Stochastic Analysis Seminar (past)

Mon, 13/05 15:45	KALLE KYTOLA (Helsinki University)	Stochastic Analysis Seminar	Oxford-Man Institute
	Random conformally invariant curves and quantum group techniques
	In this talk we consider two questions about conformally invariant random curves known as Schramm-Loewner evolutions (SLE). The first question is about the "boundary zig-zags", i.e. the probabilities for a chordal SLE to pass through small neighborhoods of given boundary points in a given order. The second question is that of obtaining explicit descriptions of "multiple SLE pure geometries", i.e. those extremal multiple SLE probability measures which can not be expressed as non-trivial convex combinations of other multiple SLEs. For both problems one needs to find solutions of a system of partial differential equations with asymptotics conditions written recursively in terms of solution of the same problem with a smaller number of variables. We present a general correspondence, which translates these problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2), and we then explicitly solve these systems. The talk is based on joint works with Eveliina Peltola (Helsinki), and with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete).

Mon, 13/05 14:15	THIERRY BODINEAU (Ecole Normale Superieure)	Stochastic Analysis Seminar	Oxford-Man Institute
	Metastability and interface motion in disordered media
	We will first review the return to equilibrium of the Ising model when a small external field is applied. The relaxation time is extremely long and can be estimated as the time needed to create critical droplets of the stable phase which will invade the whole system. We will then discuss the impact of disorder on this metastable behavior and show that for Ising model with random interactions (dilution of the couplings) the relaxation time is much faster as the disorder acts as a catalyst. In the last part of the talk, we will focus on the droplet growth and study a toy model describing interface motion in disordered media.

Mon, 29/04 15:15	HORATIO BOEDIHARDJO (University of Oxford)	Stochastic Analysis Seminar	Oxford-Man Institute
	Uniqueness of Signature
	We relate the expected signature to the Fourier transform of n-point functions, first studied by O. Schramm, and subsequentlyby J. Cardy and Simmon, D. Belyaev and J. Viklund. We also prove that the signatures determine the paths in the complement of a Chordal SLE null set. In the end, we will also discuss an idea on how to extend the uniqueness of signatures result by Hambly and Lyons (2006) to paths with finite 1<p<2variations.

Mon, 29/04 14:15	PENG HU (University of Oxford)	Stochastic Analysis Seminar	Oxford-Man Institute
	Particle methods with applications in finance
	Abstract: The aim of this lecture is to give a general introduction to the theory of interacting particle methods and an overview of its applications to numerical finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to deal with a variety of financial numerical problems such as: pricing complex path dependent European options, computing sensitivities, American option pricing or solving numerically partially observed control problems.

Mon, 22/04 15:45	MATTHIAS MEINERS (University Meunster)	Stochastic Analysis Seminar	Oxford-Man Institute
	"Generalized equations of stability".
	In many models of Applied Probability, the distributional limits of recursively defined quantities satisfy distributional identities that are reminiscent of equations of stability. Therefore, there is an interest in generalized concepts of equations of stability. One extension of this concept is that of random variables “stable by random weighted mean” (this notion is due to Liu). A random variable $X$ taking values in $\mathbb{R}^d$ is called “stable by random weighted mean” if it satisfies a recursive distributional equation of the following type: \begin{equation} \tag{1} \label{eq:1} X ~\stackrel{\mathcal{D}}{=}~ C + \sum_{j \geq 1} T_j X_j. \end{equation} Here, “ $\stackrel{\mathcal{D}}{=}$ ” denotes equality of the corresponding distributions, $(C,T_1,T_2,\ldots)$ is a given sequence of real-valued random variables, and $X_1, X_2, \ldots$ denotes a sequence of i.i.d.\;copies of the random variable $X$ that are independent of $(C,T_1,T_2,\ldots)$ . The distributions $P$ on $\mathbb{R}^d$ such that \eqref{eq:1} holds when $X$ has distribution $P$ are called fixed points of the smoothing transform (associated with $(C,T_1,T_2,\ldots)$ ). A particularly prominent instance of \eqref{eq:1} is the {\texttt Quicksort} equation, where $T_1 = 1-T_2 = U \sim \mathrm{Unif}(0,1)$ , $T_j = 0$ for all $j \geq 3$ and $C = g(U)$ for some function $g$ . In this talk, I start with the {\texttt Quicksort} algorithm to motivate the study of \eqref{eq:1}. Then, I consider the problem of characterizing the set of all solutions to \eqref{eq:1} in a very general context. Special emphasis is put on endogenous solutions to \eqref{eq:1} since they play an important role in the given setting.

