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2013 | Book

Recent Advances Related to SPDEs with Fractional Noise Read chapter Read first chapter

Seminar on Stochastic Analysis, Random Fields and Applications VII

Centro Stefano Franscini, Ascona, May 2011

Editors: Robert C. Dalang, Marco Dozzi, Francesco Russo

Publisher: Springer Basel

Book Series : Progress in Probability

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume contains refereed research or review articles presented at the 7th Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verità) in Ascona , Switzerland, in May 2011. The seminar focused mainly on: - stochastic (partial) differential equations, especially with jump processes, construction of solutions and approximations - Malliavin calculus and Stein methods, and other techniques in stochastic analysis, especially chaos representations and convergence, and applications to models of interacting particle systems - stochastic methods in financial models, especially models for power markets or for risk analysis, empirical estimation and approximation, stochastic control and optimal pricing. The book will be a valuable resource for researchers in stochastic analysis and for professionals interested in stochastic methods in finance.​

Table of Contents

Frontmatter

Stochastic Analysis and Random Fields

Frontmatter
Recent Advances Related to SPDEs with Fractional Noise
Abstract
We review the literature related to stochastic partial differential equations with spatially-homogeneous Gaussian noise, and explain how one can introduce the structure of the fractional Brownian motion into the temporal component of the noise. The Hurst parameter H is assumed to be greater than 1/2. In the case of linear equations, we revisit the conditions for the existence of a mild solution. In the nonlinear case, we point out what are the difficulties due to the fractional component of the noise. These difficulties can be avoided in the case of equations with multiplicative noise, since in this case, the solution has a known Wiener chaos decomposition. Finally, this methodology is applied to the wave equation (in arbitrary dimension d ≥ 1), driven by a Gaussian noise which has a spatial covariance structure given by the Riesz kernel.
Raluca M. Balan
On Chaos Representation and Orthogonal Polynomials for the Doubly Stochastic Poisson Process
Abstract
In an L 2-framework, we study various aspects of stochastic calculus with respect to the centered doubly stochastic Poisson process. We introduce an orthogonal basis via multilinear forms of the value of the random measure and we analyze the chaos representation property. We review the structure of non-anticipating integration for martingale random fields and in this framework we study non-anticipating differentiation. We present integral representation theorems where the integrand is explicitly given by the non-anticipating derivative.
Stochastic derivatives of anticipative nature are also considered: The Malliavin type derivative is put in relationship with another anticipative derivative operator here introduced. This gives a new structural representation of the Malliavin derivative based on simple functions. Finally we exploit these results to provide a Clark–Ocone type formula for the computation of the nonanticipating derivative.
Giulia Di Nunno, Steffen Sjursen
General Upper and Lower Tail Estimates Using Malliavin Calculus and Stein’s Equations
Abstract
Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution, and apply it to all reference distributions in the so-called Pearson class. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable’s tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation between Stein’s method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that the strategy can be extended to comparisons to a wide class of laws, including all Pearson distributions.
Richard Eden, Frederi Viens
Uniqueness and Absolute Continuity for Semilinear SPDE’s
Abstract
Given a weak solution of a semilinear stochastic partial differential equation, sufficient conditions for its uniqueness in law are presented. Moreover we characterize this law and prove that it is absolutely continuous with respect to the law of the process solving the corresponding linear stochastic partial differential equation, obtained neglecting the nonlinear term. The conditions imposed involve a ℙ-a.s. assumption on the solution process. This allows to avoid a boundeness or linear growth condition on the nonlinear term. Finally, we prove the equivalence of the laws.
