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

Convergence of Probability Measures

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

The paper proposes an original class of models for the continuous-time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentially weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that, unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent options prices.

We find a partial differential equation for the price of a European call option. Smiles and skews are found in the resulting plots of implied volatility.

Complete Models with Stochastic Volatility

An Lp-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class VMOx, which, in particular, contains all functions independent of x. Weak uniqueness of the martingale problem associated with such equations is obtained.

Parabolic and elliptic equations with VMO coefficients

This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations.

/pdf/sequential-monte-carlo-smoothing-with-application-to-53u27l6ho3.pdf

Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models

We formulate large deviations principle (LDP) for diffusion pair (X ɛ ,ξ ɛ )=(X t ɛ ,ξ t ɛ ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X ɛ ,ν ɛ ) with ν ɛ (dt, dz) being an occupation type measure corresponding to ξ t ɛ . In some sense we obtain a combination of Freidlin–Wentzell’s and Donsker–Varadhan’s results. Our approach relies on the concept of the exponential tightness and Puhalskii’s theorem.

/pdf/large-deviations-for-two-scaled-diffusions-5cqz1hiwcw.pdf

Large deviations for two scaled diffusions

We formulate large deviations principle (LDP) for diffusion pair $(X^\epsilon,\xi^\epsilon)=(X_t^\epsilon,\xi_t^\epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^\epsilon,\nu^\epsilon)$ with $\nu^\epsilon(dt,dz)$ being an occupation type measure corresponding to $\xi_t^\epsilon$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem.

The nonlinear filtering equation is said to be stable if it ``forgets'' the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure.

Stability of nonlinear filters in nonmixing case

Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$ S^\kappa_t=\frac{1}{t^\kappa}\int_0^tH(X_s)ds, \ t\to\infty $$ for an ergodic diffusion process $X_t$ under $0.5<\kappa<1$ and appropriate $H$. We mean a decomposition with ``corrector'': $$ \frac{1}{t^\kappa}\int_0^tH(X_s)ds={\rm corrector}+\frac{1}{t^\kappa}\underbrace{M_t}_{\rm martingale}. $$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$ \frac{1}{t^\kappa}M_t, t\to\infty. $'' Starting from Bayer and Freidlin, \cite{BF}, and finishing by Wu's papers \cite{Wu1}-\cite{WuH}, in the MD study Laplace's transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion'' parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ...''.

R. Liptser

Papers

Large deviations for two scaled diffusions

Large deviations for two scaled diffusions

Stability of nonlinear filters in nonmixing case

Examples of moderate deviation principle for diffusion processes

Deviation probability bound for martingales with applications to statistical estimation