Curs 2013-2014: Abstracts i Slides

18/06/14: Raluca Balan, Universitat d'Ottawa, Canadà.

Stochastic integration with respect to Lévy colored noise, with applications to SPDEs.

The purpose of this talk is to introduce a new type of noise for problems in stochastic analysis, which behaves in time like a finite-variance Lévy process without a Gaussian component. In the space variable, the noise is a stationary random distribution (in the sense introduced in Itô, 1954), whose covariance is a non-negative definite distribution $\rho$, which can be viewed as the Fourier transform of a tempered measure $\mu$. In the Gaussian case, a similar type of noise was introduced in Dalang (1999), under the assumption that the distribution $\rho$ is induced by a tempered non-negative function f (or a tempered measure $\Gamma$). We develop a theory of stochastic integration with respect to this noise without this assumption. The same theory can be developed in the Gaussian case, the motivating example (in spatial dimension d = 1) being a noise which behaves like a fractional Brownian motion in space, with Hurst index H < 1/2.

As an application of this theory, we consider the linear stochastic wave (or heat) equation with this noise. The random field solution of this equation exists if and only if the measure $\mu$ satisfies the condition:

$\int_{\mathbb{R}^d} \frac{1}{1+|\xi|^2}\mu(d\xi) < \infty,$

introduced in Dalang (1999). Studying non-linear SPDEs with this kind of noise remains an open problem, even in the Gaussian case.

04/06/14: Archil Gulisashvili, Ohio University, USA.

Asymptotic Formulas for the Implied Volatility at Extreme Strikes.

For a general call pricing model, the implied volatility, or the smile, is the value of the volatility parameter in the Black-Scholes call option pricing model, which makes the Black-Scholes call price equal to the call price in the given model. It will be explained in the lectures how the implied volatility behaves at small and large strikes. We will discuss and prove several known results describing the smile asymptotics. There results include Lee's moment formulas, the tail-wing formula of Benaim and Friz, and the asymptotic formulas with error estimates due to the author.

This is a minicourse consisting of two lectures, one on Wednesday, June 4th and the other one on Thursday, June 19th. Both lectures take place at the Aula d'IMUB at the University of Barcelona, start at 16h and last 1:30h each.

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07/05/14: Marc Noy, UPS.

Random graphs, old and new.

30/04/14: Alejandra Cabaña, UAB.

Modeling stationary data by classes of generalized Ornstein-Uhlenbeck processes.

Consider the complex Ornstein Uhlenbeck operator defined by

$OU_{\kappa}y(t)=\int_{-\infty}^t{\mathrm{e}}^{-\kappa(t-s)}dy(s),\, \kappa\in{\mathbb{C}},\mathrm{Re}(\kappa)>0$

provided that the integral makes sense. In particular, when y is replaced by a Lévy process $\Lambda$ on $\mathbb{R}$

$x(t)={OU}_{\kappa}\Lambda(t)$

is Lévy-driven Ornstein Uhlenbeck process. New stationary processes can be constructed by iterating the application of OU to Lévy processes, in particular, to the Wiener process w. We denote by OU(p) the families of processes obtained by applying successively p OU operators ${OU}_{\kappa_j}$, j=1,2,...,p to $\Lambda$. These processes can be used as models for stationary Gaussian continuous parameter processes, and their discretized version, for stationary time series. We show in particular that the series $x(0),\,x(1),\dots,\, x(n)$ obtained from

$x=\prod_{j=1}^p{\cal OU}_{\kappa_j}(\sigma w)$

satisfies a ARMA(p,p-1) model. We present an empirical comparison of the abilities of OU and ARIMA models to fit stationary data. In particular, we show examples of processes with long dependence where the fitting of the empirical autocorrelations is improved by using OU process rather than ARIMA models, and with fewer parameters in the former case.

09/04/14: Jordan Stoyanov, Newcastle, UK.

Moment Determinacy of Probability Distributions.

The emphasis will be on some recent progress in the moment analysis of distributions and their characterization as being unique (M-determinate) or non-unique (M-indeterminate) in terms of the moments. Specific topics which will be discussed are:

(a) Stieltjes classes for M-indeterminate distributions. Index of dissimilarity.

(b) New Hardy’s criterion for uniqueness. Multidimensional moment problem.

(c) Nonlinear transformations of random data and their moment (in)determinacy.

(d) Moment determinacy of distributions of stochastic processes defined by SDEs.