Benedetta Ferrario
Rate of Convergence of Wong–Zakai Approximations for Stochastic Partial Differential Equations
Abstract
In this paper we show that the rate of convergence of Wong–Zakai approximations for stochastic partial differential equations driven by Wiener processes is essentially the same as the rate of convergence of the driving processes W n approximating the Wiener process, provided the area processes of W n also converge to those of W with that rate. We consider non-degenerate and also degenerate stochastic PDEs with time dependent coefficients.
István Gyöngy, Pablo Raúl Stinga
Weak Approximations for SDE’s Driven by Lévy Processes
Abstract
In this article we briefly survey recent advances in some simulation methods for Lévy driven stochastic differential equations. We give a brief description of each method and extend the one jump scheme method for some subordinated models like the NIG process. Simulations of all the presented methods are performed and compared.
Arturo Kohatsu-Higa, Hoang-Long Ngo
Itô’s Formula for Banach-space-valued Jump Processes Driven by Poisson Random Measures
Abstract
We prove Itô’s formula for a general class of functions H: ℝ+ ×F→G of class C 1,2, where F,G are separable Banach spaces, and jump processes driven by a compensated Poisson random measure.
Vidyadhar Mandrekar, Barbara Rüdiger, Stefan Tappe
Well-posedness for a Class of Dissipative Stochastic Evolution Equations with Wiener and Poisson Noise
Abstract
We prove existence and uniqueness of mild and generalized solutions to a class of stochastic semilinear evolution equations driven by additive Wiener and Poisson noise. The non-linear drift term is supposed to be the evaluation operator associated to a continuous monotone function satisfying a polynomial growth condition. The results are extensions to the jump-diffusion case of the corresponding ones proved in [3] for equations driven by purely discontinuous noise.
Carlo Marinelli
Localization of Relative Entropy in Bose–Einstein Condensation of Trapped Interacting Bosons
Abstract
We consider a system of interacting diffusions which is naturally associated to the ground state of the Hamiltonian of a system of N pairinteracting bosons and we give a detailed description of the phenomenon of the “localization of the relative entropy”. The method is based on peculiar rescaling properties of the mean energy functional
Laura M. Morato, Stefania Ugolini
Multi-dimensional Semicircular Limits on the Free Wigner Chaos
Abstract
We show that, for sequences of vectors of multiple Wigner integrals with respect to a free Brownian motion, componentwise convergence to semicircular law is equivalent to joint convergence. This result extends to the free probability setting some findings by Peccati and Tudor (2005), and represents a multi-dimensional counterpart of a limit theorem inside the free Wigner chaos established by Kemp, Nourdin, Peccati and Speicher (2011).
Ivan Nourdin, Giovanni Peccati, Roland Speicher
Malliavin Calculus for Stochastic Point Vortex and Lagrangian Models
Abstract
We explore the properties of solutions of two stochastic fluid models for viscous flow in two dimensions. We establish the absolute continuity of the law of the corresponding solution using Malliavin calculus.
Sivaguru S. Sritharan, Meng Xu
Two Remarks on the Wasserstein Dirichlet Form
Abstract
The Wasserstein diffusion is an Ornstein–Uhlenbeck type process on the set of all probability measures with the Wasserstein metric as intrinsic metric. Sturm and von Renesse constructed in [6] this process in the case of probability measures over the unit interval using Dirichlet form theory. An essential step in this construction is the closability of a certain gradient form, defined for smooth cylindrical test functions, in the space L 2 w.r.t. the entropic measure ℚβ. In this paper we will first give an alternative proof for this closability, avoiding the striking, but elaborate integration by parts formula for ℚβ used in [6]. Second, we give explicit conditions under which certain finite-dimensional particle approximations introduced in the paper [1] by Andres and von Renesse do converge in the resolvent sense to the Wasserstein diffusion, a question that was left open in the above cited paper.
Wilhelm Stannat
Erratum
Abstract
We give some errata corresponding to the article “Statistical Inference andMalliavin Calculus” published in the Seminar on Stochastic Analysis, Random Fields and Applications VI, 2011.
Arturo Kohatsu-Higa, José Manuel Corcuera

