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In this work, we consider a pharmacokinetic (PK) model with first-order drug absorption and first-order elimination that represent the concentration of drugs in the body, including both the absorption and elimination parts, and we also add a random factor to describe the variability between patients and the environment. Using Itô’s lemma and the Laplace transform, we obtain the solutions in integral form for a single and constant dosage regimen in time. Moreover, formulas for the expected value and the variance for each case of study are presented, which allows the statistical assessment of the proposed models, as well as predicting the ideal path of drug concentration and its uncertainty. These results are important in the long-term analysis of drug concentration and the persistence of therapeutic level. Further, a numerical method for the solution of the stochastic differential equation (SDE) is introduced and developed.

We study multiplicative SDEs perturbed by an independent additive fractional Brownian motion. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients. (c) 2022 Published by Elsevier B.V.

In this work, we adapt the {\em micro-macro} methodology to stochastic differential equations for the purpose of numerically solving oscillatory evolution equations. The models we consider are addressed in a wide spectrum of regimes where oscillations may be slow or fast. We show that through an ad-hoc transformation (the micro-macro decomposition), it is possible to retain the usual orders of convergence of Euler-Maruyama method, that is to say, uniform weak order one and uniform strong order one half. We also show that the same orders of uniform accuracy can be achieved by a simple integral scheme. The advantage of the micro-macro scheme is that, in contrast to the integral scheme, it can be generalized to higher order methods.

Piezoelectric energy harvesters are electromechanical systems, capable to convert ambient dispersed mechanical vibrations into usable electrical energy. They can be used for supplying power to sensors and actuators that are wireless connected, miniaturized and remote located. In this work, we analyze piezoelectric energy harvesters for mechanical vibrations in the frequency domain. White Gaussian and colored noise models for random vibrations are considered. The governing equations for the harvester are derived from mechanical properties, the characteristic relationships of piezoelectric materials, and circuit description of the electrical load. We show that the energy harvester can be modelled by cascade connected electromechanical two-ports, and that frequency domain methods are the perfect tool for analysis. Formulas for the harvested power and power efficiency are derived. We also show that application of matching networks reduces the impedance mismatch between the mechanical and the electrical parts, significantly increasing the harvested power and power efficiency. The matching network solution is compared to others, previously proposed solutions, such as application of power-factor correction. We show that the matching network offers nine times more average power and better power efficiency than the unmatched resistive load, and increases by more than 10% the harvested power and efficiency, with respect to the power-factor corrected solution.

We apply a class of stochastic differential equations to model individual growth in a randomly fluctuating environment using cattle weight data. We have used maximum likelihood theory to estimate the parameters. However, for cattle data, it is often not feasible to obtain animal's observations at equally spaced ages nor even at the same ages for different animals and there is typically a small number of observations at older ages. For these reasons, maximum likelihood estimates can be quite inaccurate, being interesting to consider in the likelihood function a weight function associated to the elapsed times between two consecutive observations of each animal, which results in the weighted maximum likelihood method. We compare the results obtained from both methods in several data structures and conclude that the weighted maximum likelihood improves the estimation when observations at older ages are scarce and the observation instants are unequally spaced, whereas the maximum likelihood estimates are recommended when animals are weighted at equally spaced ages. For unequally spaced observations, a bootstrap estimation method was also applied to correct the bias of the maximum likelihood estimates; it revealed to be a more precise alternative, except when the available data only has young animals. info:eu-repo/semantics/acceptedVersion

In recent years, the increasing life expectancy of the world’s population, due to increased availability of prescribed medication, quality of health care services, quantity of health care institutions and quality of life, combined with a sharp decrease in birth rates over time, has proven to be a challenging problem for governments worldwide (particularly in developed countries). Both of these factors put at risk the sustainability of state-funded welfare programs (e.g., social security) and also lead to a decrease in available workforce and tax revenue (including social benefit contributions) in the near future. With the tendency for these problems to worsen in the next decades (severity varies between countries), it is of paramount importance to estimate the extension of human life in order to analyse the severity of this phenomenon. Stochastic differential equations have been used recently to model the evolution of death rates. In fact, such models have some advantages when compared to the deterministic ones since we can input random environmental fluctuations and evaluate the uncertainty in forecasts. The main goal of this paper is to apply and compare stochastic differential equations death rate models separately for each age and sex and forecast Portuguese death rates until the year 2030. info:eu-repo/semantics/publishedVersion

National audience

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs.

Markov random fields (MRFs) are well studied during the past 50 years. Their success are mainly due to their flexibility and to the fact that they gives raise to stochastic image models. In this work, we will consider a stochastic differential equation (SDE) driven by Lévy noise. We will show that the solution $ X_v $ of the SDE is a MRF satisfying the Markov property. We will prove that the Gibbs distribution of the process $ X_v $ can be represented graphically through Feynman graphs, which are defined as a set of cliques, then we will provide applications of MRFs in image processing where the image intensity at a particular location depends only on a neighborhood of pixels.

We analyze frictional motion for a laboratory fault as it passes through the stability transition from stable sliding to unstable motion. We study frictional stick-slip events, which are the lab equivalent of earthquakes, via dynamical system tools in order to retrieve information on the underlying dynamics and to assess whether there are dynamical changes associated with the transition from stable to unstable motion. We find that the lab seismic cycles exhibit characteristics of a low-dimensional system with average dimension similar to that of natural slow earthquakes (10, indicating that some regions of the phase space require a high number of degrees of freedom (dofs). Our analysis does not preclude deterministic chaos, but the lab seismic cycle is best explained by a random attractor based on rate- and state-dependent friction whose dynamics is stochastically perturbed. We find that minimal variations of 0.05% of the shear and normal stresses applied to the experimental fault influence the large-scale dynamics and the recurrence time of labquakes. While complicated motion including period doubling is observed near the stability transition, even in the fully unstable regime we do not observe truly periodic behavior. Friction's nonlinear nature amplifies small scale perturbations, reducing the predictability of the otherwise periodic macroscopic dynamics. As applied to tectonic faults, our results imply that even small stress field fluctuations (less or about 150 kPa) can induce coefficient of variations in earthquake repeat time of a few percent. Moreover, these perturbations can drive an otherwise fast-slipping fault, close to the critical stability condition, into a mixed behavior involving slow and fast ruptures.

Incompressible fluids at high Reynolds number quickly transition to turbulence. This implies that precise predictions for many flows in the physical world are extremely difficult - if not outright impossible. The field of statistical fluid mechanics aims to mitigate this difficulty by studying averaged quantities associated with turbulent flows. In the mathematical physics literature, turbulent flow is often described in the language of stochastic partial differential equations. The appropriate models are stochastic perturbations of well-known deterministic equations of fluid mechanics, so that the apparent randomness of turbulent flow is modeled via the tools of stochastic analysis. In the first part of this dissertation, this point of view of stochastic fluid mechanics is employed. We focus on the three dimensional case, with the goal of obtaining a conditional theorem for Kolmogorov's celebrated 4/5 law to hold in the presence of boundaries. The dimensionality enforces the use of a weak notion of solution to our model, in particular we work with families of \say{stationary martingale solutions} to the stochastic Navier-Stokes equations parametrized by the inverse Reynolds number. The main result of the first part of this dissertation provides a sufficient condition for a local version of the 4/5 law in the limit of infinite Reynolds number. In the second part of the dissertation, the focus is shifted to kinetic theory. Kinetic equations have played a prominent role in statistical mechanics since the 19th century; typically, the kinetic viewpoint represents an intermediate step of coarse-graining between the particle level, governed by Newtonian or Hamiltonian mechanics, and the hydrodynamic level, governed by continuum or fluid mechanics. The solution of a kinetic equation represents the normalized phase-space density of a large number of particles which might be interacting and potentially diffusing. The evolution of the density of an ensemble of particles interacting electrostatically is modeled by the Vlasov-Poisson equation. Thermal noise on the particles is modeled by the inclusion of a kinetic Fokker-Planck term. To incorporate the effect of macroscopic fluctuating force fields into kinetic modeling, we perturb the Vlasov-Poisson-Fokker-Planck equation by a stochastic kinetic transport term. We modify and exploit a popular scheme of stochastic fluid mechanics relying on the Gy\"ongy-Krylov lemma and construct local strong solutions to the stochastic Vlasov-Poisson-Fokker-Planck equation.

In this paper we are interested in fractional stochactic differential equations (SDEs) with a soft wall. What do we mean by such a type of equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now, by introducing repulsion instead of reflection, one obtains an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force.

We present a statistical inference approach to estimate the frequency noise characteristics of ultra-narrow linewidth lasers from delayed self-heterodyne beat note measurements using Bayesian inference. Particular emphasis is on estimation of the intrinsic (Lorentzian) laser linewidth. The approach is based on a statistical model of the measurement process, taking into account the effects of the interferometer as well as the detector noise. Our method therefore yields accurate results even when the intrinsic linewidth plateau is obscured by detector noise. The regression is performed on periodogram data in the frequency domain using a Markov-chain Monte Carlo method. By using explicit knowledge about the statistical distribution of the observed data, the method yields good results already from a single time series and does not rely on averaging over many realizations, since the information in the available data is evaluated very thoroughly. The approach is demonstrated for simulated time series data from a stochastic laser rate equation model with 1 / textitf-type non-Markovian noise.

Το θέμα της παρούσας εργασίας είναι η μελέτη βασικών στοχαστικών διαφορικών εξισώσεων, η επίλυση και οι εφαρμογές τις. Αρχικά, αναφέρονται χρήσιμες έννοιες για τις στοχαστικές διαδικασίες και ορίζονται οι διαδικασίες martingales και η κίνηση Brown. Στη συνέχεια γίνεται κατασκευή και γενίκευση του ολοκληρώματος του Ito, δίνεται ο τύπος του Ito και ορίζεται το ολοκλήρωμα του Stratonovich. Έπειτα, αναπτύσσεται η βασική θεωρία των στοχαστικών διαφορικών εξισώσεων και σημειώνεται η περίπτωση των γραμμικών. Επιπλέον, μελετώνται οι στατιστικές ιδιότητες των λύσεων και οι μέθοδοι εκτίμησης των παραμέτρων. Ακόμη, αναφέρονται παραδείγματα εξισώσεων Langevin, στοιχεία αρμονικής ανάλυσης του μοντέλου Langevin και η στοχαστική σταθερότητα λύσης. Τέλος περιγράφονται αριθμητικές μέθοδοι με εφαρμογές στα Χρηματοοικονομικά και στην Επιδημιολογία., The subject of this thesis is the study of basic stochastic differential equations, their solution and applications. First, useful concepts for stochastic processes are mentioned and martingales processes and Brownian motion are defined. Then the Ito integral is constructed and generalized, the Ito formula is given and the Stratonovich integral is defined. Then, the basic theory of stochastic differential equations is developed and the case of linear ones is noted. In addition, the statistical properties of the solutions and the parameter estimation methods are studied. Also, examples of Langevin equations, elements of harmonic analysis of the Langevin model, and stochastic solution stability are reported. Finally numerical methods with applications in Finance and Epidemiology are described.

Single-cell RNA sequencing, which quantifies cell transcriptomes, has seen widespread adoption, accompanied by proliferation of analysis methods. However, there has been relatively little systematic investigation of its best practices and their underlying assumptions, leading to challenges and discrepancies in interpretation. I present a set of generic, principled strategies for modeling the biological and technical components of sequencing experiments and use case studies to motivate their application to sequencing data.

We propose new numerical methods with adding a modified ordinary differential equation solver to the Milstein methods for solution of stiff stochastic systems. We study a general form of stochastic differential equations so that the Ginzburg-Landau equation and the Davis-Skodje model can be considered as special states of them. The efficiency of the method is experimented, in terms of the convergence rate and accuracy of approximate solution, employing some numerical examples, including stochastic Ginzburg-Landau equation and a paradigm of chemical reaction systems.

International audience; We consider the filtering problem of estimating the state of a continuous-time dynamical process governed by a nonlinear stochastic differential equation and observed through discrete-time measurements. As the Bayesian posterior density is difficult to compute, we use variational inference (VI) to approximate it. This is achieved by seeking the closest Gaussian density to the posterior, in the sense of the Kullback- Leibler divergence (KL). The obtained algorithm, called the continuous-discrete variational Kalman filter (CD-VKF), provides implicit formulas that solve the considered problem in closed form. Our framework avoids local linearization, and the estimation error is globally controlled at each step. We first clarify the connections between well known nonlinear Kalman filters and VI, then develop closed form approximate formulas for the CD-VKF. Our algorithm achieves state-of-the-art performances on the problem of reentry tracking of a space capsule.

Surprisingly, general estimators for nonlinear continuous time models based on stochastic differential equations are yet lacking. Most applications still use the Euler-Maruyama discretization, despite many proofs of its bias. More sophisticated methods, such as the Kessler, the Ozaki, or MCMC methods, lack a straightforward implementation and can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie-Trotter (LT) and the Strang (S) splitting schemes. We prove that S also has an $L^p$ convergence rate of order 1, which was already known for LT. We prove under the less restrictive one-sided Lipschitz assumption that the estimators are consistent and asymptotically normal. A numerical study on the 3-dimensional stochastic Lorenz chaotic system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on both precision and computational speed compared to the state-of-the-art.

This thesis comprises two different collection of results. We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô type that depends on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games without or with common noise respectively. Second part studies several portfolio management problems featuring many-player and mean-field competition and relative performance concerns under the forward performance processes (FPP) framework. In the first problem, we focus on agents using power (CRRA) type FPPs for their investment-consumption optimisation problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin “market-risk relative consumption preference”. The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle their risk-tolerance and elasticity of intertemporal substitution (EIS). This, in turn, allows a finer analysis on the agent’s consumption “income” and “substitution” regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent’s perception of consumption in a non-trivial manner in addition to a time-dependent Elasticity of Conformity of the agent to the market-risk relative consumption preference. In the follow-up problem, we solve the forward-utility finite player and mean-field investment game for the agent following exponential (CARA) type FPPs. We explicitly derive best response and equilibrium strategies in the single common stock asset and the asset specialisation with common noise. As an application, we draw on the core features of the forward utility paradigm and discuss a problem of time-consistent mean-field dynamic model selection in sequential time-horizons.

We characterize the cumulative distribution functions and copulas of two-dimensional self-similar Ito processes, with randomly correlated Wiener margins, as solutions of certain elliptic partial differential equations.

Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced by Ditlevsen and Löcherbach (Stoch. Process. Appl., 2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. In this paper, first, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity, and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.

This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations.

In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.

Mestrado Bolonha em Mathematical Finance Harvesting is a primary and fundamental activity impacting either on human society as a food supplier and on the environments as a resource consumer. An equilibrium is required to keep environments healthy and the activity attractive. Harvesting policies must aim to achieve an equilibrium through the application of a sustainable quote driven by the effort. Effort, in the fishing case, is the action plan measured by the number of vessels, number of workers, hours spent on sea, among other factors, which impose low flexibility to organisations due to heavy costs associated with a constantly changing effort. Hence, highly volatile effort policies are inapplicable but used to guide profit for comparison purposes. This project is related to the obtention of an optimal variable effort through the numerical solution of a non-linear Partial Differential Equation. Optimal variable effort leads to high profits, setted as reference, and used to study the feasibility of different strategies by comparing the outputs. A penalised effort strategy was applied to find and evaluate effort’s new behaviour and associated profit, comparing these values with the reference. Impact on profit depends on a strength parameter, ε, that penalises high effort changes. The higher (lower) the value of ε, the higher (lower) the impact on effort and, consequently, the lower (higher) the profit. Penalised effort does not oscillates as sharply as the non-penalised, verifying a reduction on oscillations’ magnitude but remaining constantly adjustments on effort. Penalised effort removes social issues imposed by low or null efforts, leaving the logistic problem from frequent oscillations to be partially solved. These tasks are studied on a research project that uses these results for further comparisons and conclusions. As atividades do setor primário são críticas, impactando na sociedade quer pela sua função de fornecedor de alimentos, quer pelas consequências da sua atividade enquanto consumidor de recursos dos meios em que operam. O equilíbrio é fundamental para manter os ambientes saudáveis e as atividades atrativas. Políticas de captura devem procurar atingir o equilíbrio através da aplicação de quotas sustentáveis guiadas pelo esforço. No caso da pesca, o esforço é medido em número de embarcações, número de trabalhadores, horas passadas no mar, entre outros factores, que impõem baixa flexibilidade a organizações com elevados custos associados com a alteração constante do nível de esforço. Uma política com alta taxa de volatilidade associada é inaplicável, mas os valores de lucro são utilizados para comparação de políticas. Este projeto está relacionado com a obtenção de políticas ótimas de esforço variável através da resolução numérica de uma equação às derivadas parciais não linear. O esforço ótimo variável conduz ao lucro mais elevado dadas as restrições, que é considerado a referência no estudo da viabilidade das diferentes estratégias. Estratégias de esforço penalizadas foram aplicadas com o objetivo de encontrar e avaliar o comportamento do esforço a estas inerente, comparando-o com os valores de referência. O impacto no lucro depende do parâmetro ε, que penaliza grandes variações de esforço. Quanto maior (menor) o valor de ε, maior (menor) o impacto no esforço e, por consequência, menor (maior) o lucro. O esforço penalizado não oscila tanto quanto o não penalizado, verificando-se uma redução na magnitude da oscilação, mantendo-se os ajustes frequentes. O esforço penalizado remove os problemas sociais causados por períodos de baixo ou nulo esforço, não resolvendo integralmente o problema logístico gerado pelos frequentes ajustes. Este trabalho está integrado num projeto de investigação que utilizará estes resultados em estudos futuros. info:eu-repo/semantics/publishedVersion

Mestrado Bolonha em Actuarial Science In recent years, the increasing life expectancy of the world population (which is most commonly seen in first-world countries), due to increased availability to prescribed medication, quality of health care services and quantity of health care institutions, combined with a sharp decrease in birth rates along time has proven to be a challenging problem for governments worldwide. Both of these factors put at risk the sustainability of state funded welfare programs (e.g. social security) and also lead to a decrease in productivity, available workforce and tax revenue in the near future. With the tendency for these problems to worsen in the next decades, it is of paramount importance to estimate the extension of human life in order to analyse the severity of this phenomena. Stochastic differential equations have been used recently to model the evolution of death rates. In fact, such models have some advantages compared to the deterministic ones since we can input random environmental fluctuations and evaluate the uncertainty in forecasts. Instead of the usual cohort analysis, we propose a cross-sectional analysis of mortality by applying stochastic differential equations models, which we wish to model to the Portuguese population, describing death rates trends for all ages and for both genders. The main goal of this work is to apply and compare stochastic differential equations death rates models (Geometric Brownian motion, Stochastic Gompertz model) separately for each age and gender with independent standard Wiener processes and forecast Portuguese death rates until the year 2030. Nos últimos anos, o aumento da esperança média de vida da população mundial (que é habitualmente observada em países de primeiro mundo), devido ao aumento do acesso a medicamentos com receita médica, qualidade dos serviços relacionados com cuidados de saúde e quantidade de instituições de saúde, em conjunto com um forte decréscimo registado nas taxas de natalidade ao longo do tempo provou ser um problema desafiante para governos de todo o mundo. Estes fatores colocam em risco a sustentabilidade dos programas de assistência social financiados pelo Estado (como por exemplo a segurança social) e podem causar uma descida na produtividade, mão-de obra disponível e receita fiscal no futuro próximo. Com a tendência destes problemas agravarem-se nas próximas décadas, é da maior importância estimar o prolongamento da vida humana com a finalidade de analisar a gravidade dos fenómenos referidos. As equações diferenciais estocásticas têm vindo a ser usadas recentemente para modelar a evolução de taxas de mortalidade. De facto, este tipo de modelos apresentam algumas vantagens quando comparados aos modelos determinísticos, visto que podemos introduzir flutuações ambientais aleatórias e avaliar a incerteza nas previsões. Em vez da habitual análise por coorte, propomos uma análise transversal da mortalidade através da aplicação de modelos de equações diferenciais estocásticas, que desejamos modelar para a população portuguesa, descrevendo as tendências das taxas de mortalidade para todas as idades e para ambos os géneros. O principal objectivo deste trabalho ´e aplicar e comparar modelos de equaçõess diferenciais estocásticas de taxas de mortalidade (movimento Browniano Geom´etrico, modelo de Gompertz estoc´astico) separadamente para cada idade e género com processos de Wiener padrão independentes e efetuar as previsões das taxas de mortalidade portuguesas até ao ano 2030. info:eu-repo/semantics/publishedVersion

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms., 38 pages, 10 figures

We describe the growth dynamics of a stock using stochastic differential equations with a generalized logistic growth model which encompasses several well-known growth functions as special cases. For each model, we compute the optimal variable effort policy and compare the expected net present value of the total profit earned by the harvester among policies. In addition, we further extend the study to include parameters sensitivity, such as the costs and volatility, and present an explicitly Crank–Nicolson discretization scheme necessary to obtain optimal policies.

In this work, we study the controllability of some Stochastic Differential Equations. First, we show some basic controllability results for this type of equations, such as the fact that stochastic evolution equations are not controllable when the control only acts on the deterministic part of the equation, thus requiring the application of a control in its stochastic part. Next, we analyze the controllability of two types of stochastic equations, namely the transport and heat equations. Using an observability estimate for reverse stochastic transport equations, we show that, for a sufficiently large time T > 0, the stochastic transport equation is exactly controllable. For the stochastic heat equation, we prove a global Carleman estimate for reverse stochastic parabolic equations and use this to obtain a unique continuation property and an observability estimate, which allow us to conclude the null and approximate controllability of the stochastic heat equation for any time T > 0. Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES Neste trabalho, faremos um estudo da controlabilidade de algumas equações diferenciais estocásticas. Primeiramente, mostramos alguns resultados básicos de controlabilidade para este tipo de equações, como por exemplo o fato que equações de evolução estocásticas não são controláveis quando o controle só atua na parte determinista da equação, sendo necessário assim a aplicação de um controle também na sua parte estocástica. Em seguida, analisaremos a controlabilidade de dois tipos de equações estocásticas, a saber, as equações de transporte e do calor. Usando uma estimativa de observabilidade para equações de transporte estocásticas reversas, mostramos que, para um tempo T > 0 suficientemente grande a equação de transporte estocástica é exatamente controlável. Para a equação do calor estocástica, provaremos uma estimativa global de Carleman para equações parabólicas estocásticas reversas, e usaremos isto para obter uma propriedade de continuação única e uma estimativa de observabilidade, as quais nos permite concluir a controlabilidade nula e aproximada da equação do calor estocástica para qualquer tempo T > 0.

The one-dimensional Kardar-Parisi-Zhang (KPZ) equation is becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening, despite previous expectations for systems with local interactions and time-dependent noise and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation., 9 pages, 8 figures, Physical Review E (in press, 2022)

Let X X be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of X X satisfies Talagrand’s inequality with respect to the uniform distance.

Актуальной проблемой развития и повышения конкурентоспособности нефтегазовой отрасли является решение задачи надежного и эффективного электроснабжения технологических объектов новых месторождений и магистральных нефтегазопроводов, территориально удаленных от центральной электрической сети. Одним из эффективных способов решения данной проблемы является применение ветроэнергетических установок. Первоочередной задачей проектирования и технико-экономического обоснования применения ветроэнергетических установок является прогноз изменения скоростей ветра в месте установки электростанции. Стохастическая природа ветра и его большая изменчивость во времени и пространстве определяют высокую сложность данной задачи, для решения которой используют методы математического моделирования. Известные модели скорости ветра на основе цепей Маркова, авторегрессии и скользящего среднего не позволяют осуществлять варьирование шага моделирования, что ограничивает возможность их применения для имитационного моделирования режимов ветроэнергетических установок и энергетических систем на их основе. В статье предложена модель скорости ветра на основе стохастического дифференциального уравнения, устраняющая данный недостаток. Цель: разработка модели скорости ветра на основе стохастического дифференциального уравнения дробного процесса Орнштейна-Уленбека c периодической функцией среднего значения, обеспечивающей моделирование статических и динамических режимов работы ветроэнергетической установки на различных временных интервалах. Методы: математическое и компьютерное моделирование с использованием программной среды MatLab/Simulink. Результаты. Предложенная модель, в отличие от известных моделей подобного типа, воспроизводит как суточные, так и сезонные вариации скорости ветра, а также долговременную корреляционную зависимость моделируемого процесса, что позволяет осуществлять моделирование рабочих режимов ветроэнергетических установок на различных временных интервалах с требуемой дискретизацией. Валидация модели выполнена с использованием данных климатических наблюдений скорости ветра из электронного архива Всероссийского института гидрометеорологической информации. Адекватность модели подтверждена хорошей степенью соответствия моделируемых траекторий скорости ветра с данными фактических наблюдений, зарегистрированных на 518 метеостанциях, расположенных на территории России. Relevance. The urgent task of the technical and economic development of the Northern and Eastern regions of Russia is to provide reliable and efficient power supply to consumers, geographically located in remote, hard-to-reach areas. The use of wind power plants is a promising way to solve this problem. The primary task of the design and feasibility study of the use of wind power plants is to predict changes in wind speeds at the site of the power plant installation. The stochastic nature of the wind and its spatiotemporal variability explain the high complexity of this problem, for the solution of which the methods of mathematical modeling are used. The known models of wind speed based on Markov chains, autoregressive moving-average and other discrete models do not allow varying the time step, which does not allow their use for simulation of operating modes of wind turbines and wind energy systems. The article proposes a model of wind speed based on the stochastic differential equation which eliminates this drawback. The aim of the study is to construct wind speed model based on fractional Ornstein-Uhlenbeck process with a periodic long-term mean, which provides modeling of static and dynamic modes of operation of a wind power plant at different time intervals. Methods: mathematical and computer modeling using Matlab/Simulink software environment. Results. The proposed model, in contrast to the known SDE-based models, is able to produce autocorrelated wind speed trajectories with long-term dependence, daily and monthly variations to perform more detailed simulation of the operating modes of wind turbines at different time intervals with the required time step. The model was tested using the data of climatic observations of wind speed obtained from electronic archive of the All-Russian Institute of Hydrometeorological Information. The model adequacy was evaluated by comparing characteristics of simulated wind speed trajectories with actual observations collected at 518 weather stations located on territory of Russia.

One of the inverse problems of dynamics in the presence of random perturbations is considered. This is the problem of the simultaneous construction of a set of first-order Ito stochastic differential equations with a given integral manifold, and a set of comparison functions. The given manifold is stable in probability with respect to these comparison functions.

Relevance. The urgent task of the technical and economic development of the Northern and Eastern regions of Russia is to provide reliable and efficient power supply to consumers, geographically located in remote, hard-to-reach areas. The use of wind power plants is a promising way to solve this problem. The primary task of the design and feasibility study of the use of wind power plants is to predict changes in wind speeds at the site of the power plant installation. The stochastic nature of the wind and its spatiotemporal variability explain the high complexity of this problem, for the solution of which the methods of mathematical modeling are used. The known models of wind speed based on Markov chains, autoregressive moving-average and other discrete models do not allow varying the time step, which does not allow their use for simulation of operating modes of wind turbines and wind energy systems. The article proposes a model of wind speed based on the stochastic differential equation which eliminates this drawback. The aim of the study is to construct wind speed model based on fractional Ornstein–Uhlenbeck process with a periodic long-term mean, which provides modeling of static and dynamic modes of operation of a wind power plant at different time intervals. Methods: mathematical and computer modeling using Matlab/Simulink software environment. Results. The proposed model, in contrast to the known SDE-based models, is able to produce autocorrelated wind speed trajectories with long-term dependence, daily and monthly variations to perform more detailed simulation of the operating modes of wind turbines at different time intervals with the required time step. The model was tested using the data of climatic observations of wind speed obtained from electronic archive of the All-Russian Institute of Hydrometeorological Information. The model adequacy was evaluated by comparing characteristics of simulated wind speed trajectories with actual observations collected at 518 weather stations located on territory of Russia.

In this paper, our objective is to design a distributed optimal control for pumping stations operating in a Water Distribution Network (WDN), where we would like to satisfy consumer demands with minimum energy consumption. The WDN has been modeled using graph theory and stochastic differential equations. This leads to a non-zero sum stochastic differential game. We have approximated the solution of the aforementioned game using Markov chain approximation and combined it with Shapley's algorithm so as to obtain Minimax mixed strategies. Minimax solution can be obtained as a distributed computation at the pumping stations without any knowledge of the costs incurred by the other pumping stations. Simulation results on the water network show convergence to an approximate Minimax solution.

In this paper, our objective is to convert the model of a water distribution network described via Stochastic differential equations (SDE) into a Markov decision process (MDP). The motivation behind this work is that while MDP's represents the underlying dynamics for dynamic programming and reinforcement learning, the actual underlying model is best described via differential equations, and therefore, we would like to convert the SDE to MDP. We have applied Kushner's Markov chain approximation (MCA) method and verified it using a novel modified Monte Carlo method which can be considered as an alternative to the well-known Kushner's MCA. Both the methods approximate the value function and simulation studies show that the obtained value functions from both the methods converge to almost the same value.

ilustraciones, diagramas, tablas Los modelos de Lotka Volterra son ecuaciones diferenciales no lineales que estudian la dinámica de poblaciones de individuos sometidos a interacciones de depredación, mutualismo, cooperación o mezclas de estas. En el caso determinista existe una amplia literatura al respecto, pero en la contraparte estocástica aún hay muchos interrogantes y preguntas abiertas. En este trabajo se hace un estudio de la existencia de la solución y de la distribución invariante de algunas ecuaciones diferenciales estocásticas de tipo Lotka Volterra siguiendo la línea del trabajo desarrollado por Mao. Finalmente, se establece una conexión entre las ecuaciones y los grafos y a partir de ahí se hace una extensión para un caso particular en el que el grafo de ecosistemas involucrados induce una forma de árbol. Lotka Volterra models are nonlinear differential equations which main objective is to study the dynamics between populations of individuals subjected to interactions as predation, mutualism, cooperation or some kind of mix between them. In deterministic case there exists a large literature about, but in the stochastic counterpart some questions and problems remains open. In this work a study is made about the existence of the solution and invariant distribution of some Lotka Volterra stochastic differential equations following the work of Mao. Finally, a connection between graphs and these equations is established and from there an extension to a particular case of tree graphs ecosystems is made. Maestría Magíster en Ciencias - Matemáticas Área Curricular en Matemáticas

Métodos de simulação numérica estão entre as mais usadas técnicas para calcular a resposta estocástica de sistemas de engenharia sujeitos a vibrações aleatórias. Em várias situações práticas os sistemas precisam ser integrados sobre longos intervalos de tempo e um grande número de amostras de trajetórias precisam ser geradas para encontrar as quantidades de interesse. Nessas circunstâncias, os métodos convencionais usualmente apresentam comportamento explosivo e sua implementação demanda muito esforço computacional. Neste trabalho propõe-se um novo método para a simulação de uma classe de sistemas vibratórios estocásticos. Trata-se de um método simplificado de convergência fraca que precisa basicamente de uma variável aleatória discreta de três pontos e do cálculo da exponencial de uma matriz por cada passo de integração para sua implementação. Demonstra-se que o método reproduz as mesmas propriedades estatísticas que caracterizam a solução exata da equação linearizada associada ao sistema em questão. Isso faz do método uma ferramenta eficiente para simulações de prazo longo. Experimentos numéricos são implementados para ilustrar a performance prática do método. Numerical simulation methods are among the most used techniques to evaluate the stochastic response of engineering systems subject to random excitations. In many practical situations the system should be integrated over long time intervals and a large number of sample trajectories need to be generated to assess the quantities of interest. In this circumstance conventional methods usually show exploding behavior or their implementation is computationally very demanding. In this work we propose a new method for simulating a class of stochastic vibrating systems. We devise a simplified weak method needing only three-point distributed random variable and a matrix exponential calculation per timestep for its implementation. Also, we prove that it reproduces the same statistical properties that characterize the exact solution to the linearized equation associated to the underlying system. This makes the method an efficient tool for stable long term simulations. Numerical experiments are carried out to illustrate the practical performance of the method.

In this paper, we establish an averaging principle on the infinite time intervals for semilinear stochastic ordinary differential equations with Levy noise. In particular, under suitable conditions we prove that if the coefficients are Poisson stable (including periodic, quasi-periodic, almost periodic, almost automorphic etc), then there exists a unique $ \mathcal L^2 $-bounded solution of the original equation, which inherits the recurrence property of the coefficients, and the recurrent solution uniformly converges to the stationary solution of the averaged equation on the whole real axis in distribution sense.

This paper is concerned with the problem of asymptotic stability for a class of stochastic differential equations with impulsive effects. A sufficient criterion on asymptotic stability is derived for such impulsive stochastic differential equations via Lyapunov stability theory, bounded difference condition and martingale convergence theorem. The results show that the impulses can facilitate the stability of the stochastic differential equations when the original system is not stable. Finally, the feasibility of our results is confirmed by two numerical examples and their simulations.

We present a methodology to connect an ordinary differential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic differential equation (SDE) intermediate model. The ODE model here presented is formulated as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.

The advent of single molecule microscopy has rev- olutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions

The topic of this work is ergodicity (stochastic stability) of various types of stochastic processes. The urge for analysis of random processes exists in every area of science and real life - medicine, biology, chemistry, physics, finance, etc., as many phenomena naturally exhibit some sort of random behaviour in their movement. Mathematical models used to describe those random movements are called stochastic differential equations (SDEs). Since the solutions of SDEs often have a very complicated structure or are impossible to obtain explicitly, it is usually hard to analyse them directly. Therefore, the emphasis has been placed on analysing their long-term stability. This includes detecting their equilibria (stacionary distributions) as well as the rate at which convergence occurs. The convergence is observed with respect to some appropriate distance function. In my work the emphasis has been placed on the quantitative aspect of this problem, namely, on finding explicit bounds on the rate of the convergence with respect to two distance functions: to total variation distance and the class of Wasserstein distances (which provides convergence in some weaker sense). As most of the existing results in this area correspond to the geometric ergodicity (that is, the case when the rate of the convergence is exponential), and known conditions ensuring sub-goemetric ergodicity are far from being optimal because there are many known examples of sub-geometrically ergodic systems that do not satisfy those conditions, the focus of my work is to find sharp and general conditions in terms of coefficients of the process that will ensure sub-geometric ergodicity of a wide range of processes. The results of my research can be divided in two parts. Firstly, I will consider classical diffusion processes - Markov processes with continuous trajectories (here, the class of processes with singular diffusion coefficients will be of special interest as they were not investigated so far). The second part of the research deals with somewhat more complicated random processes - diffusion processes with Markovian switching, which are processes that, beside the continuous, diffusive one, have a second, discrete component which changes the behaviour of the process at random times. This theory is relatively new so here we still have many interesting open questions and uninvestigated phenomena that are not characteristic for classical diffusion process. Also, in both cases, I will extend the results on a class of processes with jumps. Svrha mog rada je istražiti problem ergodičnosti (stohastičke stabilnosti) različitih tipova slučajnih procesa. Slučajni procesi iznimno su važni jer se koriste za modeliranje pojava u gotovo svim područjima znanosti i svakodnevnog života – primjenjuju se u medicini, biologiji, kemiji, fizici, financijama itd. Naime, u svim tim područjima često se dolazi do zaključka da se pojave ne mogu opisati determinističkim modelima jer su neki aspekti njihovog ponašanja slučajni. Matematički modeli koji se koriste za opisivanje ovakvih pojava su stohastičke diferencijalne jednadžbe (SDJ). Međutim, budući da rješenja SDJ-ova imaju jako kompliciranu strukturu i iznimno ih je teško analizirati direktnim putem, naglasak se stavlja na analizu njihove dugoročne stabilnosti. To uključuje određivanje njihovih ekvilibrija (stacionarnih distribucija), ali i brzine kojom konvergiraju prema tim ekvilibrijima. Konvergencija se promatra s obzirom na neku odredenu funkciju udaljenosti. U mojem istraživanju naglasak je stavljen na kvantitativni aspekt ovog problema, odnosno na eksplicitne ocjene brzine konvergencije s obzirom na dvije funkcije udaljenosti: udaljenost totalne varijacije i klasu Wassersteinovih udaljenosti (koja predstavlja konvergenciju u nešto slabijem smislu). Kako se većina dosadašnjih rezultata odnosi na geometrijsku ergodičnost (tj. slučaj kada je brzina konvergencije eksponencijalna), a poznati uvjeti za subgeometrijsku ergodičnost nisu blizu optimalnih jer su poznati mnogi sub-geometrijski ergodični sustavi koji te uvjete ne zadovoljavaju, fokus mog rada je pronaći oštre i opće uvjete u terminima koeficijenata samog procesa koji će osigurati subgeometrijsku ergodičnost široke klase procesa. Rezultate mog istraživanja mogu podijeliti u dvije cjeline. U prvoj ću proučavati klasične difuzije - Markovljeve procese neprekidnih trajektorija (gdje će od posebnog značaja biti klasa procesa sa singularnim difuzijskim koeficijentima koji do sada nisu razmatrani). Drugi dio rada proučava nešto složenije procese - difuzije sa slučajnim prebacivanjem, koji osim neprekidne, difuzijske komponente sadrže i drugu, diskretnu komponentu koja u slučajnim trenucima mijenja ponašanje procesa. Ova teorija je relativno nova pa tu ima još puno zanimljivih otvorenih pitanja i neistraženih pojava koje nisu karakteristične za klasične procese difuzija. Također, u oba slučaja, rezultate ću primijeniti na klasu procesa sa skokovima.

Software developers need information for deciding the optimal time for software release with improved software reliability. However, it is not easy for them to decide when and how to release newly developed software to the market. For a decision, the reliability and test costs of the software need to be balanced carefully for avoiding unnecessary confusion and users’ complaints. To address this need, related research has been carried out to propose an appropriate tool for such decisions. In many studies, software reliability growth models (SRGMs) were applied using the concept of confidence intervals to estimate the reliability of software. Confidence intervals were calculated on the basis of the assumption of a normal distribution showing the symmetrical occurrence of data with the mean as a center. However, the reliability data of software do not always have such symmetry for assuming the normal distribution. Therefore, it is necessary to propose a method for overcoming the mean value randomness that causes asymmetry in the related data. In previous studies, estimating variance and mean of errors of software was not considered, which led to the unreliable estimation of the confidence intervals of the mean value for decision making. Previous studies also lacked practicability in applications due to statistics from the asymmetrical data distribution. As a result, software developers could not effectively evaluate the possible risk related to the software release time. To improve the estimation, we employ the inflection S-shape model to propose the SRGM on the basis of confidence intervals assumed to come from the normal distribution. The proposed model allows determining the optimal time for software release with the consideration of its potential risk. For efficient determination, the architecture and user interface of the computation system are also proposed.

Solving the optimal control of stochastic differential equations (SDEs) using the dynamic programming method requires writing the problem in terms of the so-called value function. This paper presents conditions to assure that the value function is convex away from the origin, a concept that allows the value function be non-convex in a region close to the origin. In contrast, for regions away from the origin, the value function remains convex under some mild conditions. Stochastic ordering is used to prove this result. A numerical example illustrates the potential benefits of our approach.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art., Doctoral thesis, Mathematical Institute, University of Oxford. 231 pages

The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type dX t = S ( X t ) dt + ε dB t H , K , X 0 = x 0 , 0 ≤ t ≤ T , {\rm{d}}{{\rm{X}}_{\rm{t}}} = {\rm{S}}\left( {{{\rm{X}}_{\rm{t}}}} \right){\rm{dt + }}\varepsilon {\rm{dB}}_{\rm{t}}^{{\rm{H,K}}},\,{{\rm{X}}_{\rm{0}}} = {{\rm{x}}_{\rm{0}}},\,0 \le {\rm{t}} \le {\rm{T,}} where { B t H , K , t ≥ 0 {\rm{B}}_{\rm{t}}^{{\rm{H,K}}},{\rm{t}} \ge {\rm{0}} } is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.