There will be new results, hints for their proof, examples and counterexamples, and also open questions and conjectures.

09/04/14: Jordan Stoyanov, Newcastle, UK.

Intriguing problems from Combinatorics, Algebra, Analysis and Probability.

06/03/14: Mohammud Foondun, Loughborough University, UK.

Noise sensitivity of stochastic heat equations.

We consider a broad class of stochastic heat on a bounded domain. Under suitable conditions, we consider the growth of moments with respect to the a parameter called noise intensity, which basically measure the amount of noise being inputed in the system. The general method of proofs depends on some sharp heat kernel estimates and a localisation argument.

05/03/14: Lluís Quer-Sardanyons, UAB.

The Stratonovich heat equation: a continuity result and weak approximations.

Our main motivation has been to generalize the results obtained by Bardina {\it et al.} on weak convergence for the stochastic heat equation driven by an additive noise (which have been presented by M. Jolis in the preceding seminar) to a general multiplicative-noise setting. For this, we consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques are used to provide an almost sure continuity result for the solution with respect to the solution of the 'smooth' equation obtained by replacing the noise with an absolutely continuous process. This continuity result is then exploited to prove weak convergence results based on Donsker and Kac-Stroock type approximations of the noise.

26/02/14: Maria Jolis, UAB.

Weak convergence for the stochastic heat equation driven by Gaussian white noise.

We consider a quasi-linear stochastic heat equation on $(0,1)$, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter $n\in\mathbb{N}$ that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space $C([0,T]\times [0,1])$ of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock, which states that the family of processes $n \int_0^t (-1)^{N(n^2 s)} ds$, where $N$ is a standard Poisson process, converges in law to a Brownian motion. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

19/02/14: Luis Ortiz Gracia, CRM, Bellaterra.

A wavelet look to the discounted expected payoff pricing formula.

We present a novel method for pricing European options based on the wavelet approximation (WA) method and the characteristic function. We focus on the discounted expected payoff pricing formula, and compute it by means of wavelets. We approximate the density function associated to the underlying asset price process by a finite combination of j-th order B-splines, and recover the coefficients of the approximation from the characteristic function. Two variants for wavelet approximation will be presented, where the second variant adaptively determines the range of integration. The compact support of a B-splines basis enables us to price options in a robust way, even in cases where Fourier-based pricing methods may show weaknesses. The method appears to be particularly robust for pricing long-maturity options, fat tailed distributions, as well as staircase-like density functions encountered in portfolio loss computations.

28-30/01/14: Aleksandar Mijatovic: Imperial College London, UK.

Short PhD course: Stochastic differential equations.

The aim of this course is to develop the theory of stochastic differential equations and study certain path properties of diffusion processes using tools from semimartingale theory. Lecture 1 will be introductory while Lectures 2 and 3 will cover recent research topics. See description for more details.

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22/01/14: Alessia Ascanelli, Università di Ferrara, Itàlia.

An introduction to pseudo-differential operators.

The aim of this lecture is to introduce the basic notions of the pseudo-differential calculus. Pseudo-differential operators are a powerful natural extension of linear partial differential operators: they convert, via Fourier transform, the theory of partial differential equations with variable coefficients into an algebraic theory for the corresponding symbols.

We also give the more general concept of Fourier integral operator, and provide some examples concerning the application of pseudo-differential techniques to the solution of deterministic PDEs of elliptic and hyperbolic type.

22/01/14: Andre Suess, UB.

Random-field solutions to hyperbolic SPDEs with variable coefficients.

In this talk we investigate stochastic partial differential equations whose partial differential operator is hyperbolic, of second-order or higher-order. We work in the setting of \cite{walsh,dalang}, i.e. our aim is to get random-field solutions, where the solution u(t,x) can be evaluated as a real-valued random variable at every $(t,x)\in[0,T]\times\mathbb{R}^d$. In order to achieve this, we use the theory of pseudo-differential and Fourier integral operators, presented by Alessia Ascanelli in the previous talk, to obtain an explicit formula for the fundamental solution associated to the partial differential operator and check that it satisfies the integrability conditions necessary for the existence and uniqueness of a random field solution. The examples that we treat are: second-order strictly hyperbolic operators, second-order weakly hyperbolic operators, higher-order strictly hyperbolic operators and, if time permits, we compute the example of the stochastic wave equation in any spatial dimension explicitly and show how it fits into this theory. This is a joint work with Alessia Ascanelli of the University of Ferrara.

