Stochastic Differential Equation Formulation Research Articles

Active Inference and Artificial Spin Ice: Control Processes and State Selection.

Theory and simulations are used to demonstrate implementation of a variational Bayes algorithm called "active inference" in interacting arrays of nanomagnetic elements. The algorithm requires stochastic elements, and a simplified model based on a magnetic artificial spin ice geometry is used to illustrate how nanomagnets can generate the required random dynamics. Examples of tracking and PID control are demonstrated and shown to be consistent with the original stochastic differential equation formulation of active inference. Interestingly, nonlinear response in the form of spikes and spike trains not predicted by the original theory can appear in the nanomagnet system for certain temperature regimes. A theoretical approach using a mean-field approximation for spin systems is proposed, which describes the transition to nonlinear response. Finally, the possibility to create simple magnetic arrays using realistic models is shown with micromagnetic simulations of a simple 17 element array of nanomagnets that include magnetic anisotropies, and exchange and dipolar interactions. Possible applications are simulated to illustrate how nanomagnetic arrays can be used as the stochastic element for feedback control of processes, investigation and control of magnetic state evolution, and as a method to optimize pulsed field magnetic switching protocols.

A simple model for viral decay dynamics and the distribution of infected cell life spans in SHIV-infected infant rhesus macaques

The dynamics of HIV viral load following the initiation of antiretroviral therapy is not well-described by simple, single-phase exponential decay. Several mathematical models have been proposed to describe its more complex behavior, the most popular of which is two-phase exponential decay. The underlying assumption in two-phase exponential decay is that there are two classes of infected cells with different lifespans. However, with the exception of CD4+ T cells, there is not a consensus on all of the cell types that can become productively infected, and the fit of the two-phase exponential decay to observed data from SHIV.C.CH505 infected infant rhesus macaques was relatively poor. Therefore, we propose a new model for viral decay, inspired by the Gompertz model where the decay rate itself is a dynamic variable. We modify the Gompertz model to include a linear term that modulates the decay rate. We show that this simple model performs as well as the two-phase exponential decay model on HIV and SIV data sets, and outperforms it for the infant rhesus macaque SHIV.C.CH505 infection data set. We also show that by using a stochastic differential equation formulation, the modified Gompertz model can be interpreted as being driven by a population of infected cells with a continuous distribution of cell lifespans, and estimate this distribution for the SHIV.C.CH505-infected infant rhesus macaques. Thus, we find that the dynamics of viral decay in this model of infant HIV infection and treatment may be explained by a distribution of cell lifespans, rather than two distinct cell types.

Hypoellipticity and the Mori–Zwanzig formulation of stochastic differential equations

We develop a thorough mathematical analysis of the effective Mori–Zwanzig (EMZ) equation governing the dynamics of noise-averaged observables in stochastic differential equations driven by multiplicative Gaussian white noise. Building upon recent work on hypoelliptic operators, we prove that the EMZ memory kernel and fluctuation terms converge exponentially fast in time to a unique equilibrium state that admits an explicit representation. We apply the new theoretical results to the Langevin dynamics of a high-dimensional particle system with smooth interaction potential.

BENCHOP – SLV: the BENCHmarking project in Option Pricing – Stochastic and Local Volatility problems

ABSTRACTIn the recent project BENCHOP – the BENCHmarking project in Option Pricing we found that Stochastic and Local Volatility problems were particularly challenging. Here we continue the effort by introducing a set of benchmark problems for this type of problems. Eight different methods targeted for the Stochastic Differential Equation (SDE) formulation and the Partial Differential Equation (PDE) formulation of the problem, as well as Fourier methods making use of the characteristic function, were implemented to solve these problems. Comparisons are made with respect to time to reach a certain error level in the computed solution for the different methods. The implemented Fourier method was superior to all others for the two problems where it was implemented. Generally, methods targeting the PDE formulation of the problem outperformed the methods for the SDE formulation. Among the methods for the PDE formulation the ADI method stood out as the best performing one.

Hybrid colored noise process with space-dependent switching rates.

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit (κ→0) and fast switching limit (ε→0) depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit ε,κ→0 with ε/κ^{2} fixed.

Development of a restricted state space stochastic differential equation model for bacterial growth in rich media

In the present study, bacterial growth in a rich media is analysed in a Stochastic Differential Equation (SDE) framework. It is demonstrated that the SDE formulation and smoothened state estimates provide a systematic framework for data driven model improvements, using random walk hidden states. Bacterial growth is limited by the available substrate and the inclusion of diffusion must obey this natural restriction. By inclusion of a modified logistic diffusion term it is possible to introduce a diffusion term flexible enough to capture both the growth phase and the stationary phase, while concentration is restricted to the natural state space (substrate and bacteria non-negative). The case considered is the growth of Salmonella and Enterococcus in a rich media. It is found that a hidden state is necessary to capture the lag phase of growth, and that a flexible logistic diffusion term is needed to capture the random behaviour of the growth model. Further, it is concluded that the Monod effect is not needed to capture the dynamics of bacterial growth in the data presented.

