### Session S17 - Stochastic Systems: Analysis, Numerics and Applications

## Talks

Tuesday, July 13, 11:00 ~ 11:35 UTC-3

## Titre: Behavior of the Wishart rsndom matrix with entries in Wiener chaos

### Ciprian Tudor

#### University of Lille, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

For any $n\times d$ random matrix $X$, one can associate its Wishart matrix $W= X^{T}X$. The Wishart matrices are random matrices with many applications in various area. Their asymptotic behavior, when the dimensions $n$ and $d$ are large, is of great interest. We consider a random matrix $X$ whose entries are elements in a Wiener chaos of fixed order. These random entries are either independent or with a particular correlation structure, related to the correlation of the increments of a Hermite process. We discuss the limit behavior in distribution, under the Wasserstein distance, of its associated Wishart matrix. We use the Stein- Malliavin calculus and the characterisation of the independence on Wiener space. We also tackle the situation when the elements of the initial matrix $X$ are in an infinite sum of Wiener chaoses.

Tuesday, July 13, 11:40 ~ 12:15 UTC-3

## Stochastic Volterra Equations

### Sergio Pulido

#### ENSIIE, Universite Paris-Saclay, LaMME, Evry, France - This email address is being protected from spambots. You need JavaScript enabled to view it.

We obtain general weak existence and stability results for Stochastic Volterra Equations (SVEs) with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. The motivation to study SVEs comes from the literature on rough volatility models. Our approach relies on weak convergence in $L^p$ spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.

Joint work with Eduardo Abi Jaber (Universite Paris 1 Pantheon-Sorbonne, France), Christa Cuchiero (University of Vienna, Austria) and Martin Larsson (Carnegie Mellon University, USA).

Tuesday, July 13, 13:00 ~ 13:35 UTC-3

## PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations

### Yuri Saporito

#### School of Applied Mathematics, Getulio Vargas Foundation, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we present a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [QF, 2019, originally published in 2009], where the functional Itô calculus was developed to deal with path-dependent financial derivatives contracts. More specifically, we generalize the Deep Galerkin Method (DGM) of Sirignano and Spiliopoulos [2018] to deal with these equations. The method, which we call Path-Dependent DGM (PDGM), consists of using a combination of feed-forward and Long Short-Term Memory architectures to model the solution of the PPDE. We then analyze several numerical examples from the Financial Mathematics literature that show the capabilities of the method under very different situations.

Joint work with Zhaoyu Zhang (University of Southern California).

Tuesday, July 13, 14:00 ~ 14:35 UTC-3

## Solving non-Markovian Stochastic Control Problems driven by Wiener Functionals

### Alberto Ohashi

#### Universidade de Brasília, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we present a general methodology for stochastic control problems driven by the Brownian motion filtration including non-Markovian and non-semimartingale state processes controlled by mutually singular measures. The main result is the development of a numerical scheme for computing near-optimal controls associated with controlled Wiener functionals via a finite-dimensional approximation procedure. The theory does not require functional differentiability assumptions on the value process and ellipticity conditions on the diffusion components. Explicit rates of convergence are provided under rather weak conditions for distinct types of non-Markovian and non-semimartingale states. The analysis is carried out on suitable finite dimensional spaces and it is based on the weak differential structure introduced by the authors in previous works. The theory is applied to stochastic control problems based on path-dependent SDEs and rough stochastic volatility models, where both drift and possibly degenerated diffusion components are controlled. Optimal control of drifts for nonlinear path-dependent SDEs driven by fractional Brownian motion with exponent $H\in (0,1)$ is also discussed. Finally, we present a simple numerical example to illustrate the method.

Joint work with Dorival Leão (Estatcamp) and Francys Andrews de Souza (Estatcamp).

Tuesday, July 13, 14:40 ~ 15:15 UTC-3

## Numerical scheme for differential equation driven by fractional Brownian motion with power diffusion

### Héctor Araya

#### Universidad de Valparaíso, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.

We consider the numerical approximation of the unique solution of a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H > 1/2$ and power diffusion.

Tuesday, July 13, 15:20 ~ 15:55 UTC-3

## Optimal control for two-dimensional stochastic second grade fluids

### Nikolai V. Chemetov

#### University of São Paulo, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional.

We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.

