Stochastic initial-value problem

In both the Ito and Stratonovich formulations, the randomness in a set of SDEs is generated by an auxiliary set of statistically independent Wiener processes [12,16]. The solution of an SDE is defined by a hmiting process (which is different in different interpretations) that yields a unique solution to any stochastic initial value problem for each possible reahzation of this underlying set of Wiener processes. A Wiener process W t) is a Gaussian Markov diffusion process for which the change in value W t) — W(t ) between any two times t and t has a mean and variance... 

When this information is available, one has a well-defined stochastic initial-value problem, which one might, in principle, hope to solve by the transport equation. Since the neutron population is here assumed to have spherical symmetry, this is an integro-differential equation in four independent variables. However, a stepwise numerical integration of it is a formidable task, even for modem computing machines. 

But the major physical problem remained open Could one prove rigorously that the systems studied before 1979—that is, typically, systems of N interacting particles (with N very large)—are intrinsically stochastic systems In order to go around the major difficulty, Prigogine will take as a starting point another property of dynamical systems integrability. A dynamical system defined as the solution of a system of differential equations (such as the Hamilton equations of classical dynamics) is said to be integrable if the initial value problem of these equations admits a unique analytical solution, weekly sensitive to the initial condition. Such systems are mechanically stable. In order to... 

We saw before that we could think of the simple SDE initial value problem (6.8) as having a solution defined by a certain random process. In the same way we would like to obtain, for the Ornstein-Uhlenbeck SDE (6.19), an explicit stochastic process which is in some sense equivalent to solving the SDE. Multiply both sides of (6.19) by the integrating factor exp(yf), and observe that... 

The deterministic motion (i.e. the solution of the initial value problem) can be considered as a particular stochastic process satisfying just (5.67) and (5.68) (Lax 1966a). 

Sometimes the initial value a is also a random quantity (or vector). Then the resulting stochastic process U(t [y], a) is a function of the random variable a, as well as a functional of y. As this is only a trivial generalization of the problem with fixed initial a there is no need to treat random initial values separately. 

The problem consists of seven nodes and up to eight pipes. When the viewer is instantiated, the initial values for all the variables default to the mid-point between the lower and upper bounds. From this starting point, the user may immediately interact directly with the viewer to alter the configuration or may request the application of a stochastic optimization step. At any point, the current configuration may be exported as a GAMS input file and solved using the rigorous MINLP formulation. 

In this paper the estimation problem is solved from a point of view which is essentially different from the stochastic approach of the conventional Kalman filter. The time-variant state estimation problem is re-phrased into a time-invariant parameter estimation problem, and least-squaxes techniques, which is derived under deterministic framwork, are then used. The advantage of using least-squaxes approach is that it does not require a pHori knowledge of the noise statistics, the initial values of the estimated state, and its corresponding error covariance. Close relations are found between the Kalman filter and the least-squaxes Alter. Finally, a numerical example is provided to illustrate the feasibility of the proposed method. 

There are numerous papers that assume that demand is a Poisson process. One important early one is Li [97], who studied price and production problems with capacity investment at the beginning of the horizon. Given an initial production capacity, the demand and production rates are Poisson counting processes if there is demand in excess of production, sales are lost. Li shows that the optimal production policy under a single fixed price is a barrier or threshold policy, where production is optimal if inventory is below a certain value. Further, he characterizes the optimal policy when price is dynamic over time, and he shows that the stochastic price is always higher than the deterministic price. In extensions, he considers the case with production learning effects that is, the production rate over time becomes closer to the ideal capacity. 

Under these assumptions, the formulation of cell division as an FPT problem requires that the following be specified (i) a stochastic model for how cell size, s, increases with time, t, between divisions (ii) the function of the cell size, s, that attains critical or threshold value, 6, at division and (iii) appropriate initial conditions, including the initial distribution of cell sizes. While in all previous cases considered, every member of the ensemble (i.e., each cell in the population) was assumed to be subjected to identical initial conditions, in this section we relax that condition to allow different cells to experience different initial conditions (i.e., different freshly divided cells having different sizes). This is an added source of stochasticity, namely, extrinsic noise in addition to the intrinsic fluctuations encoded in stochastic growth for a given initial condition. 

Simulated annealing is a global, multivariate optimization technique based on the Metropolis Monte Carlo search algorithm. The method starts from an initial random state, and walks through the state space associated with the problem of interest by generating a series of small, stochastic steps. An objective function maps each state into a value in EH that measures its fitness. In the problem at hand, a state is a unique -membered subset of compounds from the n-membered set, its fitness is the diversity associated with that set, and the step is a small change in the composition of that set (usually of the order of 1-10% of the points comprising the set). While downhill transitions are always accepted, uphill transitions are accepted with a probability that is inversely proportional to... 

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