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LAMN property for hidden processes : the case of integrated diffusions

Arnaud Gloter, Emmanuel Gobet (2008)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic...

Laplace transform identities for diffusions, with applications to rebates and barrier options

Hardy Hulley, Eckhard Platen (2008)

Banach Center Publications

We start with a general time-homogeneous scalar diffusion whose state space is an interval I ⊆ ℝ. If it is started at x ∈ I, then we consider the problem of imposing upper and/or lower boundary conditions at two points a,b ∈ I, where a < x < b. Using a simple integral identity, we derive general expressions for the Laplace transform of the transition density of the process, if killing or reflecting boundaries are specified. We also obtain a number of useful expressions for the Laplace transforms...

Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections

Magdalena Kobylanski (2013)

Annales de l'I.H.P. Probabilités et statistiques

We establish a Large Deviations Principle for diffusions with Lipschitz continuous oblique reflections on regular domains. The rate functional is given as the value function of a control problem and is proved to be good. The proof is based on a viscosity solution approach. The idea consists in interpreting the probabilities as the solutions to some PDEs, make the logarithmic transform, pass to the limit, and then identify the action functional as the solution of the limiting equation.

Law of large numbers for superdiffusions : the non-ergodic case

János Engländer (2009)

Annales de l'I.H.P. Probabilités et statistiques

In previous work of D. Turaev, A. Winter and the author, the Law of Large Numbers for the local mass of certain superdiffusions was proved under an ergodicity assumption. In this paper we go beyond ergodicity, that is we consider cases when the scaling for the expectation of the local mass is not purely exponential. Inter alia, we prove the analog of the Watanabe–Biggins LLN for super-brownian motion.

Limit laws of transient excited random walks on integers

Elena Kosygina, Thomas Mountford (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider excited random walks (ERWs) on ℤ with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [15] have shown that when the total expected drift per site, δ, is larger than 1 then ERW is transient to the right and, moreover, for δ&gt;4 under the averaged measure it obeys the Central Limit Theorem. We show that when δ∈(2, 4] the limiting behavior of an appropriately centered and scaled excited random...

Limit theorems for number of diffusion processes, which did not absorb by boundaries

Aniello Fedullo, Vitalii Gasanenko (2006)

Open Mathematics

We have random number of independent diffusion processes with absorption on boundaries in some region at initial time t = 0. The initial numbers and positions of processes in region is defined by the Poisson random measure. It is required to estimate the number of the unabsorbed processes for the fixed time τ > 0. The Poisson random measure depends on τ and τ → ∞.

Limite ergodique de processus de diffusion infini-dimensionnels.

Sylvie Roelly, D. Seu (1999)

Publicacions Matemàtiques

We give a temporal ergodicity criterium for the solution of a class of infinite dimensional stochastic differential equations of gradient type, where the interaction has infinite range. We illustrate our theoretical result by typical examples.

Limiting behaviors of the Brownian motions on hyperbolic spaces

H. Matsumoto (2010)

Colloquium Mathematicae

Using explicit representations of the Brownian motions on hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity can be easily obtained. We also give a straightforward strategy to obtain explicit expressions for the limit distributions or Poisson kernels.

Linear diffusion with stationary switching regime

Xavier Guyon, Serge Iovleff, Jian-Feng Yao (2004)

ESAIM: Probability and Statistics

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : d Y t = a ( X t ) Y t d t + σ ( X t ) d W t , Y 0 = y 0 . We establish that under the condition α = E μ ( a ( X 0 ) ) &lt; 0 with μ the stationary distribution of the regime process X , the diffusion Y is ergodic. We also consider conditions for the existence of moments for the invariant law of Y when X is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that conditionally to X , Y is gaussian on the other hand, we give...

Linear diffusion with stationary switching regime

Xavier Guyon, Serge Iovleff, Jian-Feng Yao (2010)

ESAIM: Probability and Statistics

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : dYt = a(Xt)Ytdt + σ(Xt)dWt,Y0 = y0. We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of the regime process X, the diffusion Y is ergodic. We also consider conditions for the existence of moments for the invariant law of Y when X is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that...

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