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Degenerate stochastic differential equations for catalytic branching networks

Sandra Kliem (2009)

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

Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math.50 (2006) 323–383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.

Differentiability of excessive functions of one-dimensional diffusions and the principle of smooth fit

Paavo Salminen, Bao Quoc Ta (2015)

Banach Center Publications

The principle of smooth fit is probably the most used tool to find solutions to optimal stopping problems of one-dimensional diffusions. It is important, e.g., in financial mathematical applications to understand in which kind of models and problems smooth fit can fail. In this paper we connect-in case of one-dimensional diffusions-the validity of smooth fit and the differentiability of excessive functions. The basic tool to derive the results is the representation theory of excessive functions;...

Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations

Benjamin Jourdain (2010)

ESAIM: Probability and Statistics

We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix,...

Diffusions with measurement errors. I. Local asymptotic normality

Arnaud Gloter, Jean Jacod (2001)

ESAIM: Probability and Statistics

We consider a diffusion process X which is observed at times i / n for i = 0 , 1 , ... , n , each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance ρ n . There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when X is indeed a gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What is perhaps...

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