By Daniel W. Stroock
From the experiences: "This ebook is a superb presentation of the appliance of martingale concept to the idea of Markov approaches, particularly multidimensional diffusions. [...] This monograph should be advised to graduate scholars and learn employees but in addition to all attracted to Markov procedures from a extra theoretical element of view." Mathematische Operationsforschung und Statistik
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Additional resources for Multidimensional Diffusion Processes (Classics in Mathematics)
Example text
4] = 1 and for any p > 0 and T < oo lim lim sup P„ sup |x(r)-x(5)| >p 0. 0 Let D ^ [s, 00) be a countable dense set. s. for all ti,t2 € D such that t^ < t2, then (0{t), J^,, P) is a non-negative submartingale after time s. s. for all ti, tje D such that t^ < tj, then {6(t), J^,, P) is a martingale after time s. 28 1- Preliminary Material: Extension Theorems, Martingales, and Compactness Proof. Assume that s = 0. Clearly the proof boils down to showing that in either case the family {| e(t) \: te [0, T] n D} is uniformly P-integrable for all T e D. 14) implies that {9(t): t e [0, T] n D} is uniformly Pintegrable. 15) QioK^{^) — ^(<^') and x(s, co) = x(s, a>') for 0 < s < T(CO)) = 1. 5 Theorem. e a nondecreasing sequence of stopping times and for each n suppose P„ is a probability measure on (Q, ^r„)- Assume that P„ +1 equals P„ on J^r„for each n>0. If\im„^^ P„(T„ < t) = Ofor all t > 0, then there is a unique probability measure P on (Q, ^) such that P equals P„ on M^^for all n > 0. Proof. 16) P(A)= l i m P „ [ / l n { t „ > t } ] . n->oo Thus uniqueness of P on Jti for all r > 0 is proved; and therefore P, if it exists, is unique on M.