10,051 research outputs found
Switching and diffusion models for gene regulation networks
We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate 'switch plus diffusion' model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third 'switch plus ODE' model the switch remains but the diffusion is removed. The latter two models allow for multi-scale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The 'switch plus ODE' regime was proposed by Paszek (Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function, Bulletin of Mathematical Biology, 2007), who analyzed the steady state behavior, showing that the mean was preserved but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new 'switch plus diffusion' regime preserves the biologically relevant measures of mean and variance, whereas the 'switch plus ODE' model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behaviour can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis
Understanding the fine structure of electricity prices
This paper analyzes the special features of electricity spot prices derived from the physics of this commodity and from the economics of supply and demand in a market pool. Besides mean reversion, a property they share with other commodities, power prices exhibit the unique feature of spikes in trajectories. We introduce a class of discontinuous processes exhibiting a "jump-reversion" component to properly represent these sharp upward moves shortly followed by drops of similar magnitude. Our approach allows to captureâfor the first time to our knowledgeâboth the trajectorial and the statistical properties of electricity pool prices. The quality of the fitting is illustrated on a database of major U.S. power markets
Feynman-Kac representation of fully nonlinear PDEs and applications
The classical Feynman-Kac formula states the connection between linear
parabolic partial differential equations (PDEs), like the heat equation, and
expectation of stochastic processes driven by Brownian motion. It gives then a
method for solving linear PDEs by Monte Carlo simulations of random processes.
The extension to (fully)nonlinear PDEs led in the recent years to important
developments in stochastic analysis and the emergence of the theory of backward
stochastic differential equations (BSDEs), which can be viewed as nonlinear
Feynman-Kac formulas. We review in this paper the main ideas and results in
this area, and present implications of these probabilistic representations for
the numerical resolution of nonlinear PDEs, together with some applications to
stochastic control problems and model uncertainty in finance
Modeling electricity prices: international evidence
This paper analyses the evolution of electricity prices in deregulated markets. We present a general model that simultaneously takes into account the possibility of several factors: seasonality, mean reversion, GARCH behaviour and time-dependent jumps. The model is applied to equilibrium spot prices of electricity markets from Argentina, Australia (Victoria), New Zealand (Hayward), NordPool (Scandinavia), Spain and U.S. (PJM) using daily data. Six different nested models were estimated to compare the relative importance of each factor and their interactions. We obtained that electricity prices are mean-reverting with strong volatility (GARCH) and jumps of time-dependent intensity even after adjusting for seasonality. We also provide a detailed unit root analysis of electricity prices against mean reversion, in the presence of jumps and GARCH errors, and propose a new powerful procedure based on bootstrap techniques
Convergence of Gaussian quasi-likelihood random fields for ergodic L\'{e}vy driven SDE observed at high frequency
This paper investigates the Gaussian quasi-likelihood estimation of an
exponentially ergodic multidimensional Markov process, which is expressed as a
solution to a L\'{e}vy driven stochastic differential equation whose
coefficients are known except for the finite-dimensional parameters to be
estimated, where the diffusion coefficient may be degenerate or even null. We
suppose that the process is discretely observed under the rapidly increasing
experimental design with step size . By means of the polynomial-type large
deviation inequality, convergence of the corresponding statistical random
fields is derived in a mighty mode, which especially leads to the asymptotic
normality at rate for all the target parameters, and also to the
convergence of their moments. As our Gaussian quasi-likelihood solely looks at
the local-mean and local-covariance structures, efficiency loss would be large
in some instances. Nevertheless, it has the practically important advantages:
first, the computation of estimates does not require any fine tuning, and hence
it is straightforward; second, the estimation procedure can be adopted without
full specification of the L\'{e}vy measure.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1121 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Pedestrians moving in dark: Balancing measures and playing games on lattices
We present two conceptually new modeling approaches aimed at describing the
motion of pedestrians in obscured corridors:
* a Becker-D\"{o}ring-type dynamics
* a probabilistic cellular automaton model.
In both models the group formation is affected by a threshold. The
pedestrians are supposed to have very limited knowledge about their current
position and their neighborhood; they can form groups up to a certain size and
they can leave them. Their main goal is to find the exit of the corridor.
Although being of mathematically different character, the discussion of both
models shows that it seems to be a disadvantage for the individual to adhere to
larger groups. We illustrate this effect numerically by solving both model
systems. Finally we list some of our main open questions and conjectures
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