# Seminars 2011/12

### 17/10/11: Mohammud Foondun (Loughborough)

**Stochastic heat equation with spatially coloured random forcing **

The aim of this talk is to present some results concerning the long term behaviour of a class of stochastic heat equations. We will showcase some relationship between Lévy processes and the solution to these SPDEs. We will talk about both the additive noise case as well as the multiplicative case. We will also cover the case of white noise and various other coloured noise.

### 24/10/11: Sergey Nadtochiy (Oxford)

**Static hedging of barrier options: exact solutions and semi-robust extensions**

We solve the problem of static hedging of standard (call, put and digital) barrier options in models where the underlying is given by a time-homogeneous diffusion process with, possibly, independent stochastic time-change. The main result of the paper includes analytic expression for the payoff of a (single) European-type contingent claim (which pays a certain function of the underlying value at maturity, without any path-dependence), such that it has the same price as the barrier option up until hitting the barrier. We then address the issues of numerical approximation of the static hedge payoff, and, in particular, investigate the performance of the approximate static hedge consisting of vanilla (call and put) options of two strikes only. Finally, we show how the above results allow to construct static sub- and super-replicating strategies which are semi-robust with respect to implied volatility. More precisely, for each range of the implied volatility values, we construct the static sub- and super-replicating strategies which work in any continuous model for the underlying, as long as the corresponding implied volatility stays within the prescribed range.

(Joint work with Peter Carr)

### 31/10/11: Jozsef Lorinczi (Loughborough)

**Spectral Properties of some Non-Local Operators through Stochastic Methods **

Fractional Schrödinger and jump-diffusion operators provide useful tools in modelling quantum and anomalous kinetic phenomena. Driven by such applications I will formulate some problems involving spectral and analytic properties of semigroups generated by these operators. Then I will discuss some of these properties by using a probabilistic representation of the semigroups.

### 7/11/11: Alex Mijatovic (Warwick)

**Limit Distributions of Continuous-State Branching Processes with Immigration **

In this talk we describe the characterisation of the limit distributions (as time tends to infinity) of the class of continuous-state branching processes with immigration (CBI-processes). The Levy-Khintchine triplet of the limit distribution L of the process X will be given explicitly in terms of the characteristic triplets of the Levy subordinator XF and the spectrally positive Levy process XR, which describe the immigration and branching mechanism of the CBI-process X respectively. The Levy density of L is essentially given by the generator of XF applied to the scale function of the spectrally positive Levy process XR. We will also show that the class of limit distributions of CBI-processes is strictly larger (resp. smaller) than the class of self-decomposable (resp. infinitely divisible) distributions. This is joint work with Martin Keller-Ressel from TU Berlin.

### 14/11/11: Peter Mörters

**Shifting Brownian motion **

Let \(\{B_t : t\in\mathbb{R}\}\) be a standard linear Brownian motion. A stopping time T is called an unbiased shift if \(\{B_{T+t}-B_T : t\in\mathbb{R}\}\) is a Brownian motion independent of \(B_T\). We solve the Skorokhod embedding problem for unbiased shifts and discuss optimality of our solution. The talk is based on joint work with Günter Last (Karlsruhe) and Hermann Thorisson (Reykjavik).

### 18/11/11: Louigi Addario-Berry (McGill University)

**Cutting down trees with a Markov chainsaw **

Let \(T_n\) be a uniformly random rooted tree on labels \(1,\ldots,n\). Removing an edge from \(T_n\) separates \(T_n\) into two components. Throw away the component not containing the root. Repeat this process until all that is left is the root, and call the number of edge removals (cuts) \(C_n\). In 2003, Panholzer showed that \(C_n/\sqrt{n}\) converges in distribution to a Rayleigh random variable; Janson (2005) generalized the result to all critical conditioned Galton-Watson trees with finite variance offspring distribution.

We present a new, coupling-based proof of Janson's result, based on a modification of the Aldous-Broder algorithm for generating spanning trees. Our approach also yields a family of novel transformations from Brownian excursion to Brownian bridge, and uncovers how to reconstruct a CRT from the sawdust created by the Aldous--Pitman fragmentation.

Based on joint work with Nicolas Broutin, Cecilia Holmgren, and Gregory Miermont.

### 28/11/11: Marion Hesse

**Branching Brownian motion in a strip: Survival near criticality **

We consider a branching Brownian motion in which particles are killed on exiting a strip and study the evolution of the process as the width of the strip shrinks to the critical value at which survival is no longer possible. We obtain asymptotics for the survival probability near criticality and a quasi-stationary limit result for the process conditioned on survival.