Stochastic Methods in Financial Models

Frontmatter
Stochastic Modeling of Power Markets Using Stationary Processes
Abstract
We make a survey over recent developments in stochastic modelling of power markets, with a particular focus on the application of stationary processes. We analyse the class of Lévy semistationary processes proposed by Barndorff-Nielsen, Benth and Veraart [1] for modelling electricity spot prices. We suggest and analyse different numerical methods for simulating the paths of these processes, a particulary important question for risk management studies in power markets. Finally, we discuss the aspect of pricing forward contracts based on a class of stationary models, and review some implications.
Fred Espen Benth, Heidar Eyjolfsson
Evaluating Hybrid Products: The Interplay Between Financial and Insurance Markets
Abstract
A current issue in the theory and practice of insurance and reinsurance markets is to find alternative ways of securitizing risks. Insurance companies have the possibility of investing in financial markets and therefore hedge against their risks with financial instruments. Furthermore they can sell part of their insurance risk by introducing insurance linked products on financial markets. Hence insurance and financial markets may no longer be considered as disjoint objects, but can be viewed as one arbitrage-free market. Here we provide an introduction to how mathematical methods for pricing and hedging financial claims such as the benchmark approach and local risk minimization can be applied to the valuation of hybrid financial insurance products, as well as to premium determination, risk mitigation and claim reserve management.
Francesca Biagini
f-Divergence Minimal Equivalent Martingale Measures and Optimal Portfolios for Exponential Lévy Models with a Change-point
Abstract
We study exponential Lévy models with change-point which is a random variable, independent from initial Lévy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point model and we give the conditions for the existence of f-divergence minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration. We illustrate our results considering the Black–Scholes model with change-point.
S. Cawston, L. Vostrikova
Optimal Investment-consumption for Partially Observed Jump-diffusions
Abstract
We deal with an optimal consumption-investment problem under restricted information in a financial market where the risky asset price follows a non-Markovian geometric jump-diffusion process. We assume that agents acting in the market have access only to the information flow generated by the stock price and that their individual preferences are modeled through a power utility. We solve the problem with a two steps procedure. First, by using filtering results we reduce the partial information problem to a full information one involving only observable processes. Next, by using dynamic programming, we characterize the value process and the optimal-consumption strategy in terms of solution to a backward stochastic differential equation.
Claudia Ceci
Stochastic Control and Pricing Under Swap Measures
Abstract
This paper relates to an approach described in [6] which, for the pricing of bonds and bond derivatives, is alternative to the classical approach that involves martingale measures and is based on the solution of a stochastic control problem, thereby avoiding a change of measure. It turns out that this approach can be extended to various situations where traditionally a change of measure is involved via a change of numeraire. In the present paper we study this extension for the case of Swap measures that are relevant in the classical approach to the pricing of Swaps and Swaptions.
R. Cogo, A. Gombani, W. J. Runggaldier
Affine Variance Swap Curve Models
Abstract
This paper provides a brief overview of the stochastic modeling of variance swap curves. Focus is on affine factor models. We propose a novel drift parametrization which assures that the components of the state process can be matched with any pre-specified points on the variance swap curve. This should facilitate the empirical estimation for such stochastic models. Moreover, sufficient and yet flexible conditions that guarantee positivity of the rates are readily available. We finally discuss the relation and differences to affine yield-factor models introduced by Duffie and Kan [8]. It turns out that, in contrast to variance swap models, their yield factor representation requires imposing constraints on systems of nonlinear equations that are often not solvable in closed form.
Damir Filipović
Efficient Second-order Weak Scheme for Stochastic Volatility Models
Abstract
Stochastic volatility models can be seen as a particular family of two-dimensional stochastic differential equations (SDE) in which the volatility process follows an autonomous one-dimensional SDE. We take advantage of this structure to propose an efficient discretization scheme with order two of weak convergence. We prove that the order two holds for the asset price and not only for the log-asset as usually found in the literature. Numerical experiments confirm our theoretical result and we show the superiority of our scheme compared to the Euler scheme, with or without Romberg extrapolation.
Benjamin Jourdain, Mohamed Sbai
Bid-Ask Spread Modelling, a Perturbation Approach
Abstract
Our objective is to study liquidity risk, in particular the so-called “bid-ask spread”, as a by-product of market uncertainties. “Bid-ask spread”, and more generally “limit order books” describe the existence of different sell and buy prices, which we explain by using different risk aversions of market participants. The risky asset follows a diffusion process governed by a Brownian motion which is uncertain. We use the error theory with Dirichlet forms to formalize the notion of uncertainty on the Brownian motion. This uncertainty generates noises on the trajectories of the underlying asset and we use these noises to expound the presence of bid-ask spreads. In addition, we prove that these noises also have direct impacts on the mid-price of the risky asset. We further enrich our studies with the resolution of an optimal liquidation problem under these liquidity uncertainties and market impacts. To complete our analysis, some numerical results will be provided.
Thomas Lim, Vathana Ly Vath, Jean-Michel Sahut, Simone Scotti
Optimal Portfolio in a Regime-switching Model
Abstract
In this paper we derive the solution of the classical Merton problem, i.e., maximizing the utility of the terminal wealth, in the case when the risky assets follow a diffusion model with switching coefficients. We show that the optimal portfolio is a generalisation of the corresponding one in the classical Merton case, with portfolio proportions which depend on the market regime. We perform our analysis via the classical approach with the Hamilton–Jacobi– Bellman equation. First we extend the mutual fund theorem as presented in [5] to our framework. Then we show explicit solutions for the optimal strategies in the particular cases of exponential, logarithm and power utility functions.
Adrian Roy L. Valdez, Tiziano Vargiolu

Public Lecture

Frontmatter
Can there Be Excessive Mathematization of the World?
Abstract
If we consider the evolution of ideas regarding chance since Antiquity (Aristotle, Cicero), the appearance of the calculations during the 17th century (Pascal, Fermat), to the theory of hedging on financial markets, we see: a) an extraordinary development of mathematics to manipulate randomness b) the increasing use in this language in economics in the 20th century c) a gradual eviction in the backyard of all that concerns the interpretation of phenomena. The shift to a collective work involving interpretation is an urgent need in the contemporary controversies: financial crises, long-term, biodiversity, but it faces a passive resistance due to the comfort of the agreement on mathematics.
The question on which we focus here is on what philosophical bases and under what circumstances can there be excessive mathematization of the world? This question is asked repeatedly about the economy. To elucidate this difficult problem we address it in a broader scope than just the economy, for knowledge in general. We discuss when and how to diagnose excessive mathematization and what it means. This leads us to ask: why normal science and revolutions in jolts? Why orthodox economics and crises?
Nicolas Bouleau
Metadata
Title
Seminar on Stochastic Analysis, Random Fields and Applications VII
Editors
Robert C. Dalang
Marco Dozzi
Francesco Russo
Copyright Year
2013
Publisher
Springer Basel
Electronic ISBN
978-3-0348-0545-2
Print ISBN
978-3-0348-0544-5
DOI
https://doi.org/10.1007/978-3-0348-0545-2