Mon, 22/04 14:15	DAVID KELLY (University of Warwick)	Stochastic Analysis Seminar	Oxford-Man Institute
	Ito's formula via rough paths.
	Abstract: Non-geometric rough paths arise when one encounters stochastic integrals for which the the classical integration by parts formula does not hold. We will introduce two notions of non-geometric rough paths - one old (branched rough paths) and one new (quasi geometric rough paths). The former (due to Gubinelli) assumes one knows nothing about products of integrals, instead those products must be postulated as new components of the rough path. The latter assumes one knows a bit about products, namely that they satisfy a natural generalisation of the "Ito" integration by parts formula. We will show why they are both reasonable frameworks for a large class of integrals. Moreover, we will show that Ito's formula can be derived in either framework and that this derivation is completely algebraic. Finally, we will show that both types of non-geometric rough path can be re-written as geometric rough paths living above an extended version of the original path. This means that every non-geometric rough differential equation can be re-written as a geometric rough differential equation, hence generalising the Ito-Stratonovich correction formula.

Mon, 11/03 15:45	NIKOLAOS ENGLEZOS (University of Piraeus)	Stochastic Analysis Seminar	Oxford-Man Institute
	Random FBSDEs: Burgers SPDEs, Rational Expectations / Consol Rate Models, Control for Large Investors, and Stochastic Viscosity Solutions.
	Abstract: Burgers equation is a quasilinear partial differential equation (PDE), proposed in 1930's to model the evolution of turbulent fluid motion, which can be linearized to the heat equation via the celebrated Cole-Hopf transformation. In the first part of the talk, we study in detail general versions of stochastic Burgers equation with random coefficients, in both forward and backward sense. Concerning the former, the Cole-Hopf transformation still applies and we reduce a forward stochastic Burgers equation to a forward stochastic heat equation that can be treated in a “pathwise" manner. In case of deterministic coefficients, we obtain a probabilistic representation of the Cole-Hopf transformation by associating the backward Burgers equation with a system of forward-backward stochastic differential equations (FBSDEs). Returning to random coefficients, we exploit this representation in order to establish a stochastic version of the Cole-Hopf transformation. This generalized transformation allows us to find solutions to a backward stochastic Burgers equation through a backward stochastic heat equation, subject to additional constraints that reflect the presence of randomness in the coefficients. In both settings, forward and backward, stochastic Feynman-Kac formulae are derived for the solutions of the respective stochastic Burgers equations, as well. Finally, an application that illustrates the obtained results is presented to a pricing/hedging problem arising from mathematical finance. In the second part of the talk, we study a class of stochastic saddlepoint systems, represented by fully coupled FBSDEs with infinite horizon, that gives rise to a continuous time rational expectations / consol rate model with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness and dependence on a parameter of adapted solutions. Making further the connection with quasilinear backward stochastic PDEs (BSPDEs), we are led to the notion of stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of a large investor is also provided as an application to this framework. This is joint work with N. Frangos, X.- I. Kartala and A. N. Yannacopoulos*

Mon, 11/03 14:15	SANDIE DAVIE (University of Edinburgh)	Stochastic Analysis Seminar	Oxford-Man Institute
	Pathwise approximation of SDE solutions using coupling
	The standard Taylor series approach to the higher-order approximation of vector SDEs requires simulation of iterated stochastic integrals, which is difficult. The talk will describe an approach using methods from optimal transport theory which avoid this difficulty in the case of non-degenerate diffusions, for which one can attain arbitrarily high order pathwise approximation in the Vaserstein 2-metric, using easily generated random variables.

Mon, 04/03 15:45	SAMUEL COHEN (University of Oxford)	Stochastic Analysis Seminar	Oxford-Man Institute
	Uniformly Uniformly-Ergodic Markov Chains and applications
	If one starts with a uniformly ergodic Markov chain on countable states, what sort of perturbation can one make to the transition rates and still retain uniform ergodicity? In this talk, we will consider a class of perturbations, that can be simply described, where a uniform estimate on convergence to an ergodic distribution can be obtained. We shall see how this is related to Ergodic BSDEs in this setting and outline some novel applications of this approach.

