Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

Necessary and Sufficient Condition for Oscillations of Neutral Differential Equation

2000 Mathematics Subject Classification: 34K15, 34C10.

Symbolic Solving of Partial Differential Equation Systems and Compatibility Conditions

An algorithm is produced for the symbolic solving of systems of partial differential equations by means of multivariate Laplace–Carson transform. A system of K equations with M as the greatest order of partial derivatives and right-hand parts of a special type is considered. Initial conditions are input. As a result of a Laplace–Carson transform of the system according to initial condition we obtain an algebraic system of equations. A method to obtain compatibility conditions is discussed.

An impulsive differential equation model for Marek's disease

Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn. In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model. Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution. Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.

Transcriptional regulation os phenylalaline biosynthesis and utilization

Conifer trees divert large quantities of carbon into the biosynthesis of phenylpropanoids, particularly to generate lignin, an important constituent of wood. Since phenylalanine is the precursor for phenylpropanoid biosynthesis, the precise regulation of phenylalanine synthesis and utilization should occur simultaneously. This crucial pathway is finely regulated primarily at the transcriptional level. Transcriptome analyses indicate that the transcription factors (TFs) preferentially expressed during wood formation in plants belong to the MYB and NAC families. Craven-Bartle et al. (2013) have shown in conifers that Myb8 is a candidate regulator of key genes in phenylalanine biosynthesis involved in the supply of the phenylpropane carbon skeleton necessary for lignin biosynthesis. This TF is able to bind AC elements present in the promoter regions of these genes to activate transcription. Constitutive overexpression of Myb8 in white spruce increased secondary-wall thickening and led to ectopic lignin deposition (Bomal et al. 2008). In Arabidopsis, the transcriptional network controlling secondary cell wall involves NAC-domain regulators operating upstream Myb transcription factors. Functional orthologues of members of this network described have been identified in poplar and eucalyptus, but in conifers functional evidence had only been obtained for MYBs. We have identified in the P. pinaster genome 37 genes encoding NAC proteins, which 3 NAC proteins could be potential candidates to be involved in vascular development (Pascual et al. 2015). The understanding of the transcriptional regulatory network associated to phenylpropanoids and lignin biosynthesis in conifers is crucial for future applications in tree improvement and sustainable forest management. This work is supported by the projects BIO2012-33797, BIO2015-69285-R and BIO-474 References: Bomal C, et al. (2008) Involvement of Pinus taeda MYB1 and MYB8 in phenylpropanoid metabolism and secondary cell wall biogenesis: a comparative in planta analysis. J Exp Bot. 59: 3925-3939. Craven-Bartle B, et al. (2013) A Myb transcription factor regulates genes of the phenylalanine pathway in maritime pine. Plant J, 74: 755-766. Pascual MB, et al. (2015) The NAC transcription factor family in maritime pine (Pinus pinaster): molecular regulation of two genes involved in stress responses. BMC Plant Biol, 15: 254.

Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).

Atom lasers, coherent states, and coherence. I. Physically realizable ensembles of pure states

A laser, be it an optical laser or an atom laser, is an open quantum system that produces a coherent beam of bosons (photons or atoms, respectively). Far above threshold, the stationary state rho(ss) of the laser mode is a mixture of coherent-field states with random phase, or, equivalently, a Poissonian mixture of number states. This paper answers the question: can descriptions such as these, of rho(ss) as a stationary ensemble of pure states, be physically realized? Here physical realization is as defined previously by us [H. M. Wiseman and J. A. Vaccaro, Phys. Lett. A 250, 241 (1998)]: an ensemble of pure states for a particular system can be physically realized if, without changing the dynamics of the system, an experimenter can (in principle) know at any time that the system is in one of the pure-state members of the ensemble. Such knowledge can be obtained by monitoring the baths to which the system is coupled, provided that coupling is describable by a Markovian master equation. Using a family of master equations for the (atom) laser, we solve for the physically realizable (PR) ensembles. We find that for any finite self-energy chi of the bosons in the laser mode, the coherent-state ensemble is not PR; the closest one can come to it is an ensemble of squeezed states. This is particularly relevant for atom lasers, where the self-energy arising from elastic collisions is expected to be large. By contrast, the number-state ensemble is always PR. As the self-energy chi increases, the states in the PR ensemble closest to the coherent-state ensemble become increasingly squeezed. Nevertheless, there are values of chi for which states with well-defined coherent amplitudes are PR, even though the atom laser is not coherent (in the sense of having a Bose-degenerate output). We discuss the physical significance of this anomaly in terms of conditional coherence (and hence conditional Bose degeneracy).

Dynamic programming for a Markov-switching jump–diffusion

We consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a Markov-switching jump–diffusion stochastic differential equation. Our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems. More specifically, we provide a detailed proof of Bellman’s optimality principle (or dynamic programming principle) and obtain the corresponding Hamilton–Jacobi–Belman equation, which turns out to be a partial integro-differential equation due to the extra terms arising from the Lévy process and the Markov process. As an application of our results, we study a finite horizon consumption– investment problem for a jump–diffusion financial market consisting of one risk-free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state Markov process. We provide a detailed study of the optimal strategies for this problem, for the economically relevant families of power utilities and logarithmic utilities.

Large deviation for BSDE with subdifferential operator

In this paper we prove that the solution of a backward stochastic differential equation, which involves a subdifferential operator and associated to a family of reflecting diffusion processes, converges to the solution of a deterministic backward equation and satisfes a large deviation principle.

Unraveling the complex network of cuticular structure and function.

A hydrophobic cuticle is deposited at the outermost extracellular matrix of the epidermis in primary tissues of terrestrial plants. Besides forming a protective shield against the environment, the cuticle is potentially involved in several developmental processes during plant growth. A high degree of variation in cuticle composition and structure exists between different plant species and tissues. Lots of progress has been made recently in understanding the different steps of biosynthesis, transport, and deposition of cuticular components. However, the molecular mechanisms that underlie cuticular function remain largely elusive.

Weak approximations. A Malliavin calculus approach

We introduce a variation of the proof for weak approximations that issuitable for studying the densities of stochastic processes which areevaluations of the flow generated by a stochastic differential equation on a random variable that maybe anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations fordensities and distributions can also be achieved. We apply theseideas to the case of stochastic differential equations with boundaryconditions and the composition of two diffusions.

Slow dynamis in the 3D gonihedric model

We provide analytical evidence of stochastic resonance in polarization switching vertical-cavity surface-emitting lasers (VCSELs). We describe the VCSEL by a two-mode stochastic rate equation model and apply a multiple time-scale analysis. We were able to reduce the dynamical description to a single stochastic differential equation, which is the starting point of the analytical study of stochastic resonance. We confront our results with numerical simulations on the original rate equations, validating the use of a multiple time-scale analysis on stochastic equations as an analytical tool.