### 5/12/11: Alex Watson

**Censored stable processes **

We describe a method for exploiting the self-similarity of stable processes in order to obtain some new hitting identities.

Our technique is to construct a positive, self-similar Markov process which we call the censored stable process, and from this obtain a new Levy process via the Lamperti transformation. It happens that we can find the Wiener-Hopf factorisation of this Levy process explicitly, which allows us to compute a number of key quantities for it. These translate to identities of the original stable process.

This is joint work with Andreas Kyprianou and Juan-Carlos Pardo (CIMAT).

### 12/12/11: Perla Sousi (Cambridge)

**The effect of drift on the volume of the Wiener sausage and the dimension of the Brownian path **

The Wiener sausage at time t is the algebraic sum of a Brownian path on [0,t] and a ball. Does the expected volume of the Wiener sausage increase when we add drift? How do you compare the expected volume of the usual Wiener sausage to one defined as the algebraic sum of the Brownian path and a square (in 2D) or a cube (in higher dimensions)? We will answer these questions using their relation to the detection problem for Poisson Brownian motions, and rearrangement inequalities on the sphere.

### 14/12/11: Amandine Veber (CMAP, École Polytechnique)

**Large-scale behaviour of the spatial Lambda-Fleming-Viot process **

The SLFV process is a population model in which individuals live in a continuous space. Each of them also carries some heritable genetic type or allele. We shall describe the long-term behaviour of this measure-valued process and that of the corresponding genealogical process of a sample of individuals in two cases : one that mimics the evolution of nearest-neighbour voter model (but in a spatial continuum), and one that allows some individuals to send offspring at very large distances. This is a joint work with Nathanaël Berestycki and Alison Etheridge.

### 9/1/12: Malwina Luczak (Sheffield)

**The supermarket model with arrival rate tending to 1 **

There are \(n\) queues, each with a single server. Customers arrive in a Poisson process at rate \(\lambda n\), where \(0 < \lambda = \lambda (n) < 1\). Upon arrival each customer selects \(d = d(n) \ge 1\) servers uniformly at random, and joins the queue at a least-loaded server among those chosen. Service times are independent exponentially distributed random variables with mean 1.

We will review the literature, including results of Luczak and McDiarmid (2006), for the case where \(\lambda\) and \(d\) are constants independent of \(n\).

We will then investigate the speed of convergence to equilibrium and the maximum length of a queue in the equilibrium distribution when \(\lambda (n) \to 1\) and \(d(n) \to \infty\) as \(n \to \infty\). This is joint work with Graham Brightwell.

### 30/1/12: Mathew Penrose

**Connectivity of \(G(n,r,p)\) **

Consider a graph on \(n\) vertices placed uniformly independently at random in the unit square, in which any two vertices distant at most \(r\) apart are connected by an edge with probability \(p\). This generalises both the classical random graph and the random geometric graph. We discuss the chances of its being disconnected without having any isolated vertices, when \(n\) is large, for various choices of the other parameters.

### 6/2/12: Martin Barlow (UBC)

**The uniform spanning tree in two dimensions **

This talk will discuss properties of the UST on the Euclidean lattice, and in particular with the relation between distance in the tree and Euclidean distance. These results can then be applied to study SRW on the UST. (Martin Barlow and Robert Masson)

### 13/2/12: Antal Jarai

**Minimal configurations and sandpile measures **

We give a brief introduction to the Abelian sandpile model, and its relationship to uniform spanning trees. We give a new simple construction of the sandpile measure on an infinite graph \(G\), under the sole assumption that each tree in the Wired Uniform Spanning Forest on \(G\) has one end almost surely. For so called generalized minimal configurations the limiting probability on \(G\) exists even without this assumption. We also give determinantal formulas for minimal configurations on general graphs in terms of the transfer current matrix. (Joint work with N. Werning)

### 20/2/12: Emmanuel Jacob (Paris VI)

**Second order reflections of the integrated Brownian motion **

### 27/1/12: Nathalie Eisenbaum (Paris VI)

**Characterization of positively correlated squared Gaussian processes **

When does a centered Gaussian vector have the property of positive correlation (also called association or positive association) ? The answer is found by Loren Pitt in 1982. In 1991 Steve Evans raises the problem of the characterization of the centered Gaussian vectors \((\eta_1, \eta_2,..., \eta_d)\) such that \((\eta_1^2, \eta_2^2,..., \eta_d^2)\) is positively correlated. This talk will present a solution to that problem.