Mon, 04/03 14:15	IOAN MANOLESCU (University of Cambridge)	Stochastic Analysis Seminar	Oxford-Man Institute
	Bond Percolation on Isoradial Graphs
	The star-triangle transformation is used to obtain an equivalence extending over a set bond percolation models on isoradial graphs. Amongst the consequences are box-crossing (RSW) inequalities and the universality of alternating arms exponents (assuming they exist) for such models, under some conditions. In particular this implies criticality for these models. (joint with Geoffrey Grimmett)

Mon, 25/02 15:45	JOHANNES RUF (University of Oxford)	Stochastic Analysis Seminar	Oxford-Man Institute
	Nonnegative local martingales, Novikow's and Kazamaki's criteria, and the distribution of explosion times
	I will give a new proof for the famous criteria by Novikov and Kazamaki, which provide sufficient conditions for the martingale property of a nonnegative local martingale. The proof is based on an extension theorem for probability measures that can be considered as a generalization of a Girsanov-type change of measure. In the second part of my talk I will illustrate how a generalized Girsanov formula can be used to compute the distribution of the explosion time of a weak solution to a stochastic differential equation

Mon, 25/02 14:15	SAM FINCH (University of Copenhagen)	Stochastic Analysis Seminar	Oxford-Man Institute
	Poisson random forests and coalescents in expanding populations.
	Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one. Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly. We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D. We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.

Mon, 18/02 15:45	GIDI AMIR (Bar-Ilan University)	Stochastic Analysis Seminar	Oxford-Man Institute
	A continuum of exponents for the rate of escape of random walks on groups
	Abstract: A central question in the theory of random walks on groups is how symmetries of the underlying space gives rise to structure and rigidity of the random walks. For example, for nilpotent groups, it is known that random walks have diffusive behavior, namely that the rate of escape, defined as the expected distance of the walk from the identity satisfies E\|Xn\|~=n^{1/2}. On nonamenable groups, on the other hand we have E\|Xn\| ~= n. (~= meaning upto (multiplicative) constants ) In this work, for every 3/4 <= \beta< 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point after n steps is n^\beta (up to constants). This answers a question of Vershik, Naor and Peres. In other examples, the speed exponent can fluctuate between any two values in this interval. Previous examples were only of exponents of the form 1-1/2^k or 1 , and were based on lamplighter (wreath product) constructions.(Other than the standard beta=1/2 and beta=1 known for a wide variety of groups) In this lecture we will describe how a variation of the lamplighter construction, namely the permutational wreath product, can be used to get precise bounds on the rate of escape in terms of return probabilities of the random walk on some Schreier graphs. We will then show how groups of automorphisms of rooted trees, related to automata groups , can be constructed and analyzed to get the desired rate of escape. This is joint work with Balint Virag of the University of Toronto. (Paper available at http://arxiv.org/abs/1203.6226) No previous knowledge of random walks,automaton groups or wreath products is assumed.

Mon, 18/02 14:15	NICOLAS PERKOWSKI (Humboldt University, Berlin)	Stochastic Analysis Seminar	Oxford-Man Institute
	Rough paths, controlled distributions, and nonlinear SPDEs
	Abstract: Hairer recently had the remarkable insight that Lyons' theory of rough paths can be used to make sense of nonlinear SPDEs that were previously ill-defined due to spatial irregularities. Since rough path theory deals with the integration of functions defined on the real line, the SPDEs studied by Hairer have a one-dimensional spatial index variable. I will show how to combine paraproducts, a notion from functional analysis, with ideas from the theory of controlled rough paths, in order to develop a formulation of rough path theory that works in any index dimension. As an application, I will present existence and uniqueness results for an SPDE with multidimensional spatial index set, for which previously it was not known how to describe solutions. No prior knowledge of rough paths or paraproducts is required for understanding the talk. This is joint work with Massimiliano Gubinelli and Peter Imkeller.

Mon, 11/02 15:45	Camilo Andres Garcia Trillos (University of Nice Sophia-Antipolis)	Stochastic Analysis Seminar	Oxford-Man Institute
	Numerical Solution of FBSDEs Using a Recombined Cubature Method
	(Joint work with P.E. Chaudru de Raynal and F. Delarue) Several problems in financial mathematics, in particular that of contingent claim pricing, may be casted as decoupled Forward Backward Stochastic Differential Equations (FBSDEs). It is then of a practical interest to find efficient algorithms to solve these equations numerically with a reasonable complexity. An efficient numerical approach using cubature on Wiener spaces was introduced by Crisan and Manoralakis [1]. The algorithm uses an approximation scheme requiring the calculation of conditional expectations, a task achieved through the cubature kernel approximation. This algorithm features several advantages, notably the fact that it is possible to solve with the same cubature tree several decoupled FBSDEs with different boundary conditions. There is, however, a drawback of this method: an exponential growth of the algorithm's complexity. In this talk, we introduce a modification on the cubature method based on the recombination method of Litterer and Lyons [2] (as an alternative to Tree Branch Based Algorithm proposed in [1]). The main idea of the method is to modify the nodes and edges of the cubature trees in such a way as to preserve, up to a constant, the order of convergence of the expectation and conditional expectation approximations obtained via the cubature method, while at the same time controlling the complexity growth of the algorithm. We have obtained estimations on the order of convergence and complexity growth of the algorithm under smoothness assumptions on the coefficients of the FBSDE and uniform ellipticity of the forward equation. We discuss that, just as in the case of the plain cubature method, the order of convergence of the algorithm may be degraded as an effect of solving FBSDEs with rougher boundary conditions. Finally, we illustrate the obtained estimations with some numerical tests. References [1] Crisan, D., and K. Manolarakis. “Solving Backward Stochastic Differential Equations Using the Cubature Method. Application to Nonlinear Pricing.” In Progress in Analysis and Its Applications, 389–397. World Sci. Publ., Hackensack, NJ, 2010. [2] Litterer, C., and T. Lyons. “High Order Recombination and an Application to Cubature on Wiener Space.” The Annals of Applied Probability 22, no. 4 (August 2012): 1301–1327. doi:10.1214/11-AAP786.

Mon, 11/02 14:15	ERIC CATOR (Delft University of Technology)	Stochastic Analysis Seminar	Oxford-Man Institute
	A randomluy forced Burgers equation on the real line
	In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used. This is joined work Yuri Bakhtin and Konstantin Khanin.

Mon, 04/02 14:15	MARTIN LARSSON (EPFL Swiss Finance Institute)	Stochastic Analysis Seminar	Oxford-Man Institute
	Filtration shrinkage, strict local martingales and the Follmer measure
	Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Mon, 28/01 15:45	CHRISTOPHE GARBAN (ENS Lyon)	Stochastic Analysis Seminar	Oxford-Man Institute
	Near-critical Ising mode.
	In this talk, I will present two results on the behavior of the Ising model on the planar lattice near its critical point: (i) In the first result (joint work with F.Camia and C. Newman), we will fix the temperature to be the critical temperature T_c and we will vary the magnetic field h \geq 0. Our main result states that in the plane Z^2, the average magnetization at the origin behaves up to constants like h^{1/15}. This result is interesting since the classical computa- tion of the average magnetization by Onsager requires the external magnetic field h to be exactly 0 . (ii) In the second result (joint work with H. Duminil-Copin and G. Pete), we focus on the correlation length of the Ising model when h is now fixed to be zero and one varies instead the temperature T around T_c. In rough terms, if T

Mon, 28/01 14:15	OMER ANGEL (University of British Colombia)	Stochastic Analysis Seminar	Oxford-Man Institute
	Half planar random maps
	Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

Mon, 21/01 15:45	RONGCHAN ZHU (Bielefeld University)	Stochastic Analysis Seminar	Oxford-Man Institute
	The stochastic quasi-geostrophic equation
	In this talk we talk about the 2D stochastic quasi-geostrophic equation on T2 for general parameter _ 2 (0; 1) and multiplicative noise. Weprove the existence of martingale solutions and Markov selections for multiplicative noise for all _ 2 (0; 1) . In the subcritical case _ > 1=2, we prove existence and uniqueness of (probabilistically) strong solutions. We obtain the ergodicity for _ > 1=2 for degenerate noise. We also study the long time behaviour of the solutions tothe 2D stochastic quasi-geostrophic equation on T2 driven by real linear multiplicative noise and additive noise in the subcritical case by proving the existence of a random attractor.

Forthcoming Seminars

Mon, 20/05/2013 - 2:15pm

Eigenvalues of large random matrices, free probability and beyond.

Speaker: CAMILLE MALE
Mon, 20/05/2013 - 2:15pm

Four-manifolds, surgery and group actions

Speaker: Ian Hambleton
Mon, 20/05/2013 - 3:45pm

Fibering 5-manifolds with fundamental group Z over the circle

Speaker: Yang Su
Mon, 20/05/2013 - 3:45pm

Random Wavelet Series

Speaker: STEPHANE JAFFARD
Mon, 20/05/2013 - 4:00pm

The Hasse Principle for the Equation Polynomial=Norm: An Approach Using Sieve Theory

Speaker: Alastair Irving
Mon, 20/05/2013 - 5:00pm

Analysis of some nonlinear PDEs from multi-scale geophysical applications

Speaker: Bin Cheng
Tue, 21/05/2013 - 12:00pm

Quantum information processing in spacetime

Speaker: Ivette Fuentes (Nottingham)

Stochastic Analysis Seminar (past)

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