34 research outputs found
Rates for branching particle approximations of continuous-discrete filters
Herein, we analyze an efficient branching particle method for asymptotic
solutions to a class of continuous-discrete filtering problems. Suppose that
is a Markov process and we wish to calculate the measure-valued
process , where and is a distorted, corrupted, partial
observation of . Then, one constructs a particle system with
observation-dependent branching and initial particles whose empirical
measure at time , , closely approximates . Each particle
evolves independently of the other particles according to the law of the signal
between observation times , and branches with small probability at an
observation time. For filtering problems where is very small, using
the algorithm considered in this paper requires far fewer computations than
other algorithms that branch or interact all particles regardless of the value
of . We analyze the algorithm on L\'{e}vy-stable signals and give
rates of convergence for , where
is a Sobolev norm, as well as related convergence
results.Comment: Published at http://dx.doi.org/10.1214/105051605000000539 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
CONVERGENCE OF MARKOV CHAIN APPROXIMATIONS TO STOCHASTIC REACTION DIFFUSION EQUATIONS
In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger class of reaction functions. Our alterations also feature dramatically slower Markov chain state change rates often yielding a ten to one hundred fold simulation speed increase over the previous version of the method as evidenced in our computer implementations. On a weighted L2 Hilbert space chosen to symmetrize the elliptic operator, we consider existence of and convergence to pathwise unique mild solutions of our stochastic reaction-diffusion equation. Our main convergence result, a quenched law of large numbers, establishes convergence in probability of our Markov chain approximations for each fixed path of our driving Poisson measure source. As a consequence, we also obtain the annealed law of large numbers establishing convergence in probability of our Markov chains to the solution of the stochastic reaction-diffusion equation while considering the Poisson source as a random medium for the Markov chains.
EXPLICIT STRONG SOLUTIONS OF MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS
Herein, we characterize strong solutions of multidimensional stochastic differential equations (formula) that can be represented locally as (formula) where W is an multidimensional Brownian motion and U, (symbole) are continuous functions. Assuming that (symbole) is continuously differentiable, we find that (symbole) must satisfy a commutation relation for such explicit solutions to exist and we identify all drift terms b as well as U and (symbole) that will allow X to be represented in this manner. Our method is based on the existence of a local change of coordinates in terms of a diffeomorphism between the solutions X and the strong solutions to a simpler Ito integral equation.Diffeomorphism, Ito processes, explicit solutions.
On almost sure limit theorems for detecting long-range dependent, heavy-tailed processes
Marcinkiewicz strong law of large numbers, almost surely with , are developed for
products , where the are two-sided linear process
with coefficients and i.i.d. zero-mean
innovations . The decay of the coefficients
as , can be slow enough for to have
long memory while can have heavy tails. The long-range dependence and
heavy tails for are handled simultaneously and a decoupling property
shows the convergence rate is dictated by the worst of long-range dependence
and heavy tails, but not their combination. The results provide a means to
estimate how much (if any) long-range dependence and heavy tails a sequential
data set possesses, which is done for real financial data. All of the stocks we
considered had some degree of heavy tails. The majority also had long-range
dependence. The Marcinkiewicz strong law of large numbers is also extended to
the multivariate linear process case.Comment: 28 pages, 1 Figur