Internal Probability Seminars

1st Semester 2011/2012

• 5 Dec
  2011
  A Wiener-Hopf Monte Carlo simulation method for Levy processes
  K. van Schaik
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  The talk will be about a new Monte Carlo simulation method for meromorphic Levy processes based on the Wiener-Hopf factorisation. We will first (shortly) introduce meromorphic Levy processes and their (pretty explicit) Wiener-Hopf factors. Next we will construct the simulation method and explain/give some numerical evidence that for path functionals such as the first passage time over a level this new simulation method performs better than 'standard' Monte Carlo. Finally we will discuss a convergence proof.

• 28 Nov
  2011
  Variance of the giant component of a random graph
  P. Neal
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Abstract will appear here

• 14 Nov
  2011
  Calculating a probability involving Brownian Motion
  R. Doney
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Abstract will appear here

• 17 Oct
  2011
  Windings of planar Brownian motion and Bougerol's identity
  S. Vakeroudis
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  We 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 Bougerol's celebrated identity. We develop some identities in law in terms of planar Brownian motion, which are equivalent to Bougerol's identity. This allows us to characterize the laws of the hitting times of the bounadries of a cone. We also obtain a new non-computational proof of Spitzer's asymptotic theorem and some integrability properties for these hitting times. Finally, we extend our results to the case of (planar) complex-valued Ornstein-Uhlenbeck processes

• 10 Oct
  2011
  Quickest detection of a hidden target and extremal surfaces
  G. Peskir
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Abstract will appear here

• 3 Oct
  2011
  Stochastic calculus for fractional Brownian motion in Banach spaces
  E. Issoglio
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  The main aim of this talk is to introduce a suitable notion of fractional Brownian motion (fBm) on Banach spaces and the corresponding stochastic integration theory. We adopt cylindrical stochastic processes, which are more general objects than stochastic processes and therefore allow us to work in general separable Banach spaces. Doing so, we are able to define the stochastic integral with respect to a fBm in a Banach space as a cylindrical process. This integral is derived using stochastic integration with respect to real-valued fBms and results in relatively simple conditions on the integrand. The special case of fBm in Hilbert spaces is considered and it is compared with the existing literature.

2nd Semester 2010/2011

• 4 Apr
  2011
  Pricing of Credit Linked Note
  Z. Imdad
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 21 Mar
  2011
  The American barrier option
  L. Al-Fagih
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 7 Mar
  2011
  Pricing an American Put Option on a Stochastic Zero Coupon Bond in the HJM model
  T. De Angelis
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 28 Feb
  2011
  Optimal detection of a hidden target: The median rule
  G. Peskir
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 21 Feb
  2011
  First Passage times for Levy processes
  R. Doney
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 7 Feb
  2011
  Inside trading in the market with rational expected price
  F. Gong
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

1st Semester 2010/2011

• 13 Dec 2010
  
  Bubbles and crashes
  M. Riedle
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

• 22 Nov 2010
  
  Some problems in insurance mathematics
  P. Patie
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Abstract: In this talk, we present the solutions to three problems arising in insurance mathematics. We first consider an insurance risk process modeled by a mean-fleeing Ornstein-Uhlenbeck type process with a subordinator as the background driving process. In this model the company earns interest on positive surplus and pays interest (at the same rate) when the surplus is negative. It is possible that the company gets absolutely ruined, which is the event where the premium income can no longer compensate for the interest payments. We provide simple expressions for the Laplace transform in space of both the finite- and infinite-time absolute ruin probability leading to some explicit representations for the finite-time absolute ruin probability in the risk process under interest force. By means of the spectral theory, we give under some conditions a spectral representation for the finite-time absolute ruin probabilities. We proceed by solving the so-called two-sided exit problem introduced by Dickson and Gray. Finally, we study the distribution of the present value of a perpetuity subjected to a rate of interest driven by a spectrally negative Lévy process. In particular, we show that its law is absolutely continuous with a smooth density which admits a power series representation, generalizing works from Dufresne and Goovaerts et al. The methodologies developped to solve these problems rely on techniques ranging from stochastic calculus, potential theory to functional and complex analysis.

• 25 Oct 2010
  
  A Universal Signal Process for Optimal Stopping and Singular Control
  J. Sexton
  
  2pm - Alan Turing Building - Frank Adams Room
  Abstract

  Abstract: With any optional process of class D one may associate a non-decreasing signal process Y. Certain level crossing times of Y have been shown in [1] and [2] to characterise the optimal stopping times of American options written on X as well as the solution to a singular control problem with convex running costs. We formulate a dual problem for a convex singular control problem by deriving a new version of the maximum principle valid on a infinite time horizon. The adjoint equation which solves this dual problem is the largest semimartingale with concave drift dominating the intervention costs. This semimartingale can be identified by studying a stochastic analogue of the concave envelope of the intervention costs. Moreover, the adjoint equation may be explicitly represented as a stopping problem. This signal processes are also used to provide a proof of the result in Ekstrom & Peskir [3] using martingale methods. References: [1] Bank, P. (2005). Optimal Control under a Dynamic Fuel Constraint. SIAM J. Control Optim. 44, (1529-1541). [2] Bank, P. and H. Follmer (2003). American options, multi-armed bandits, and optimal consumption plans: a unified view. In Paris-Princeton Lectures in Financial Mathematics, Vol 1814, Springer, (1-42). [3] Ekstrom, E. and Peskir, G. (2008). Optimal stopping games for Markov processes. SIAM J. Control Optim. 47 (684-702).

Further information

For further information contact Dr John Moriarty