Optimal premium policy of an insurance firm: Full and partial information

Herein, we study the optimization problem faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels by selecting an appropriate premium policy. Our problem is totally new and has three distinguishable features: (1) both full and partial information cases are investigated here; (2) the state is subject to terminal constraint; (3) a forward–backward stochastic differential equation formulation is given which is more systematic and mathematically advanced. This formulation also enables us to continue further research in a generalized stochastic recursive control framework (see Duffie and Epstein (1992), El Karoui et al. (2001), etc.). The optimal premium policy with the associated optimal objective functional are completely and explicitly derived. In addition, a backward separation technique adaptive to forward–backward stochastic systems with the state constraint is presented as an efficient and convenient alternative to the traditional Wonham’s ( 1968) separation principle in our partial information setup. Some concluding remarks are also given here.

Optimal Portfolio and Consumption Policies Subject to Rishel's Important Jump Events Model: Computational Methods

At important events or announcements, there can be large changes in the value of financial portfolios. Events and their corresponding jumps can occur at random or scheduled times. However, the amplitude of the response in either case can be unpredictable or random. While the volatility of portfolios is often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling the response to important external events that significantly affect the prices of financial assets. Discontinuous jump processes are modeled by compound Poisson processes for random events or by quasi-deterministic jump processes for scheduled events. In both cases, the responses are randomly distributed and are modeled in a stochastic differential equation formulation. The objective is the maximal, expected total discounted utility of terminal wealth and instantaneous consumption. This paper was motivated by a paper by Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for more realistic computational considerations with constraints and parameter values. The problem is illustrated for a canonical risk-adverse power utility model. However, the usual explicit canonical solution is not strictly valid. Fortunately, iterations about the canonical solution result in computationally feasible approximations.

Coordinate invariance in stochastic dynamical systems

Stochastic dynamical systems have been used to model a broad range of atmospheric and oceanicphenomena. Previous work has focused on the stochastic differential equation formulation of thesesystems, has largely remained in a single coordinate system, and has highlighted the role of nonnormalityof the deterministic dynamics. Here, the coordinate independent properties of stochasticdynamical systems are studied. The properties previously attributed to non-normality, which can beremoved by a coordinate transformation, are more fundamentally seen to be coordinate-dependentmanifestations of violations of detailed balance. Systems violating detailed balance can both amplifyand rectify the random forcing.Newcoordinate-invariant measures of noise amplification are introducedand shown to achieve their lower bound when detailed balance is satisfied. Rectification results in acoherent phase space velocity which gives rise to a structured nonzero flux of all physically importantquantities such as energy and momentum. The qualitative and quantitative features of these fluxesprovide new predictions which can be used to further validate previously proposed stochastic modelsof geophysical systems.

A biomathematical model of time-delayed feedback in the human male hypothalamic-pituitary-Leydig cell axis.

We develop, implement, and test a feedback and feedforward biomathematical construct of the male hypothalamic [gonadotropin-releasing hormone (GnRH)]-pituitary [luteinizing hormone (LH)]-gonadal [testosterone (Te)] axis. This stochastic differential equation formulation consists of a nonstationary stochastic point process responsible for generating episodic release of GnRH, which is modulated negatively by short-loop (GnRH) and long-loop (Te) feedback. Pulsatile GnRH release in turn drives bursts of LH secretion via an agonistic dose-response curve that is partially damped by Te negative feedback. Circulating LH stimulates (feedforward) Te synthesis and release by a second dose response. Te acts via negative dose-responsive feedback on GnRH and LH output, thus fulfilling conditions of a closed-loop control system. Four computer simulations document expected feedback performance, as published earlier for the human male GnRH-LH-Te axis. Six other simulations test distinct within-model coupling mechanisms to link a circadian modulatory input to a pulsatile control node so as to explicate the known 24-h variations in Te and, to a lesser extent, LH. We conclude that relevant dynamic function, internodal dose-dependent regulatory connections, and within-system time-delayed coupling together provide a biomathematical basis for a nonlinear feedback-feedforward control model with combined pulsatile and circadian features that closely emulate the measurable output activities of the male hypothalamic-pituitary-Leydig cell axis.

Forest height growth modelling

There are many functional forms which may be used to model height against age, and a variety of methods by which the model parameters may be estimated. There are four main questions that the forest height modeller needs to decide upon. 1. (1) Should each height-age trajectory be modelled separately? 2. (2) What error structure model should be adopted? 3. (3) How should the mortality process be treated? 4. (4) Should measurement errors be taken into account? Possible answers to these questions are considered in the context of the much used Chapman-Richards family of growth models. The appropriate method of estimation follows from the choices that the modeller makes in relation to these questions. Most published work on height growth modelling are regression based, but do not explicitly model error structures and use the corresponding fitting criterion. In contrast, there is the approach which takes growth trajectories as realisations of a stochastic process, with the error structure to be used in the fitting process arising naturally out of the stochastic formulation. Two (essentially equivalent) approaches to stochastic growth curve modelling are reviewed; a diffusion model formulation, and a stochastic differential equation formulation, in which the expected growth curve is the integrated form of the Chapman-Richards growth model. A simpler formulation is used to obtain a descriptive representation of a complete set of height growth trajectories obtained from Sitka spruce sample plots. The simple model adopted ‘overcomes’ the complications of correlated errors that traditionally arise in the analysis of longitudinal data but avoids the sophisticated model specification of the stochastic models.

Likelihood methods and the analysis of fatigue crack growth

Models based on a stochastic differential equation formulation of the fatigue crack growth problem are shown to give fit data well. Moreover, the models show how data may be pooled and how the effects of conditions on crack growth rates may be measured.