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

Building on the line of work [DIRT15a], [DIRT15b], [NS17a], [DT17], [HLS18], [HS18] we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures, and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the "times of fragility" of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells "synchronize") explicitly in terms of the dynamics of the driving processes, the current distribution of the particles' values, and the topology of the underlying network (represented by its Perron-Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes: i.e., the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder's fixed-point theorem for the Skorokhod space with the M1 topology, and the application of the max-plus algebra to the equilibrium version of the network flow problem.

Mean field systems on networks, with singular interaction through hitting times

We develop a general theory of intertwined diffusion processes of any dimension. Our main result gives an SDE characterization of all possible intertwinings of diffusion processes and shows that they correspond to nonnegative solutions of hyperbolic partial differential equations. For example, solutions of the classical wave equation correspond to the intertwinings of two Brownian motions. The theory allows us to unify many older examples of intertwinings, such as the process extension of the beta-gamma algebra, with more recent examples such as the ones arising in the study of two-dimensional growth models. We also find many new classes of intertwinings and develop systematic procedures for building more complex intertwinings by combining simpler ones. In particular, `orthogonal waves' combine unidimensional intertwinings to produce multidimensional ones. Connections with duality, time reversals and Doob's h-transforms are also explored.

Intertwining diffusions and wave equations

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investor's initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.

Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

We prove a Large Deviations Principle (LDP) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the approriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to the porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a LDP for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for a certain tilted version of the porous medium equation.

Large Deviations for Diffusions Interacting Through Their Ranks

Nous etudions des systemes de particules browniennes sur l’axe reel qui interagissent de facon asymetrique en fonction de leurs temps locaux de collision. Nous prouvons l’existence et l’unicite au sens fort de tels processus et les identifions avec un systeme de particules constitue d’une famille de processus browniens ordonnes ou les coefficients de derive et de diffusion, ainsi que les temps locaux de collision entre les particules, dependent de leurs rangs. Ces systemes browniens peuvent etre compris comme des generalisations de processus stochastiques issus des marches boursiers et de la gestion de portefeuilles. Nous pouvons pallier certaines lacunes de ces modeles tout en conservant la possibilite d’effectuer des calculs explicites. Nous montrons aussi, qu’en plus de leur interet intrinseque, de tels systemes de particules browniennes apparaissent comme des limites universelles de processus de sauts sur un reseau avec des interactions locales. Une etape clef dans la preuve est l’analyse d’une generalisation des applications de Skorokhod qui inclut les temps locaux a l’intersection des faces de l’orthant positif. Le resultat generalise la convergence du processus d’exclusion simple totalement asymetrique (TASEP) vers sa version continue. Finalement, nous identifions parmi les systemes de particules browniennes ceux qui possedent une structure probabiliste determinantale.

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Mykhaylo Shkolnikov

Papers

Mean field systems on networks, with singular interaction through hitting times

Intertwining diffusions and wave equations

Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

Large Deviations for Diffusions Interacting Through Their Ranks

Systems of Brownian particles with asymmetric collisions