A.N. Shiryaev and Contemporary Probability Theory

Liste des Contributions

11. Estimation of volatility in presence of high activity jumps, noise and irregular sampling

Prof. Jean Jacod (Université Pierre-et-Marie Curie)

30/11/2015 10:00

We consider an Itô semimartingale which is observed along a discrete time grid, within a fixed time interval. The observations are contaminated by noise, and the semimartingale has jumps with a degree of activity bigger than 1. Our aim is to revisit the estimation of the integrated volatility in such a setting: we use a mixture of the pre-averaging method (to eliminate noise) and of the...

1. Two price valuation theory

Prof. Ernst Eberlein (University of Freiburg)

30/11/2015 11:00

In classical economic theory the law of one price prevails and market participants trade freely in both directions at the same price. This approach is appropriate for highly liquid markets. In the absence of perfect liquidity, the law of one price has to be replaced by a two price economy where market participants continue to trade freely with the market but the terms of trade now depend...

9. Moral hazard and second order backward SDEs

Prof. Nizar Touzi (Ecole Polythecnique)

30/11/2015 11:50

We consider a general formulation of the Principal-Agent problem with a lump-sum payment on a finite horizon. Our main result is a reduction of this problem to a standard stochastic control problem, so that the principal's problem is solved by the standard tools of control theory. Our proofs rely on the Backward Stochastic Differential Equations approach to non-Markovian stochastic control,...

0. Optimization problem for a portfolio with an illiquid asset: Lie group analysis

Prof. Ljudmila Bordag (University of Applied Sciences Zittau/Görlitz)

30/11/2015 14:30

Management of a portfolio that includes an illiquid asset is an important problem of modern mathematical finance. One of the ways to model illiquidity among others is to build an optimization problem and assume that one of the assets in a portfolio can not be sold until a certain finite, infinite or random moment of time. This approach arises a certain amount of models that are actively...

6. Mixed Gaussian processes: a filtering approach

Prof. Marina Kleptsyna (Université du Maine)

30/11/2015 15:20

We present a new approach to the analysis of mixed processes: for $t\in [0,T]$ $ \hspace{6cm} X_t = B_t + G_t, $ where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel $ \hspace{5cm} K(s,t) = \frac{\partial^2}{\partial s\partial...

8. Probability Distributions: Characterizations by Their Moments

Prof. Jordan Stoyanov (Newcastle University & University of Ljubljana)

30/11/2015 16:20

The talk is on selected new results on uniqueness and non-uniqueness of distributions in terms of their moments. The results cover distributions of random variables, random vectors and stochastic processes, including solutions of SDEs. A couple of open questions will be outlined.

13. Inversion, duality and h-processes of self-similar Markov processes

Prof. Loïc Chaumont (Université d'Angers)

30/11/2015 17:00

We show that any $\mathbb{R}^d$-valued self-similar Markov process $X$ with index $\alpha>0$ absorbed at 0, can be represented as a path transformation of some Markov additive process (MAP) $(\theta,\xi)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Then we prove that the same transformation of the dual MAP in the weak sense of $(\theta,\xi)$...

7. On the Martingale Property of Local Martingales: When a Local Martingale is a True Martingale?

Prof. Hans-Jürgen Engelbert (Friedrich Schiller-University of Jena, Institute of Mathematics)

01/12/2015 10:00

For many problems in the theory of stochastic processes and its applications it is of great importance to know effective criteria ensuring that a given local martingale is a true martingale or even a uniformly integrable martingale. This question is closely related to absolute continuity of probability measures and change of measure and has been the subject of research over many decades....

10. Valuation, put-call parity and bubbles

Prof. Martin Schweizer (ETH Zurich)

01/12/2015 11:00

We discuss a systematic approach to the valuation of general European contingent claims in general continuous-time financial markets. We want to provide bounds on economically reasonable valuations that do not depend too much on precise assumptions on the underlying primary assets. This allows us to provide a general result on put-call parity and to give an explanation for some pricing...

4. On one application of the Cherny-Shiryaev criterion of stochastic integrability

Prof. Yuri Kabanov (Université Franche-Comté)

01/12/2015 11:50

The Cherny-Shiryaev criterion provides a description of predictable processes which are integrable with respect to a vector-valued semimartingale in terms of its local characteristics. We provide an example how this result can be used in the problem of existence of local martingale numéraire in a model of financial market which has no asymptotic arbitrage opportunities of the first kind.

3. Probabilistic interpretation for Fully nonlinear Stochastic PDEs

Prof. Anis Matoussi (Université du Maine)

01/12/2015 14:30

We present an overview on different classes of nonlinear stochastic partial differential equations (SPDEs in short). In particular, we focus on providing a probabilistic representation of solution of Fully nonlinear SPDEs (stochastic Viscosity solutions) by means of solution of the associated class of Second order BSDEs. This presentation includes the numerical study of quasilinear and...

2. Existence and uniqueness of viscosity solutions for second order integro-differential equations without monotonicity condition

Prof. Said Hamadène (Université du Maine)

01/12/2015 15:20

In this talk, we discuss a new existence and uniqueness result of a continuous viscosity solution for integro-partial differential equation (IPDE in short). The novelty is that we relax the so-called monotonicity assumption on the driver which is classically assumed in the literature of viscosity solution of equation with a non local term. Our method is based on the link of those IPDEs with...

12. On exponential functionals for processes with independent increments

Prof. Lioudmila Vostrikova (Université d'Angers)

01/12/2015 16:20

We study the exponential functionals of the processes with independent increments which are integrable semi-martingales with absolutely continuous characteristics. We give necessary and sufficient conditions for existence of Laplace exponent, and also the sufficient conditions of finiteness of the moments of exponential functionals. We derive a recurrent integral equation for its Mellin...