计量经济分析 = Econometric analysis

Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Markov Chains and Stochastic Stability

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons.

/pdf/an-introduction-to-mcmc-for-machine-learning-22slt5b6x5.pdf

An introduction to MCMC for machine learning

This paper introduces a new nonparametric jump test for continuous-time asset pricing models. It distinguishes jump arrival times and realized jump sizes in asset prices as precisely as at intra-day levels. We demonstrate the likelihood of misclassification of jumps in discrete data becomes negligible when we use high-frequency returns. We explore real-time jump dynamics using intra-day U.S. individual equity prices through the test, and find empirical evidence that jump arrivals are associated with both pre-scheduled earnings announcements and unscheduled real-time news release.

/pdf/jumps-in-real-time-financial-markets-a-new-nonparametric-27cmh1b0i2.pdf

Jumps in Real-Time Financial Markets: A New Nonparametric Test and Jump Dynamics

The connection between large and small deviation results for the signed square root statistic $R$ is studied, both for likelihoods and for likelihood-like criterion functions. We show that if $p - 1$ Barlett identities are satisfied to first order, but the $p$th identity is violated to this order, then $\operator{cum}_q(R) = O(n^{-q/2})$ for $3 \leq q < p$, whereas $\operator{cum}_p(R) = \kappa_p n^{-(p-2)/2} + O(n^{-p/2})$. We also show that the large deviation behavior of $R$ is determined by the values of $p$ and $\kappa_p$. The latter result is also valid for more general statistics. Affine (additive and/or multiplicative) correction to $R$ and $R^2$ are special cases corresponding to $p = 3$ and 4. The cumulant behavior of $R$ gives a way of characterizing the extent to which $R$-statistics derived from criterion functions other than log likelihoods can be expected to behave like ones derived from true log likelihoods, by looking at th number of Bartlett identities that are satisfied. Empirical and nonparametric survival analysis type likelihoods are analyzed from this perspective via the device of “dual criterion functions.”

/pdf/bartlett-identities-and-large-deviations-in-likelihood-1pxvnxrf0n.pdf

Bartlett identities and large deviations in likelihood theory

http://galton.uchicago.edu/~mykland/paperlinks/mlepmgmm-hs-092205.pdf

An analysis of Hansen–Scheinkman moment estimators for discretely and randomly sampled diffusions

Conservative delta hedging permits traders to pass from nonparametric bounds on interest rates and volatilities to trading strategies. The uncertainty reflected in the bounds, however, provides relatively high starting prices for these strategies. We here show how the effect of uncertainty to a substantial extent can be offset by interpolation, i.e., the hedging in auxiliary market traded securities. As before, the technology does not involve the specification of models for the term structure of volatilities and interest rates, and should therefore be particularly appropriate from the point of view of risk management.

/pdf/the-interpolation-of-options-5gmoblqdhv.pdf

The interpolation of options

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.

Per A. Mykland

Papers

Jumps in Real-Time Financial Markets: A New Nonparametric Test and Jump Dynamics

Bartlett identities and large deviations in likelihood theory

An analysis of Hansen–Scheinkman moment estimators for discretely and randomly sampled diffusions

The interpolation of options

Embedding and asymptotic expansions for martingales