Also we will discuss the uniqueness result.

Bibliography:

Chemetov N.V., Cipriano F., Optimal control for two-dimensional stochastic second grade fluids. Stochastic Processes and their Applications, 128, n. 8, 2710-2749 (2018).

Chemetov N.V., Cipriano F., Well-posedness of stochastic second grade fluids, J. Mathematical Analysis and Applications, 454, n.2, 585-616 (2017).

Chemetov N.V., Cipriano F., Injection-Suction Control for Two-Dimensional Navier--Stokes Equations with Slippage, SIAM Journal on Control and Optimization, 56, n. 2, 1253-1281 (2018).

Joint work with F. Cipriano (Universidade Nova de Lisboa, Portugal).

Friday, July 16, 10:50 ~ 11:20 UTC-3

## Mean square analysis of fractional linear differential equations

### Laura Villafuerte Altuzar

#### University of Texas at Austin, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We study an extension of a class of fractional linear differential equations (in the Caputo sense) to the random framework. The analysis is based on the mean square calculus. We construct a solution stochastic process, via a generalized power series, which is mean square convergent. We provide explicit approximations of the expectation and variance functions of the solution. To complete the random analysis and from this latter key information, we take advantage of the Principle of Maximum Entropy to calculate approximations of the first probability density function of the solution.

Joint work with Juan Carlos Cortés (Universidad Politécnica de Valencia) and Clara Burgos (Universidad Politécnica de Valencia).

Friday, July 16, 11:20 ~ 11:55 UTC-3

## Convex topological algebras via linear vector fields and Cuntz algebras.

### Mikhail Neklyudov

#### UFAM, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we describe realization by linear vector fields for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is in analogue to the classical Jordan-–Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schr\"odinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra. In the end, we give explicit formula of immersion of arbitrary locally convex finite dimensional topological algebra into certain class of dynamical systems and discuss applications to quantization and stochastic interacting particle systems.

Bibliography: W. Bock, M. Neklyudov, V. Futorny, {\em Convex topological algebras via linear vector fields and Cuntz algebras}, JPAA, Vol. 225, Issue 3 (2021). https://doi.org/10.1016/j.jpaa.2020.106535

Joint work with Wolfgang Bock (TU Kaiserslautern Germany) and Vyacheslav Futorny (USP, Brasil).

Friday, July 16, 12:00 ~ 12:35 UTC-3

## Stratonovich type integration with respect to fractional Brownian motion with Hurst parameter less than 1/2

### Jorge A. León

#### Departamento de Control Automático, Cinvestav-IPN, Mexico - This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk, we introduce a Stratonovich type integral with respect to fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. Then, we study an It\^o's type formula, the relation between this integral and an extension of the divergence operator, and the existence of a unique solution to some Stratonovich stochastic differential equations. Towards this end, roughly speaking, we only need to use the norm of the space $L^2(\Omega\times[0,T])$ instead of a norm of a Sobolev space given by the Malliavin calculus.

Friday, July 16, 12:40 ~ 13:15 UTC-3

## Yule's "nonsense correlation" for Gaussian random walks

### Frederi Viens

#### Michigan State University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.

We provide an exact formula for the second moment of the empirical correlation of two independent Gaussian random walks, as well as implicit formulas for higher moments, and some practical theorems and numerical results in the in-fill-asymptotics regime.

This empirical correlation $\rho_n$, defined for two related series of data of length $n$ using the standard Pearson correlation statistic which is appropriate for i.i.d. data with two moments, is known as Yule's "nonsense correlation" in honor of the statistician G. Udny Yule who described in 1926 the phenomenon by which random walks and other time series are not appropriate for use in this statistic to gauge independence of data series. He observed empirically that its distribution is not concentrated around 0 but diffuse over the entire interval $(-1,1)$. This well-documented effect was roundly ignored by many scientists over the decades, up to the present day, even sparking recent controversies in important areas like climate-change attribution. Since the 1960s, probability theorists wanted to close any possible ambiguity about the issue by computing the variance of the continuous-time version $\rho$ of Yule's nonsense correlation, based on the paths of two independent Brownian motions. This problem eluded the best minds until it was finally closed by Philip Ernst and two co-authors 90 years after Yule's observation, in a paper published in 2017 in the Annals of Statistics. The more practical question of what happens with $\rho_n$ in discrete time remained, which we address here by computing its moments in the case of Gaussian data, the second moment being explicit. We also relate $\rho$ and $\rho_n$, by estimating the speed of convergence of the second moment of its difference, which we find tends to zero at the rate $1/n^2$, an important result in practice since it could help justify using statistical properties of $\rho$ when devising tests for pairs of time series of moderate length.

