Pénalisations de marches aléatoires

Abstract : The subject of my thesis is the theory of penalisation originaly developed by B .Roynette, P. Vallois and M. Yor in the case of the brownian motion. In a few words, it consists in putting a weight on the probability measure to favorise trajectories with probability measure equals to zero. The first part of my thesis is the discrete counterpart of their work : let (Omega,(Xn,,n>=0),Fn,n>=0, P) the symmetric random walk and Fn is the canonical filtration. For some adapted and positive functionals G:N*Omega->R+, I study for all n in N, for all An in Fn, the limit when p goes to infinity of the quantity: Ex[An Gp] / Ex[Gp] When this limit exists, it is equal to Q(An):=Ex[An Mn] where (M_n,n>=0) is a positive non uniformly integrable martingale. The definition of Q induces a new probability on (Omega, F) and then I study (Xn,n>=0) under Q. In a second part, I try to expend this theory to birth and death Markov processes. Recall that these processes have the property that, after an exponential random length of time, only transitions to neighbouring states are possible. Precisely, I penalize the distribution of the transient birth and death process by the number of visits at the state 0 (which is like local time type penalization). When I force the process to visit an infinitely often the state zero, I prove that, under the new probability measure induced by penalization, the process behaves as a recurrent birth and death process.
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Pierre Debs. Pénalisations de marches aléatoires. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 2007. Français. ⟨NNT : 2007NAN10091⟩. ⟨tel-01748273v1⟩

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