15/01/14: Robert Dalang, EPFL, Suïssa.

Multiple points of the Brownian sheet in critical dimensions.

It is well-known that an N-parameter d-dimensional Brownian sheet has no k-multiple points when (k-1)d > 2kN, and does have such points when (k-1)d < 2kN. Existing results on hitting probabilities for general Gaussian processes do not provide information concerning the critical dimensions. We complete the study of the existence of k-multiple points by showing that in the critical cases where (k-1)d = 2kN, there are a.s. no k-multiple points. This is joint work with Carl Mueller (University of Rochester)

11/12/13: Markus Riedle, King's College, UK.

Cylindrical Lévy processes in Banach spaces and Hilbert spaces.

The objective of this talk is the introduction of cylindrical Lévy processes and their stochastic integrals in Hilbert spaces.

The degree of freedom of models in infinite dimensions is often reflected by the request that each mode along a dimension is independently perturbed by the noise. In the Gaussian setting, this leads to the cylindrical Wiener process including from a model point of view the very important possibility to model a Gaussian noise in both time and space in a great flexibility space-time white noise. Up to very recently, there has been no analogue for Lévy processes.

Based on the classical theory of cylindrical processes and cylindrical measures we introduce cylindrical Lévy processes as a natural generalisation of cylindrical Wiener processes. We continue to characterise the distribution of cylindrical Lévy processes by a cylindrical version of the Lévy-Khintchine formula.

In Hilbert spaces we introduce a stochastic integral for operator-valued stochastic processes with respect to cylindrical Lévy processes. We apply the developed theory to derive the existence of a solution for a Cauchy problem and to consider spatial and temporal regularity and irregularity properties of the solution.

(parts of this talk are based on joint work with D. Applebaum or A. Jakubowski)

[SLIDES]

26 - 27/11/13: Two-Day Workshop on Finance and Stochastics.

Anthony Reveillac: Non-classical BSDEs arising in the utility maximization problem with random horizon.

In this talk we will present a class of non-standard BSDE which come into play in the context of the utility maximization problem with random horizon. We will explain in which sense these equations do not belong to the classical theory, and we will present existence and uniqueness type results for them. This talk is based on a joint work with Monique Jeanblanc and Nam Hai Nguyen.

Asma Kheder: Weak stationarity of Ornstein-Uhlenbeck processes with stochastic speed of mean reversion.

When modeling energy prices with the Ornstein-Uhlenbeck process, it was shown in Barlow, Gusev, and Lai [1] and Zapranis and Alexandris [2] that there is a large uncertainty attached to the estimation of the speed of mean-reversion and that it is not constant but may vary considerably over time. In this paper we generalise the Ornstein-Uhlenbeck process to allow for the speed of mean reversion to be stochastic. We suppose that the mean-reversion is a Brownian stationary process. We apply Malliavin calculus in our computations and we show that this generalised Ornstein-Uhlenbeck process is stationary in the weak sense. Moreover we compute the instantaneous rate of change in the mean and in the squared uctuations of the genaralised Ornstein-Uhlenbeck process given its initial position. Finally, we derive the chaos expansion of this generalised Ornstein-Uhlenbeck process. (Joint work with Fred Espen Benth)

References:

[1] Barlow, M., Gusev. Y., and Lai, M. (2004). Calibration of multifactor models in electricity markets. Intern. J. Theoret. Appl. Finance, 7, (2), pp. 101-120.

[2] Zapranis, A., and Alexandridis, A. (2008). Modelling the temperature time-dependent speed of mean reversion in the context of weather derivatives pricing. Appl. Math. Finance 15(4), pp. 355-386.

Salvador Ortiz-Latorre: A pricing measure to explain the risk premium in power markets.

In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump Lévy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving Lévy processes, while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this talk we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we also can slow down the speed of mean reversion and generate stochastic risk premiums with stochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity.

[SLIDES]

Albert Ferreiro-Castilla: Euler-Poisson schemes for Lévy processes.

In this talk we will contextualize the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Lévy processes from Kuznetsov et al. [4] into a more general framework allowing us to use the same technique in a larger set of problems. We will briefly show how the scheme can be used to approximate Lévy driven SDEs, be enhanced with a multilevel Monte Carlo scheme, or approximate different types of path-quantities.

