Continuous time discrete state stochastic models

Continuous time discrete state stochastic models... 

Why are the arguments for the usual continuous time discrete state stochastic model so good for reaction kinetics here ... 

Each of the optional dynamical models mentioned above involves a homogeneous Markov process Xt = Xt teT in either continuous or discrete time on some state space X. The motion of Xt is given in terms of the stochastic... 

One possibility to simplify a continuous stochastic system is the reduction to a description in terms of a few discrete states. The system s behavior is then specified by the transition times between these discrete states. For example when investigating a neurons behavior, the important aspect are often only the times when a spike is emitted and not the complex evolution of the membrane potential [16]. In a doublewell potential system, depending on the questions asked, it may be sufficient to know in which of the two wells the system is located, neglecting the fluctuations in the wells as well as the actual dynamics when crossing from one well to the other. In these cases a reduction to a discrete description can be considered as an appropriate simplification. We first review the two state description of bistable systems [10] and then introduce a phenomenological discrete model for excitable dynamics. 

At least eight different kinetic models can be defined, depending on the specification of time (X), state-space (T) and nature of determination (Z). As was explained earlier, time can be discrete (D) or continuous (C), the state-space can be also discrete (D) or continuous (C), and the nature of determination is deterministic (D) or stochastic (S). 

The proposed procedure is shown in Figure 1 and consists of five basic steps. First a standard ODE model is derived from first engineering principles and the constitutive equations containing unknown functional relations are identified. The ODE model is then translated into a stochastic state space model consisting of a set of SDE s describing the dynamics of the system in continuous time and a set of discrete time measurement equations signifying how the available experimental data was obtained. 

The use of the embedded Discrete Time Markov Chain in a continuous stochastic process for deter-mining the events probability makes assumption that the system is in a stationary state characterizing by stationary distribution probabihties over its states. But the embedded DTMC is not limited to Continuous Time Markov Chain a DTMC can also be defined from semi-Markov or under some hypothesis from more generally stochastic processes. Another advantage to use the DTMC to obtain the events probability is that the probability of an event is not the same during the system evolution, but can depends on the state where it occurs (in other words the same event can be characterized by different occurrence probabilities). The use of the Arden lemma permits to formally determine the whole set of events sequences, without model exploring. Finally, the probability occurrence for relevant or critical events sequences and for a sublanguage is determined. 

Nassar et al. [10] employed a stochastic approach, namely a Markov process with transient and absorbing states, to model in a unified fashion both complex linear first-order chemical reactions, involving molecules of multiple types, and mixing, accompanied by flow in an nonsteady- or steady-state continuous-flow reactor. Chou et al. [11] extended this system with nonlinear chemical reactions by means of Markov chains. An assumption is made that transitiions occur instantaneously at each instant of the discretized time. 

To define the problem of nonlinear filtering (Ristic, Arulampalam, Gordon, 2004), let us consider the state vector ii , where n is the dimension ofthe state vectorand k e N. Here the index k is attributed to a continuous-time instant tf.. The state vector evolves according to the following discrete-time stochastic model ... 

The monomolecular reaction systems of chemical kinetics are examples of linear coupled systems. Since linear coupled systems are the simplest systems with many degrees of freedom, their importance extends far beyond chemical kinetics. The linear coupled systems in which we are interested may be characterized, in general terms, as arising from stochastic or Markov processes that are continuous in time and discrete in an appropriate space. In addition, the principle of detailed balancing is observed and the total amount of material in the system is conserved. The system is characterized by discrete compartments or states and material passes between these compartments by first order processes. Such linear systems are good models for a large number of processes. 

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