Past Internal Probability Seminars

Internal Probability Seminars 2009/10

• 17 May
  2010
  A duality principle for the Legendre transform
  G. Peskir
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 10 May
  2010
  On uniqueness of Dirichlet operators to Gibbs measures on a path space with exponential interactions
  H. Kawabi
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 26 Apr
  2010
  Stochastic stability of the Ekman spiral
  W. Stannat
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 15 Mar
  2010
  Stochastic Integration for Levy processes in Banach Spaces
  M. Riedle
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

  In the past few years, there has been an increasing interest on stochastic integration in Banach spaces. In this talk, a stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Levy processes is defined. There are no conditions on the Banach spaces nor on the Levy processes. The integral is defined analogously to the Pettis integral. We apply this stochastic Pettis integral for the Levy-Ito decomposition of Levy processes in Banach spaces. Although this decomposition is well known, see Linde [3], the small jumps are only represented as a limit. It turns out that the stochastic Pettis integral allows a representation of the small terms as a stochastic integral with respect to the compensated random Poisson measure as in the finite-dimensional situation. This part of the talk is based on the publication [1]. In the second part of the talk we relate the integrability of a function with respect to a Levy process in a Banach space with properties of a corresponding operator. For the analogue stochastic Pettis integral but with respect to a Wiener process in a Banach space the integrability of a function is equivalent to the condition that the embedding of a certain Hilbert space into the Banach space is g-radonifying, where g is the standard Gaussian cylindrical measure on the Hilbert space. It turns out, that in the situation of a Levy process one can find an analogue cylindrical measure m on a Hilbert space such that the integrability of a function is equivalent to the condition that the embedding radonifies the cylindrical measure m, i.e. the image measure extends to a Radon measure. We give a detailed introduction on these relations in the case of a Wiener process and a Levy process. Finally, we use the requirement on the embedding to be m-radonifying to derive simple conditions for the integrability of a function in some specific situations. In particular, we consider the case of Hilbert spaces and Banach spaces of type p or cotype q. Important for many applications is the situation where the distribution of the Levy process is stable. In this situation, we present further conditions which are easy to verify and which guarantee the integrability of a function. The second part of the talk is based on [2]. References [1] M. Riedle and O. van Gaans, Stochastic integration for Levy processes with values in Banach spaces, Stochastic processes and their applications, 119, 6, 1952-1974, (2009). [2] M. Riedle and O. van Gaans, Stochastic integrals and radonifying operators, Preprint, 119, (2010). [3] W. Linde, Probability in Banach spaces - stable and infinitely divisible distributions, Chichester: John Wiley & Sons, (1986).

• 15 Feb
  2010
  An SPDE with the distributions of Levy processes as its invariant measures
  B. Xie
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

  It is well known that the Wiener measure (i.e. the distribution of a Brownian motion) is the invariant measure of the stochastic heat equation driven by a space-time Gaussian noise. So, it is natural to ask to whether the distribution of one dimensional Levy process will be invariant under a stochastic heat equation? In this talk, we will first construct a singular noise and then consider a linear heat equation on a half line with this noise to answer the above question. Our assumption on the corresponding Levy measure is, in general, mild except that we need its integrability to show that the distributions of Levy processes are the only invariant measures of the stochastic heat equation. The distribution of a stable process is an example.

• 8 Feb
  2010
  Local behaviour of passage times
  R. Doney
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 23 Nov
  2009
  Optimal Portfolio Selection in the Carbon Emissions Market
  J. Sexton
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  In 2008 new phase of an emissions trading scheme was launched in the EU in order to regulate the quantity of carbon dioxide emitted by key industries. We show using a Merton-type model that certain types of regulated firm should consider both abatement and active portfolio management in response to this legislation. We will also discuss briefly the problems encountered when extending this model to an incomplete market setting.
  Keywords: New markets; Utility indifference pricing; Portfolio optimisation; Stochastic control.

• 9 Nov
  2009
  First exit time of Brownian motion with constant drift from a wedge
  J. Moriarty
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  The reflection principle may be applied to the logarithm of a complex Brownian motion to obtain the distribution of the first exit time of Brownian motion with constant drift from a wedge. In this way, new explicit results for this problem are obtained.

