There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

We consider competing particle systems in $\mathbb{R}^d$, i.e. random locally finite upper bounded configurations of points in $\mathbb{R}^d$ evolving in discrete time steps. In each step i.i.d. increments are added to the particles independently of the initial configuration and the previous steps. Ruzmaikina and Aizenman characterized quasi-stationary measures of such an evolution, i.e. point processes for which the joint distribution of the gaps between the particles is invariant under the evolution, in case $d=1$ and restricting to increments having a density and an everywhere finite moment generating function. We prove corresponding versions of their theorem in dimension $d=1$ for heavy-tailed increments in the domain of attraction of a stable law and in dimension $d\geq 1$ for lattice type increments with an everywhere finite moment generating function. In all cases we only assume that under the initial configuration no two particles are located at the same point. In addition, we analyze the attractivity of quasi-stationary Poisson point processes in the space of all Poisson point processes with almost surely infinite, locally finite and upper bounded configurations.

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Competing Particle Systems Evolving by I.I.D. Increments

We study two-dimensional stochastic differential equations (SDEs) of McKean–Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed in (Probab. Theory Related Fields 71 (1986) 501–516), typically to produce an Ito process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. We prove the strong existence of stationary solutions for these SDEs as well as their strong uniqueness in an important special case. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding.

Inverting the Markovian projection, with an application to local stochastic volatility models

We propose two structural models for stochastic losses given default which allow to model the credit losses of a portfolio of defaultable financial instruments. The credit losses are integrated into a structural model of default events accounting for correlations between the default events and the associated losses. We show how the models can be calibrated and analyze the impact of correlations between the occurrences of defaults and recoveries by testing our models for a representative sample portfolio.

Two Models of Stochastic Loss Given Default

Interacting particle systems at the edge of multilevel Dyson Brownian motions

Nous accedons a l’extremite du spectre des ensembles beta gaussiens avec perturbation de rang un par l’entremise de grandes puissances des matrices tridiagonales qui y sont associees. Pour les valeurs traditionnelles $\beta =1,2,4$, ceci correspond a l’etude des fluctuations des valeurs propres maximales des matrices aleatoires GOE/GUE/GSE assujetties a une perturbation additive de rang un. En dimensions infinies, nos resultats nous menent vers une famille de semi-groupes de type Feynman–Kac qui, dans un cas extreme, correspond au stochastic Airy semigroup introduit par Gorin et Shkolnikov (Ann. Probab. 46 (2018) 2287–2344). De plus, nos resultats ont pour corollaire des formules de Feynman–Kac pour les spiked stochastic Airy operators introduits par Bloemendal et Virag (Probab. Theory Related Fields 156 (2013) 795–825). Ces formules sont exprimees a l’aide de certaines fonctionnelles du mouvement brownien reflechi et de ses temps locaux. Ce faisant, les operateurs en question peuvent etre etudies a l’aide du calcul stochastique. Nous obtenons un premier resultat dans cette lignee en demontrant une nouvelle identite decrivant la distribution du mouvement brownien reflechi ayant ete conditionne sur son temps local a zero. La principale innovation de notre demonstration consiste en la preuve d’un nouveau resultat sur l’approximation forte du mouvement brownien reflechi et de son temps local par une marche aleatoire non negative en utilisant la methode de reflexion de Skorokhod.

Mykhaylo Shkolnikov

Papers

Competing Particle Systems Evolving by I.I.D. Increments

Inverting the Markovian projection, with an application to local stochastic volatility models

Two Models of Stochastic Loss Given Default

Interacting particle systems at the edge of multilevel Dyson Brownian motions

Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions