Mathematical Finance and Stochastics: A Conference in Honor of David Nualart
Conference in honor of Prof. David Nualart, San Sebastian, May, 29-31, 2023
Theme : Financial Mathematics and Stochastics
The work of David Nualart has had influence in different fields, in particular in Financial Mathematics and in Mathematical Statistics. The aim of this workshop is to put together international leaders, as well as young researchers in these fields to honor him in his retirement.
Program
Monday 29th of May 2023
15.00 - 15.30 : Registration
15.30 - 16.00 : Opening by Jose Manuel Corcuera and Wim Schoutens
16.00 - 16.40 : Dilip Madan : Risk Conscious Investment
Abstract: The risk conscious investor is defined as the maximizer of a conservative valuation or dynamically a nonlinear expectation. Both the static and dynamic problems are addressed using distortions of tail probabilities or distortions of tail measures. The multivariate static problem is solved in the context of the multivariate bilateral gamma model. For the dynamic model state transitions are modeled in two ways. The first uses continuous time finite state Markov chains embedded in a response to price movements. The second develops OU equations for all of the first four power variations with drift terms that are modeled using Tempered Fractional Lévy Processes (TFLP). Numerical solutions for policy functions are implemented in trading 768 equity assets over seven years ending December 2021.
16.40 - 17.20 : Giovanni Peccati: Around fourth moment theorems
Abstract: In February 2003, during a memorable month-long visit to the University of Barcelona, David and I discovered a surprising phenomenon, that is: for sequences of normalized random variables living in a fixed Wiener chaos, a CLT takes place if and only the sequence of the corresponding fourth cumulants converges to zero. Such a result, which is now known as the "Fourth moment theorem", has been used, extended a refined by several communities over more than two decades. The aim of my talk is to illustrate some of the most impactful ramifications of the Fourth moment theorem, with special emphasis on discrete and continuum random geometric structures.
17.20 - 18.00 : Yahozong Hu: Nonlinear stochastic wave equation driven by rough noise
Abstract: In this talk , we present a result obtained jointly with Shuhui Liu and Xiong Wang on the existence and uniqueness of the strong solution to one (spatial) dimensional stochastic wave equation $\frac{\partial^2 u(t,x)}{\partial t^2}=\frac{\partial^2 u(t,x)}{\partial x^2}+\sigma(t,x,u(t,x))\dot{W}(t,x)$ assuming $\sigma(t,x,0)=0$, where $\dot W$ is a mean zero Gaussian noise which is white in time and fractional in space with Hurst parameter $H\in(1/4, 1/2)$. The main idea is to find a decomposition of the wave kernel analogous to that for heat kernel. The estimates about the decomposition plays important role in our method.
18.00 - 19.30 : Welcome Reception
Tuesday 30th of May 2023
8.45 - 9.00 : Welcome
9.00 - 9.40 : Arturo Kohatsu-Higa : A Multi level method to study the law of the supremum of a stable process
Abstract: There are very few methods to study the law of the supremum of a jump process using Malliavin Calculus. One of them was proposed by Bouleau and Denis using the lent particle method. This method could be used to obtain the existence of the law. In this presentation we are interested in obtaining the regularity and optimal estimates of the joint law of the stable process and its supremum. Clearly the problems are not only related to the differentiability of the functional given by the supremum process but also related to the restricted amount of finite moments that the stable process has.
David in many articles studied the maximum of continuous processes using the Garsia-Rodemich-Rumsey approach which can not be used in the present situation because the underlying process has jumps.
We used a combination of three effective simulation methods for the maximum of the stable process together with an interpolation methodology (similar to one introduced to V. Bally and L. Caramellino) to obtain almost optimal upper bounds for the joint law of the stable law and its supremum:
1. The convex majorant approach to supremum of Levy processes
2. The Chambers-Mallows-Stuck simulation method for stable laws
3. The Multi Level Monte Carlo method
This is joint work with Jorge Gonzalez-Cazares and Alex Mijatovic (Warwick University and Turing Institute).
9.40 - 10.20 : Salvador Ortiz : SPDE bridges with observation noise and their spatial approximation
Abstract: In this talk, the notion of SPDE bridge with observation noise is first introduced. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. Then, a general framework for the spatial discretization of these bridge processes is introduced. Finally, an example of these discretizations is presented.
This is a joint work with Giulia Di Nunno (UiO) and Andreas Petersson (UiO).
10.20 - 10.50 : Coffee Break
10.50 - 11.30 : Ivan Nourdin: David's influence on my career
Abstract: In this talk, I will reflect on the evolution of my research work, from my beginnings as a doctoral student until today, and will try to highlight the important influence that David has had (and continues to have!) on my research trajectory.
11.30 - 12.10 : Mark Podolskij : On estimation of the maximal rank of stochastic volatility
Abstract: In this talk we address the question of how many Brownian motions are required to model a multivariate price process of diffusion type. It turns out that this question is equivalent to estimation of the maximal rank of the volatility component. We solve this mathematical problem in the high frequency regime and provide a formal estimation and testing procedure. If the time allows we touch upon high dimensional aspects of the problem.
12.10 - 12.40 : Gero Junike : Sequential decompositions at their limit
Abstract: Sequential updating (SU) decompositions are a well-known technique for creating profit and loss (P&L) attributions, e.g., for a bond portfolio, by dividing the time horizon into subintervals and sequentially editing each risk factor in each subinterval, e.g., FX, IR or CS. We show that SU decompositions converge for increasing number of subintervals if the P&L attribution can be expressed by a smooth function of the risk factors. We further consider average SU decompositions, which are invariant with respect to the order or labeling of the risk factors. The averaging is numerically expensive, and we discuss several ways in which the computational complexity of average SU decompositions can be significantly reduced.
12.40 - 15.00 : Lunch break
15.00 - 15.40 : Nakahiro Yoshida : Some recent developments in asymptotic expansion
Abstract: Asymptotic expansion has been established for independent and weakly dependent time series models, as a basic technique to construct the modern fields of theoretical statistics. In the central limit cases, asymptotic expansion was studied in the 1990s for martingales and mixing Markov processes. Statistical inference for non-ergodic models experienced the second trend in the 1990s, synchronized with finance with high frequency data. Stimulated by the success in the inferential theory with asymptotic expansion in the classical settings, asymptotic expansion of a martingale in the non-ergodic statistics was studied. A correction term involving the Malliavin derivatives appears in the formula although the object is in the Ito calculus (Yoshida 2013). Asymptotic expansion for Skorohod integrals was obtained in Nualart and Yoshida (2019), inspired by the idea of interpolation introduced by Nourdin, Nualart and Peccati (2016), and applied to a quadratic form of a fractional Brownian motion with random weights. In applications of the general scheme, we need a systematic way of assessing the order of a randomly weighted sum of products of multiple Wiener integrals. This talk gives an overview of some recent developments in the theory of asymptotic expansion.
15.40 - 16.10 : Asma Khedher : A characterisation and analysis of infinite-dimensional Wishart processes
Abstract: In this talk, we provide a characterisation result for the existence of Wishart processes in the cone of positive trace class operators. Moreover, we derive the Fourier-Laplace transform for the purely real and purely imaginary cases as well as certain joint cases and also derive the equations for the eigenvalues and eigenvectors of our Wishart processes.
This is based on a joint work with Christa Cuchiero and Sonja Cox.
16.10 - 16.50 : Coffee Break
16.50 - 17.30 : Guangqu Zheng : CLT on Wiener chaoses and for SPDEs
Abstract: In this talk, I will present several seminal contributions of David that intersects with my research.
I will begin with a brief introduction to the famous/powerful fourth moment theorem by David
and Giovanni Peccati (2005), then David’s further contribution to the recent Malliavin-Stein approach.
Next, I will speak about recent progress on the study of spatial averages of SPDEs, which was initiated
by David with Jingyu Huang and Lauri Viitasaari (2018).
17.30 - 18.00 : Benjamin Robinson : A regularized Kellerer theorem in arbitrary dimension
Abstract: We present a multidimensional extension of Kellerer's theorem on the existence of mimicking Markov martingales for peacocks, a term derived from the French for stochastic processes increasing in convex order. For a continuous-time peacock in arbitrary dimension, after Gaussian regularization, we show that there exists a strongly Markovian mimicking martingale Itô diffusion. A novel compactness result for martingale diffusions is a key tool in our proof. Moreover, we provide counterexamples to show, in dimension d≥2, that uniqueness may not hold, and that some regularization is necessary to guarantee existence of a mimicking Markov martingale.
20.00 - 23.00: Conference Dinner
Wednesday 31th of May 2023
8.45 - 9.00 : Welcome
9.00 - 9.40 : Giulia Di Nunno : Pricing in sandwiched Volterra volatility models
Abstract: We introduce a new model of financial market with stochastic volatility driven by an arbitrary Ho ̈lder continuous Gaussian Volterra process. The distinguishing feature of the model is the form of the volatility equation which ensures the solution to be “sandwiched” between two arbitrary Ho ̈lder continuous functions chosen in advance. We discuss the structure of local martingale measures on this market, investigate integrability and Malliavin differentiability of prices and volatilities as well as study absolute continuity of the corresponding probability laws. Malliavin calculus ia rhino used to develop an algorithm of pricing options with discontinuous payoffs.
9.40 - 10.20 : Tommi Sottinen : Completely correlated mixed fractional Brownian motion
Abstract: The completely correlated mixed fractional Brownian motion is a process that is driven by a mixture of Brownian motion and a completely correlated fractional Brownian motion (with H>1/2) that is constructed from the Brownian motion via the Molchan–Golosov representation. Thus, there is a single Brownian motion driving the mixed process. We provide a transfer principle for the completely correlated mixed fractional Brownian motion and use it to construct the Cameron–Martin–Girsanov–Hitsuda theorem and prediction formulas.
10.20 - 10.50 : Coffee Break
10.50 - 11.30 : Peter Imkeller : Geometric properties of some rough curves: SBR measure and local time
Abstract: We investigate geometric properties of graphs of Takagi type functions, represented by series based on smooth functions. They are Holder continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pullback attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by time reversal. A suitable version of approximate self similarity for deterministic functions allows to ”telescope”
small scale properties from macroscopic ones. As a consequence, absolute continuity of the SBR measure is seen to be dual to the existence of local time. The link between the rough curves considered and smooth dynamical systems can be generalized in various ways. Applications to regularization of singular ODE by rough signals are on our agenda. This is joint work with O. Pamen (U Liverpool and AIMS Ghana).
11.30 - 12.00 : Some words by David Nualart
12.00 - 12.30 : Closing by Jose Manuel Corcuera