction accuracy; (2) the trajectories generated by the proposed model achieve higher accuracy in both the noise-free condition (by approximately 14.9%) and the noisy condition (by approximately 17.8%) compared with the state-of-the-art methods in terms of the similarity to the demonstration; and (3) the proposed approach can learn smoother trajectories even if the observations are contaminated by noises.

Impacts of climate change on vicennial spatial behaviors of Trichiurus japonicus in the East China Sea

Understanding fish's spatial behaviors are important for conservation biology under global warming, as one of the direct responses of fish to climate change is the alterations in distribution. We chose Trichiurus japonicus, which is an ecologically and economically important species in the East China Sea (ECS), to study the vicennial responses of demersal fish to climate change. Eight environmental variables (sea bottom temperature, sea bottom salinity, chlorophyll-a concentration, sea surface height, bathymetry, distance to the coast, slope data, and Terrain ruggedness index) combined with the annual survey data were fitted using integrated nested Laplace approximations and stochastic partial differential equations approach under the Bayesian framework to investigate its distribution shifts in spring during 1999–2019. Based on the distribution model, three spatial behavior indicators (center of gravity, Inertia, and Isotropy) were calculated and correlated with six regional climate indicators (Kuroshio index, Taiwan Warm Current Water index, meridional wind, zonal wind, Pacific Decadal Oscillation, and El Niño/Southern Oscillation) to study the potential cause of its spatial behaviors. During the study periods, the biomass of T. japonicus declined gradually, with a brief increase in 2003, 2007–2008, and 2015–2017. T. japonicus preferred high salinity, warm and deep-water area. The relatively persistent hotspots of T. japonicus in spring were in the offshore waters of eastern Zhejiang Province. The change in the center of gravity during the study periods was not obvious, which implied that the distribution shifts were not necessarily vulnerable for T. japonicus in response to climate change and it has strong adaptability to the environment. We also found that strong Kuroshio Current and Taiwan Warm Current Water as well as cold phased of El Niño/Southern Oscillation were accompanied by weakened spatial aggregation of T. japonicus. Overall, for these types of fish with relatively persistent hotspots, the establishment of seasonally protected areas will contribute to the sustainable utilization of the species.

Rethinking Generalized Beta family of distributions

We approach the Generalized Beta (GB) family of distributions using a mean-reverting stochastic differential equation (SDE) for a power of the variable, whose steady-state (stationary) probability density function (PDF) is a modified GB (mGB) distribution. The SDE approach allows for a lucid explanation of Generalized Beta Prime (GB2) and Generalized Beta (GB1) limits of GB distribution and, further down, of Generalized Inverse Gamma (GIGa) and Generalized Gamma (GGa) limits, as well as describe the transition between the latter two. We provide an alternative form to the "traditional" GB PDF to underscore that a great deal of usefulness of GB distribution lies in its allowing a long-range power-law behavior to be ultimately terminated at a finite value. We derive the cumulative distribution function (CDF) of the "traditional" GB, which belongs to the family generated by the regularized beta function and is crucial for analysis of the tails of the distribution. We analyze fifty years of historical data on realized market volatility, specifically for S\&P500, as a case study of the use of GB/mGB distributions and show that its behavior is consistent with that of negative Dragon Kings.

Spatial modeling of two mosquito vectors of West Nile virus using integrated nested Laplace approximations

AbstractThe abundance of Culex restuans and Culex pipiens in relation to ecological predictors is poorly understood in regions of the United States where their ranges overlap. It is suspected that these species play different roles in spreading West Nile virus (WNV) in these regions, but few studies have modeled these species separately or accounted for spatial correlation using Bayesian methods. We used mosquito surveillance data collected by the Pennsylvania Department of Environmental Protection from 2002 to 2016 and integrated nested Laplace approximations with the stochastic partial differential equation approach to predict C. restuans and C. pipiens abundance in relation to several ecological predictors. We then made a predictive risk surface of abundance for each species at locations that were not sampled. Explanatory variables in the models included ecological variables previously described to be important predictors of the abundance of these mosquito species. Developed habitat, temperature, and precipitation were important predictor variables for the abundance of C. restuans, whereas developed habitat, snow water equivalent, and normalized difference water index were important predictor variables for the abundance of C. pipiens. The abundance of C. restuans had a negative relationship with developed habitat in contrast to C. pipiens abundance, which had a positive relationship with developed habitat. Julian date was modeled as a temporal trend for both species and showed C. restuans to be more abundant from late April through late June and C. pipiens to be more abundant from July through September. A seasonal crossover was observed between these two species on Julian day 185, 4 July. We observed different spatial patterns of abundance in the predictive risk maps of each of the species. Our results indicate that modeling the abundance of these species spatially and separately in regions where these two mosquito vectors coexist can help gain further insight into understanding the epidemiology of WNV in human and susceptible animal populations.