In this presentation, we provide ideas of the proofs of our results, based on a symbolically tractable integro-differential representation formula for the moments of any order in a class of empirical correlations, which were first established and investigated in the aforementioned paper by Ernst et al., and a 2019 arXiv preprint by Ernst, L.C.G. Rogers, and Quan Zhou. It is only because we succeeded in computing moment generating functions of the various objects used in defining $\rho_n$ that we succeeded in estimating the speed of convergence of its variance. We conjecture that the speed $1/n^2$ applies because of the random-walk structure (independence of increments), while for other types of time series, such as mean-reverting ones, the speed increases to $1/n$.

This work is partially supported by the US National Science Foundation award DMS-1811779.

Joint work with Philip Ernst (Rice University, Houston, TX, USA) and Dongzhou Huang (Rice University, Houston, TX, USA).

Friday, July 16, 13:20 ~ 13:55 UTC-3

## Cutoff thermalization for Ornstein-Uhlenbeck systems with small Lévy noise in the Wasserstein distance

### Michael Hoegele

#### Universidad de los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk presents recent results on cutoff thermalization (also known as the cutoff phenomenon) for a general class of asymptotically exponentially stable Ornstein-Uhlenbeck systems under $\varepsilon$-small additive Lévy noise. The driving noise processes include Brownian motion, $\alpha$-stable Lévy flights, finite intensity compound Poisson processes and red noises and may be highly degenerate. Window cutoff thermalization is shown under generic mild assumptions, that is, we see an asymptotically sharp $\infty / 0$-collapse of the renormalized Wasserstein distance from the current state to the equilibrium measure $\mu^\varepsilon$ along a time window centered in a precise $\varepsilon$-dependent time scale $t_\varepsilon$ . In many interesting situations such as reversible (Lévy) diffusions it is possible to prove the existence of an explicit, universal, deterministic cutoff thermalization profile. The existence of this limit is characterized by the absence of non-normal growth patterns in terms of an orthogonality condition on a computable family of generalized eigenvectors of the drift matrix Q. With this piece of theory at hand we provide a complete discussion of the cutoff phenomenon for the classical linear oscillator with friction subject to $\varepsilon$-small Brownian motion or $\alpha$-stable Lévy flights. Furthermore, we cover the highly degenerate case of a linear chain of oscillators in a generalized heat bath at low temperature.

Joint work with Gerardo Barrera (University of Helsinki, Finland) and Juan Carlos Pardo (CIMAT, México).

Friday, July 16, 14:15 ~ 14:50 UTC-3

## Decomposition of stochastic flows generated by Stratonovich SDEs with jumps

### Paulo Ruffino

#### University of Campinas, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Melo, Morgado and Ruffino (Disc Cont Dyn Syst- B 2016, 21(9)) it is proved that if a semimartingale $X_t$ has a finite number of jumps in compact intervals then, up to a stopping time $\tau$, a stochastic flow of local diffeomorphisms in $M$ driven by $X_t$ can be decomposed into a process in the Lie group of diffeomorphisms which preserve the horizontal foliation composed with a process in the Lie group of diffeomorphisms which preserve the vertical foliation. SDEs of these processes are shown. The stochastic flows with jumps are generated by the classical Marcus equation (as in Kurtz, Pardoux and Protter, Annal. I.H.P., section B, 31 (1995)). Here we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps. Our technique is based in an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and others topological obstructions for the decomposition are also considered, e.g. sufficient conditions for the existence of global decomposition for all $t\geq 0$.

Joint work with Lourival Lima (University of Campinas, Brazil).

Friday, July 16, 14:55 ~ 15:30 UTC-3

## Initial-boundary value problem for stochastic transport equations

### Wladimir Neves

#### Universidade Federal do Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.

This talk concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Itô’s formulation for bounded domains. