Parameter estimation in a simple stochastic differential equation for phytoplankton modelling

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one‐sided estimate type that is formulated in terms of so‐called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1‐complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so‐called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration.

Optimal Control of Stochastic Systems

Stability in distribution of neutral stochastic functional differential equations with Markovian switching

Probabilistic ordinary differential equation (ODE) solvers have been introduced over the past decade as uncertainty-aware numerical integrators. They typically proceed by assuming a functional prior to the ODE solution, which is then queried on a grid to form a posterior distribution over the ODE solution. As the queries span the integration interval, the approximate posterior solution then converges to the true deterministic one. Gaussian ODE filters, in particular, have enjoyed a lot of attention due to their computational efficiency, the simplicity of their implementation, and their provable fast convergence rates. In this article, we extend the methodology to stochastic differential equations (SDEs) and propose a probabilistic simulator for SDEs. Our approach involves transforming the SDE into a sequence of random ODEs using piecewise differentiable approximations of the Brownian motion. We then apply probabilistic ODE solvers to the individual ODEs, resulting in a pathwise probabilistic solution to the SDE. We establish worst-case strong 1.5 local and 1.0 global convergence orders for a specific instance of our method. We further show how we can marginalise the Brownian approximations, by incorporating its coefficients as part of the prior ODE model, allowing for computing exact transition densities under our model. Finally, we numerically validate the theoretical findings, showcasing reasonable weak convergence properties in the marginalised version.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

This work focuses on modeling of film porosity in thin film deposition using stochastic differential equations. A deposition process is modeled via kinetic Monte Carlo (kMC) simulation on a triangular lattice. The microscopic process events involve atom adsorption and migration and the film growth allows for vacancies and overhangs to develop inside the film. Appropriate definitions of film site occupancy ratio (SOR), i.e., fraction of film sites occupied by particles over total number of film sites, and its fluctuation are introduced to describe film porosity. Deterministic and stochastic ordinary differential equation (ODE) models are also derived to describe the time evolution of film SOR and its fluctuation. The coefficients of the ODE models are estimated on the basis of data obtained from the kMC simulator of the deposition process using least-square methods. Simulation results demonstrate the applicability and effectiveness of the proposed film porosity modeling methods in the context of the deposition process under consideration.

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.

Stochastic physics-informed neural ordinary differential equations

A numerical method for some stochastic differential equations with multiplicative noise

Operator-valued stochastic differential equations in the context of Kurzweil-like equations

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

Roughly speaking, stochastic differential equations are differential equations driven by Gaussian white noise. Here, we are using the term “stochastic differential equations” in a restricted sense and not merely to denote differential equations with some probabilistic aspects. The importance of. stochastic differential equations is largely due to the fact that the solution of such an equation is a sample-continuous Markov process, and conversely, a large and important class of sample-continuous Markov processes can be modeled by the solutions of stochastic differential equations. From the point of view of applications, this is a direct benefit of using white noise as a noise model, and this fact accounts for its popularity. After all, white noise is, at best, a tolerable abstraction and is never a completely faithful representation of a physical noise source. Its raison d’être is the simplicity of analysis that it brings about. We have seen this in connection with Kalman filtering, and we shall see it again in the Markovian nature of solutions to stochastic differential equations.

Comparison theorems for stochastic differential equations in finite and infinite dimensions

We discuss Lie-point symmetries of stochastic (ordinary) differential equations, and the interrelations between these and analogous symmetries of the associated Fokker-Planck equation for the probability measure.