• 26 Oct
  2009
  Some Inequalities for Brownian Motion
  G. Peskir
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 12 Oct
  2009
  Random walks and Levy processes conditioned to stay positive
  R. Doney
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 28 Sept
  2009
  Probability of extinction: multiple populations
  P. Neal
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

Internal Probability Seminars 2008/9

• 11 May
  2009
  Anticipating stochastic equations
  T. Zhang
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 27 Apr
  2009
  Inverses of Levy processes
  R. Doney
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 16 Mar
  2009
  A Sojourn Time Problem for Brownian Motion with Drift
  J. Du Toit
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 9 Mar
  2009
  An Affine Stochastic Functional Differential Equation Model of An Inefficient Financial Market
  S. Hossain
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 2 Mar
  2009
  The British Put-Call Symmetry
  G. Peskir
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

  I will review recent results/problems arising in the British pricing mechanism. This involves optimal stopping with non-monotone free boundaries. Keywords: British put option; British call option; the British put-call symmetry; optimal stopping; non-monotone free boundary.

• 23 Feb
  2009
  The British Option Continued
  F. Samee
  
  12pm - Alan Turing Building - Frank Adams Room
  Abstract

• 15 Dec
  2008
  Radonifying Operators and Stochastic Integration in Banach Spaces
  M. Riedle
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  NA
  
  

• 8 Dec
  2008
  NO SEMINAR
  
  
  Abstract

  NA
  
  

• 1 Dec
  2008
  Characteristics of Semimartingales in Hilbert Spaces and Applications to Levy Processes
  M. Liu
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  NA
  
  

• 24 Nov
  2008
  Local times of Levy processes
  R. Doney
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  NA
  
  

• 17 Nov
  2008
  Epidemic outbreaks amongst network populations
  P. Neal
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Most of the `classical' results for stochastic epidemic models are asymptotic results as the population size becomes infinitely large. However, the usual assumption the the population is homogeneously mixing becomes increasingly unrealistic as the population size grows. Therefore over the last fifteen years considerable attention has been devoted to the analysis of the spread of infectious diseases in heterogeneously mixing populations. One particular example of which is the two level mixing model where individuals make both global infections, uniformly at random with members of the entire population and local infections, within their locality where locality is model specific. In this talk I shall consider random graphs to model the local acquaintance structure. The asymptotic final size of the epidemic will be studied and a central limit theorem will be obtained for the size of a major epidemic outbreak with explicit calculations of the variance. The variance calculations are new and exploit the exchangeability of vertices in the construction of the random graph. This is accompanied by the derivation of a central limit theorem which is applicable to (random) graph valued random variables. Moreover, the ideas can be extended to epidemics upon random graphs in the absence of global infection.
  
  

• 3 Nov
  2008
  Optimal Stopping Games
  G. Peskir
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  I will present some results on optimal stopping games for Markov processes (Nash vs Stackelberg equilibrium).
  
  

• 13 Oct
  2008
  Stochastic Partial Differential Equations with Reflection
  T. Zhang
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  NA
  
  

• 29 Sept
  2008
  Exit problems associated with Coxeter groups
  J. Moriarty
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  I will explain how certain boundary value problems can be solved using reflection groups. Several examples will be given, including first exit times of Brownian motion from both Euclidean and non-Eucliean domains.

Internal Probability Seminars 2007/8

• 21 Apr
  2008
  When to Sell a Stock
  J Du Toit
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  None
  
  

• 14 Apr
  2008
  Coupons, Birthdays and Species
  J Moriarty
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  The coupon collector problem and the birthday problem provide a natural framework for many combinatorial questions, and their generalisations enjoy many applications. We will introduce the problems and some of their literature and present two recent results, which were motivated by a study in ecology.
  
  

• 10 Mar
  2008
  Law of the Iterated Logarithm for Levy Processes at Small Times
  M Savov
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  None
  
  

• 3 Mar
  2008
  Analytical and topological aspects of signatures
  SCP Yam
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  None
  
  

• 18 Feb
  2008
  More about the density of the supremum of a Stable process
  R Doney
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  None
  
  

• 4 Feb
  2008
  The two-dimensional stochastic Navier-Stokes Equation
  Tiange Xu
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  The incompressible Navier-Stokes equation is a well accepted model for atmosphere and ocean dynamics, and the stochastic Navier-Stokes equation has a long history as a model to understand external random force. In the first part of this talk, we introduce some backgrouds of 2-D stochastic Navier-Stokes equation, including the existence and uniqueness of the solution. In the second part, we will present the recent result of the small time asymptotic behaviour of the solution.
  
