Introduction to the Theory of Random Processes

By Nicolai V. Krylov. 2002

451 citations recorded by the source corpus.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cd4cb0b89eeae2b2e57f1c88404bcdf1ac7ef726

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Citations in the corpus

A FUNCTIONAL VERSION OF THE BIRKHOFF ERGODIC THEOREM FOR A NORMAL INTEGRAND: A VARIATIONAL APPROACH (2003, background): “[For real-valued functions, this corresponds exactly to the Neveu (1964), Definition III-4-4, page 86 and Gikhman and Skorokhod (1969), Definition 1, page 157 definitions of a measurable random real function.]”
Ergodic Markov processes and Poisson equations (lecture notes) (2016, background): “Since this is a course about ergodic properties, we do not recall the definitions of what are Markov, strong Markov, homogeneous Markov processes (MP) which are assumed to be known to the reader: consult any of the textbooks [4, 10, 11, 12, 20, 27, 38, 49] if in doubt.”
Sequential Monte Carlo Methods in Practice (2001, background): “As described in (Gikhman and Skorokhod 1969), X(t) is a Markov jump process with jump measure q(s, A), if X(t) is a Markov process having a transition probability Pt(s, A) and satisfies: (i) X(t) is continuous in probability, Le.”
Functional Analysis for Probability and Stochastic Processes: An Introduction (2005, unknown): “8 Notes to Chapter 8 The vast literature on Markov processes that covers various aspects of the theory includes [4], [20], [21], [29], [35], [34], [38], [41], [42], [46], [47], [48], [57], [58], [61], [62], [84], [88], [92], [93], [99], [100], [102], [105], [107], [109], [113], [114].”
On the ergodicity and mixing of max-stable processes (2008, background): “PROOF: If X is continuous in probability, then it has a measurable modification (see e.g. Gihman and Skorohod (1969)).”
Control: A perspective (2014, background): “Other books on control of systems with random processes are [674,615,412,163].”
Linear Stochastic Systems: A Geometric Approach to Modeling, Estimation and Identification (2016, background): “Stationary increments processes in continuous time are discussed in Chapter 1 of [39].”
On the Lamperti stable processes (2008, unknown): “…f(ξ), we have( er(f(ξ)−θr(α+1))/2 − 1 )2 ≤ ( er(f(ξ)−(α+1))/2 − 1 )2 ∨ ( erf(ξ)/2 − 1 )2 (5.43) Using a Taylor expansion again and (5.43), it is clear that there exists a constant R > 0 such that if r R, then ( er(f(ξ)−θr(α+1))/2 − 1 )2 ≤ K(f2(ξ) + 1)r2, (5.44) where K is a positive constant.”