### 5/3/12: Stephan Luckhaus (Leipzig)

**Gibbs Gradient Young measures, a stochastic version of twoscale convergence in nonlinear elasticity **

### 12/3/12: Vitali Wachtel (LMU, Munich)

**Random walks in cones **

We study the asymptotic behaviour of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone.

### 19/3/12: Christian Mönch

**Sublogarithmic distances in preferential attachment networks **

Preferential attachment networks with power law degree sequence undergo a phase transition when the power law exponent \(\tau\) changes. For \(\tau>3\) typical distances in the network are logarithmic in the size of the network and for \(\tau < 3\) they are doubly logarithmic. I will discuss the latter case for a sublinear preferential attachment model with Poisson outdegrees and then turn to the critical behaviour at \(\tau=3\).

### 26/3/12: Max von Renesse (LMU, Munich)

**Hamiltonian Mechanics on Wasserstein Space and Quantum Fluid Models **

Otto's Riemannian Framework for the Wasserstein space of probablity measures allows not only for first order gradient flows but also for second order Hamiltonian ODEs. As a result we give a concise representation of the SchrÃ¶dinger equation for wave functions as an instance of Newton's classical law of motion on Wasserstein space, the two representations being related by a natural sympelctic morphism. Introducting friction leads to dissipative quantum fluid models such as the Quantum Navier Stokes equation, which was derived as a model for a tagged particle in a many body quantum system.

Partially based on joint works with A. Jüngel and P. Fuchs (Vienna)

### 16/4/12: Richard Cowan (Sydney)

**A topological identity for the convex hulls of \(n\) arbitrary points in \(\mathbb{R}^d\) - and its role in the study of convex hulls of random points **

The author will speak about a new topologically identity that he discovered in 2007. It concerns convex hulls of \(n\) arbitrarily positioned points in \(d\)-dimensional Euclidean space. The identity gives insights into the more specialised situation where the points are randomly placed. Whilst the talk is being presented to the Probability Group, it should be of interest to others in the department.

### 21/5/12: Rafal Lochowski (Warsaw School of Economics)

**Truncated variation of a stochastic process - its optimality for processes with cadlag trajectories and its characterisations for semimartingales, diffusions and Wiener process.**

For a given function \(f : [a; b] \to R\), we define its truncated variation at the level \(c >0\), \(TV^c (f , [a; b])\), as the smallest possible and attainable total variation of a function uniformly approximating \(f\) with accuracy \(c/2\). It appears, that for \(f\) being a cadlag function its truncated variation is always finite, in opposite to total variation, which is a limit value of \(TV^c (f , [a; b])\) as \(c\) tends to \(0\) and may be infinite.

Together with \(TV^c (f , [a; b])\), we define two other functional related - upward and downward truncated variations. This may be viewed as a generalization of Hahn-Jordan decomposition of a function with finite total variation.

In the second part of my talk I will present briefly results on behaviour of \(TV^c (X, [0; T])\) for semimartingales, diffusions and Wiener process. For \(X\) being a semimartingale, I will present results on a.s. functional convergence of the process \(c TV^c (X, [0; t])\) on the interval \(0 \leq t \leq T\) as \(c\) tends to \(0\); for \(X\) being a diffusion, I will present results on weak functional convergence of the process \(TV^c (X, [0; t]) - _t/c\) on the interval \(0 \leq t \leq T\) as c tends to 0 and for Wiener process with drift I will present full characterisation of \(TV^c (X, [0; T])\) via its Laplace transform. The majority of my talk will be based on the join work with Piotr Milos.

### 28/5/12: Christoph Höggerl

**Model-independent no-arbitrage conditions for American Put options **

Suppose European Put options with fixed maturity and for a finite number of strikes are traded in the market and these prices are consistent with no (model-independent) arbitrage We investigate necessary and sufficient conditions on the American Put option prices corresponding to the absence of arbitrage in the extended market, where not only European, but also American options can be traded.

### 13/6/12: Stavros Vakeroudis (Manchester)

**Windings of planar processes**

We first study the distribution of several first hitting times for the continuous winding process associated with the planar Brownian motion. To obtain analytical results, we use and Bougerol's celebrated identity in law. This allows us to characterize the laws of the hitting times of the bounadries of a cone for planar Brownian motion. We give some applications, namely: a new non-computational proof of Spitzer's asymptotic theorem, some integrability properties for these hitting times, similar results for the (planar) complex-valued Ornstein-Uhlenbeck processes case. Finally, we shall discuss the windings of planar stable processes, where the approach is different than in the Brownian motion case.