  

• 28 Jan
  2008
  Endemic equilibrium of the SIS Great circle epidemic model
  P. Neal
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  None
  
  

• 3 Dec
  2008
  A rapid optimisation methodology for demand-side management
  S.D. Howell
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  The optimal timing of temperature control exemplifies many intractable problems; the underlying variables change in continuous time, discretely and/or continuously, deterministically and/or stochastically (e.g. external temperature, internal temperature, marginal price of electricity, stage of the daily cycle, user's periods of occupation). Realism requires outputs at (of the order of) one million points in the state space. In this paper, we show that such a system can be accu- rately modelled using ideas based on financial mathematics concepts, which lead concisely to a partial differential equation (PDE), which can be rapidly solved numerically on a modern PC. The addition of suitable indicator functions to the PDE can model many of the system's physical and/or economic behaviours. Compared to simulation approaches (as used extensively in the past to compute systems of this type), we can report a speedup of approximately one million times. This significant increase in computational speed thereby permits an op- timisation of the control policy across the entire state space simultaneously, such that the solution subtly exploits the dynamics of the system, and beats a naive strategy on cost and comfort. Joint work with P.V. Johnson and P.W. Duck.
  
  

• 26 Nov
  2008
  The British Option
  Goran Peskir
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  We present a new put/call option where the buyer may exercise at any time prior to maturity whereupon his payoff is the `best prediction' of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is the protection feature which is key to the British option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements), he can substitute the true drift with the contract drift and minimise his losses. With the contract drift properly selected the British put option becomes a more `buyer friendly' alternative to the American put: when stock price movements are favourable, the buyer may exercise rationally to very comparable gains; when price movements are unfavourable he is afforded the unique protection described above. Moreover, the British put option is always cheaper than the American put. In the final part we present a brief review of optimal prediction problems which preceded the development of the British option. This is a joint work with F. Samee (Manchester).
  
  

• 5 Nov
  2008
  On the supremum of a stable process
  Ron Doney
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  NA
  
  

• 15 Oct
  2008
  The Ito-Levy decomposition for Levy processes in Banach spaces
  Markus Riedle
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  Levy processes play an important role in models of random evolutionary phenomena if the noise exhibits jumps. In finite dimensions the influence of the jumps on the dynamic can be modeled most sensitive by using the pathwise Ito-Levy decomposition of the Levy process. But in many models the complexity of the dynamics under consideration is often captured more effectively using stochastic processes with values in infinite dimensional spaces. For finite-dimensional Levy processes the pathwise Ito-Levy decomposition into its continuous and jump part is well-known and often used. In this decomposition, the small jumps are represented by a stochastic integral with respect to the compensated random Poisson measure. But for infinite-dimensional Levy processes in Banach spaces this integral can not be generalised directly. In this talk we introduce a new stochastic integral which allows to derive the pathwise decomposition for Levy process in Banach spaces without any further condition on the underlying Banach space or Levy process. This pathwise decomposition enables us to study stochastic evolutionary equations where the noise operator acts differently on the continuous and jump part of the Levy process.
  
  

• 8 Oct
  2008
  Reflected Brownian motion in a wedge: sum-of-exponential stationary densities
  John Moriarty
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  Reflected Brownian motion (RBM) in a two-dimensional wedge is a well-known stochastic process. With an appropriate drift, it is positive recurrent and has a stationary distribution / invariant measure, which is absolutely continuous with respect to Lebesgue measure. I will give necessary and sufficient conditions for the stationary density to be written as a finite sum of exponentials with linear exponents. Such densities are a natural generalisation of the stationary density of one-dimensional RBM. Using geometric ideas reminiscent of the reflection principle, I will give an explicit formula for the density in such cases, which can be written as a determinant.
  
