# Past Stochastic Analysis Seminar

10 May 2004
14:15
Touzi Nizar
Abstract
We formulate a problem of super-hedging under gamma constraint by taking the portfolio process as a controlled state variable. This leads to a non-standard stochastic control problem. An intuitive guess of the associated Bellman equation leads to a non-parabolic PDE! A careful analysis of this problem leads to the study of the small time behaviour of double stochastic integrals. The main result is a characterization of the value function of the super-replication problem as the unique viscosity solution of the associated Bellman equation, which turns out to be the parabolic envelope of the above intuitive guess, i.e. its smallest parabolic majorant. When the underlying stock price has constant volatility, we obtain an explicit solution by face-lifting the pay-off of the option.
• Stochastic Analysis Seminar
3 May 2004
15:45
Svante Janson
Abstract
The Brownian snake (with lifetime given by a normalized Brownian excursion) arises as a natural limit when studying random trees. This may be used in both directions, i.e. to obtain asymptotic results for random trees in terms of the Brownian snake, or, conversely, to deduce properties of the Brownian snake from asymptotic properties of random trees. The arguments are based on Aldous' theory of the continuum random tree. I will discuss two such situations: 1. The Wiener index of random trees converges, after suitable scaling, to the integral (=mean position) of the head of the Brownian snake. This enables us to calculate the moments of this integral. 2. A branching random walk on a random tree converges, after suitable scaling, to the Brownian snake, provided the distribution of the increments does not have too large tails. For i.i.d increments Y with mean 0, a necessary and sufficient condition is that the tails are o(y^{-4}); in particular, a finite fourth moment is enough, but weaker moment conditions are not.
• Stochastic Analysis Seminar
3 May 2004
14:15
Ma Zhi-Ming
Abstract
The celebrated Levy-Khintchine formula provides us an explicit structure of Levy processes on $R^d$. In this talk I shall present a structure result for quasi-regular semi-Dirichlet forms, i.e., for those semi-Dirichlet forms which are associated with right processes on general state spaces. The result is regarded as an extension of Levy-Khintchine formula in semi-Dirichlet forms setting. It can also be regarded as an extension of Beurling-Deny formula which is up to now available only for symmetric Dirichlet forms.
• Stochastic Analysis Seminar
8 March 2004
15:45
Abstract
Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every self-intersection to the simple random walk path. The Edwards model (EM) is obtained by giving a penalty proportional to the square integral of the local times to the Brownian motion path. Both measures significantly reduce the amount of time the motion spends in self-intersections. The above models serve as caricature models for polymers, and we will give an introduction polymers and probabilistic polymer models. We study the WSAW and EM in dimension one. We prove that as the self-repellence penalty tends to zero, the large deviation rate function of the weakly self-avoiding walk converges to the rate function of the Edwards model. This shows that the speeds of one-dimensional weakly self-avoiding walk (if it exists) converges to the speed of the Edwards model. The results generalize results earlier proved only for nearest-neighbor simple random walks via an entirely different, and significantly more complicated, method. The proof only uses weak convergence together with properties of the Edwards model, avoiding the rather heavy functional analysis that was used previously. The method of proof is quite flexible, and also applies to various related settings, such as the strictly self-avoiding case with diverging variance. This result proves a conjecture by Aldous from 1986. This is joint work with Frank den Hollander and Wolfgang Koenig.
• Stochastic Analysis Seminar
8 March 2004
14:15
Philippe Biane
Abstract
We give a construction of Brownian motion in a Weyl chamber, by a multidimensional generalisation of Pitman's theorem relating one dimensional Brownian motion with the three dimensional Bessel process. There are connections representation theory, especially to Littelmann path model.
• Stochastic Analysis Seminar
1 March 2004
14:15
Olaf Wittich
Abstract
We consider Brownian motion on a manifold conditioned not to leave the tubular neighborhood of a closed riemannian submanifold up to some fixed finite time. For small tube radii, it behaves like the intrinsic Brownian motion on the submanifold coupled to some effective potential that depends on geometrical properties of the submanifold and of the embedding. This characterization can be applied to compute the effect of constraining the motion of a quantum particle on the ambient manifold to the submanifold.
• Stochastic Analysis Seminar
23 February 2004
15:45
Stanislav Volkov
Abstract
We consider a queuing system with three queues (nodes) and one server. The arrival and service rates at each node are such that the system overall is overloaded, while no individual node is. The service discipline is the following: once the server is at node j, it stays there until it serves all customers in the queue. After this, the server moves to the &quot;more expensive&quot; of the two queues. We will show that a.s. there will be a periodicity in the order of services, as suggested by the behavior of the corresponding dynamical systems; we also study the cases (of measure 0) when the dynamical system is chaotic, and prove that then the stochastic one cannot be periodic either.
• Stochastic Analysis Seminar
16 February 2004
15:45
Thomas Mountford
Abstract
We discuss estimating the growth exponents for positive solutions to the random parabolic Anderson's model with small parameter k. We show that behaviour for the case where the spatial variable is continuous differs markedly from that for the discrete case.
• Stochastic Analysis Seminar
16 February 2004
14:15
Etienne Pardoux
Abstract
The probabilistic approach to homogenization can be adapted to fully degenerate situations, where irreducibility is insured from a Doeblin type condition. Using recent results on weak sense Poisson equations in a similar framework, obtained jointly with A. Veretennikov, together with a regularization procedure, we prove the homogenization result. A similar approach can also handle degenerate random homogenization.
• Stochastic Analysis Seminar
9 February 2004
15:45
Martijn Pistorius
Abstract
Consider a spectrally one-sided Levy process X and reflect it at its past infimum I. Call this process Y. We determine the law of the first crossing time of Y of a positive level a in terms of its 'scale' functions. Next we study the exponential decay of the transition probabilities of Y killed upon leaving [0,a]. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.
• Stochastic Analysis Seminar