References:

[1] Ferreiro-Castilla, A., Kyprianou, A.E., Scheichl, R. and Suryanarayana, G. (2013) Multi-level Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorization. Stoch. Proc. Appl. (To appear).

[2] Ferreiro-Castilla, A., Kyprianou, A.E. and Scheichl, R. (2013) An Euler-Poisson scheme for numerical approximation of Lévy driven SDEs. Preprint: http://arxiv.org/abs/1309.1839

[3] Ferreiro-Castilla, A. and van Schaik, K. (2013) Applying the Wiener-Hopf Monte Carlo simulation technique for Lévy processes to path functionals. J. Appl. Probab. (To appear)

[4] Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2011) A Wiener-Hopf Monte Carlo simulation technique for Lévy process. Ann. App. Probab. 21(6), 2171-2190.

[SLIDES]

Michael Tehranchi: An HJM approach to equity derivatives.

There has been recent interest in applying the Heath-Jarrow-Morton interest rate framework to other areas of financial modelling. Unfortunately, there are serious technical challenges in implementing the approach for modelling the dynamics of the implied volatility surface of a given stock. By a suitable change of parametrisation, we derive an HJM-style SPDE and discuss its existence theory. We survey some recent negative results to illustrate some of the technical challenges.

[SLIDES]

Olivier Menoukeu Pamen: Maxiumm principles of Markov regime-switching forward-backward stochastic differential equations with jumps and partial information.

This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward-backward stochastic differential equations with jumps (FBSDEJs). A general suffcient maximum principle for optimal control for a system driven by a Markov regime-switching forward and backward jump-diffusion model is developed. After, an equivalent maximum principle is proved. Malliavin calculus is employed to derive a general stochastic maximum principle for non-Markovian system. The latter does not required concavity of Hamiltonian. Applications of the stochastic maximum principle to non-concave Hamiltonian and recursive utility maximization is also discussed.

Steffen Sjursen: BSDEs and optimal control for time-changed Lévy processes.

We study backward stochastic differential equations (BSDE's) for time-changed Lévy noises when the time-change is independent of the Lévy process. We prove existence and uniqueness of the solution and we obtain an explicit formula for linear BSDE's and a comparison principle. BSDE's naturally appear in control problems. Here we prove sufficient and necessary maximum principles for a general optimal control problem of a system driven by a time-changed Lévy noise. As an application we solve the mean-variance portfolio selection problem.

[SLIDES]

Elisa Alòs: On the closed-form approximation of short-time random strike options..

We propose a general technique to develop first and second order closed-form approximation formulas for short-time options with random strikes. Our method is based on Malliavin calculus techniques and allows us to obtain simple closed-form approximation formulas depending on the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches on two-assets and three-assets spread options as Kirk' s formula or the decomposition mehod presented in Alòs, Eydeland and Laurence (2011).

[SLIDES]

M.D. Ruiz-Medina: Cox-Ingersoll-Ross model in infinite-dimensional spaces.

Infinite-dimensional Chi-squared distributions in the central and non-central case are introduced through their characteristic functional (see, for example, Proposition 1.2.8 of Da Prato and Zabczyk, 2002). Conditions for their decomposition in terms of infinite sums of independent real-valued central and non-central Chi-squared distributions are respectively derived. These results are applied to obtaining orthogonal systems with respect to such infinite-dimensional probability measures. In particular, in the recurrent case, the space $L^2(H; B(H))$ generated by the invariant distribution of Cox-Ingersoll-Ross Model in Hilbert spaces (i.e., for Hilbert-valued random variables) is orthogonally decomposed.

References:

Da Prato, G. and Zabczyk, J. (2002) Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press, Cambridge.

Thibaut Mastrolia: Density analysis of BSDEs.

In this paper we investigate existence and smoothness of densities for the solution of Backward Stochastic Differential Equations. This question has been very few studied in the literature and the existing results mainly focus on the Y component. In this talk, we will focus on the Z component since it finds applications in Finance. Our approach relies on the Malliavin calculus and on BSDEs techniques. This talk is based on a work in progress with Dylan Possamaï and Anthony Réveillac.

[SLIDES]

Lluis Quer-Sardanyons: Gaussian upper density estimates for spatially homogeneous SPDEs.

We consider a general class of SPDEs in $\mathbb{R}^d$ driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with $d\in \{1,2,3\}$. In these particular cases, the condition on the spectral measure turns out to be optimal.