  

• 1 Oct
  2008
  Numerical schemes for stochastic Volterra equations
  Conghua Wen
  
  12pm - Alan Turing - Frank Adams Room
  Abstract

  NA
  
  

Internal Probability Seminars 2006/7

• 2 Oct
  2006
  Local limit Theorems
  R. Doney
  
  1.00pm - MSS B.10
  Abstract

  None





• 9 Oct
  2006
  Optimal stopping and free-boundary problems
  G. Peskir
  
  1.00pm - Ferranti C.18
Abstract

I will review and discuss the contents of the book "Optimal stopping and free-boundary problems" (Co-author: A. N. Shiryaev), Lectures in Mathematics, ETH Zurich, Birkhauser, 2006, (500 pp).



• 16 Oct
  2006
  The principle of smooth fit for killed diffusions
  F. Samee
  
  1.00pm - Ferranti C.18
  Abstract

  We consider the case of an optimal stopping problem with discounted gain. It is well known that discounting the gain is equivalent to killing the paths of the underlying diffusion process. In the case of the undiscounted problem, we know that the principle of 'smooth fit' (which states that the value function in the optimal stopping problem sits smoothly on the gain function in the sense that the derivatives agree at the optimal stopping boundary) holds if, crucially, the scale function is differentiable at the optimal stopping boundary. We consider necessary and sufficient conditions for smooth fit to hold in the case of a killed diffusion, when a scale function can no longer be defined.
  
  

• 23 Oct
  2006
  Stochastic and deterministic SIS household epidemics
  P. Neal
  
  1.00pm - Ferranti C.18
  Abstract

  The SIS (Susceptible -> Infective -> Susceptible) epidemic model is the simplest epidemic model which exhibits endemic behaviour. Most of the `classical' results for stochastic epidemic models are asymptotic results as the population size becomes infinitely large. Therefore the implicit homogeneously mixing assumption of the SIS epidemic becomes increasingly unrealistic, and so, over the last fifteen years considerable attention has been devoted to the analysis of the spread of infectious diseases in heterogeneously mixing populations. The prime example is the household epidemic model which has been studied extensively for the SIR epidemic model. However very little has progress has been made with the SIS epidemic model.
  
  In this talk I shall consider the SIS household epidemic model. We shall study both the stochastic and deterministic model. In particular we show how Markov chains can be utilised to obtain the equilibrium distribution of the deterministic model. We shall also consider the fluctuations of the stochastic model about the deterministic trajectory.
  
  

• 6 Nov
  2006
  Numerical integration of forward-backward stochastic differential equations
  G. Milstein (MIMS Visitor)
  
  1.00pm - Ferranti C.18
  Abstract

  None
  
  

• 20 Nov
  2006
  Predicting the ultimate supremum of a stable Levy process
  G. Peskir
  
  1.00pm - Ferranti C.18
  Abstract

  I will discuss recent results on optimal prediction of the ultimate supremum for stable Levy processes.
  
  Key words and phrases: Stable Levy process (with no negative jumps), optimal prediction, optimal stopping, ultimate supremum, fractional Laplacian, Riemann-Liouville fractional derivative, Caputo fractional derivative, Volterra integral equations of the first and second kind, free-boundary problem, local time-space calculus, smooth fit, curved boundary.
  
  

• 27 Nov
  2006
  Optimal Scaling for Random walk Metropolis algorithms
  P. Neal
  
  1.00pm - Ferranti C.18
  Abstract

  None
  
  

• 4 Dec
  2006
  Strassen functional limit laws
  M. Savov
  
  1.00pm - Ferranti C.18
  Abstract

  None
  
  

• 11 Dec
  2006
  Muntz kernels and transformations of Brownian motions
  L. Alili (University of Warwick)
  
  1.00pm - Ferranti C.18
  Abstract

  None
  
  

• 19 Mar
  2007
  The law of the supremum of a stable Levy process
  G. Peskir
  
  2.00pm - Ferranti C.29
  Abstract

  None
  
  

• 23 Apr
  2007
  Dirichlet boundary problems of elliptic operators with measurable
  T. Zhang
  
  2.00pm - Ferranti C.29
  Abstract

  None
  
  

• 30 Apr
  2007
  About the scale function for spectrally negative Levy processes
  R. Doney
  
  2.00pm - Ferranti C.29
  Abstract

  None
  
  

• 17 May
  2007
  Weak convergence of some conditioned random walks
  E. Jones
  
  11.00am - Ferranti C.29
  Abstract

  None
  
  



• Probability and statistics group seminar series

Further information

For further information contact Dr John Moriarty