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Extra resources for Algebraic Theory of the Bianchi Groups

Example text

5. 4 Much more general stochastic programming problems can be dealt with in this way, such as multistage stochastic programs with recourse and stochastic programs with chance constraints. Approximation of Stochastic Programming Problems 47 2 Preliminaries on Epi-convergence In this Section, we deﬁne the concept of epi-convergence. Its main properties are brieﬂy recalled in Appendix A. For a more complete treatment of the subject, we refer the reader to the monographs [3] or [9]. Let (E, d) be a metric space and φ : E → R = [−∞, +∞] be a function from X into the extended reals.

Our numerical tests showed evidence that the resulting distributions and thereof constructed weak approximations are of very high quality. References [Bal95] Paolo Baldi. Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. , 23(4):1644–1670, 1995. M. P. Petersen. Solving Dirichlet problems numerically using the Feynman-Kac representation. BIT, 43(3):519–540, 2003. [BS02] Andrei N. Borodin and Paavo Salminen. Handbook of Brownian motion— facts and formulae.

Is said to be stationary if the random vectors (X1 , . . , Xn ) and (Xk+1 , . . , Xn+k ) have the same distribution for all integers n, k ≥ 1. Any stationary sequence X1 , X2 , . . g. 11 in [6]). The transformation T : Ω → Ω on (Ω, A, Q) is said n−1 to be asymptotically mean stationary (ams) if the sequence n1 j=0 Q T −j A is convergent for all A ∈ A (see [13]). It is known from the Vitali-Hahn-Saks n−1 Theorem that lim n1 j=0 QT −j is a probability measure that we indicate n→∞ as P and call asymptotic mean of Q.