In this talk I will present recent results on ergodicity of Markov semigroups in large dimensional spaces including interacting Levy type systems as well as some R-D models.
In this talk, we present an extension of the theory of rough paths to partial differential equations. This allows a robust approach to stochastic partial differential equations, and in particular we can replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. This is joint work with Peter Friz in Cambridge.
One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating individuals in a sample of neutral genes from a panmictic population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees before describing a new approach (joint work with Nick Barton, IST Austria) to modelling the evolution of populations distributed in a spatial continuum.
I talk about a recent article of mine that aims at giving an alternative proof to a formula by Carne on random walks. Consider a discrete, reversible random walk on a graph (not necessarily the simple walk); then one has a surprisingly simple formula bounding the probability of getting from a vertex x at time 0 to another vertex y at time t, where it appears a universal Gaussian factor essentially depending on the graph distance between x and y. While Carne proved that result in 1985, through‘miraculous’ (though very pretty!) spectral analysis reasoning, I will expose my own ‘natural' probabilistic proof of that fact. Its main interest is philosophical, but it also leads to a generalization of the original formula. The two main tools we shall use will be techniques of forward and backward martingales, and a tricky conditioning argument to prevent a random walk from being `’too transient'.
A new general approach to optimal stopping problems in L\'evy models, regime switching L\'evy models and L\'evy models with stochastic volatility and stochastic interest rate is developed. For perpetual options, explicit solutions are found, for options with finite time horizon, time discretization is used, and explicit solutions are derived for resulting sequences of perpetual options.
The main building block is the option to abandon a monotone payoff stream. The optimal exercise boundary is found using the operator form of the Wiener-Hopf method, which is standard in analysis, and interpretation of the factors as {\em expected present value operators} (EPV-operators) under supremum and infimum processes.
Other types of options are reduced to the option to abandon a monotone stream. For regime-switching models, an additional ingredient is an efficient iteration procedure.
L\'evy models with stochastic volatility and/or stochastic interest rate are reduced to regime switching models using discretization of the state space for additional factors. The efficiency of the method for 2 factor L\'evy models with jumps and for 3-factor Heston model with stochastic interest rate is demonstrated. The method is much faster than Monte-Carlo methods and can be a viable alternative to Monte Carlo method as a general method for 2-3 factor models.
Joint work of Svetlana Boyarchenko,University of Texas at Austin and Sergei Levendorski\v{i},
University of Leicester
We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information.
We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy
Particle current is the net number of particles that pass an observer who moves with a deterministic velocity V. Its fluctuations in time-stationary interacting particle systems are nontrivial and draw serious attention. It has been known for a while that in most models diffusive scaling and the corresponding Central Limit Theorem hold for this quantity. However, such normal fluctuations disappear for a particular value of V, called the characteristic speed.
For this velocity value, the correct scaling of particle current fluctuations was shown to be t1/3 and the limit distribution was also identified by K. Johansson in 2000 and later by P. L. Ferrari and H. Spohn in 2006. These results use heavy combinatorial and analytic tools, and their application is limited to a few particular models, one of which is the totally asymmetric simple exclusion process (TASEP). I will explain a purely probabilistic, more robust approach that provides the t2/3-scaling of current variance, but not the limit distribution, in (non-totally) asymmetric simple exclusion (ASEP) and some other particle systems. I will also point out a key feature of the models which allows the proof of such universal behaviour.
Joint work with Júlia Komjáthy and Timo Seppälläinen)
The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, we prove that, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss the link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on the Poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes (joint work with Zhonggen Su).
We present the ideas of Malliavin calculus in the context of rough differential equations (RDEs) driven by Gaussian signals. We then prove an analogue of Hörmander's theorem for this set-up, finishing with the conclusion that, for positive times, a solution to an RDE driven by Gaussian noise will have a density with respect to Lebesgue measure under Hörmander's conditions on the vector fields.
We consider one-dimensional Brownian motion conditioned (in a suitable
sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.5860... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). I will also describe other conditionings of Brownian motion in which this principle of entropic repulsion manifests itself.
Joint work with Itai Benjamini.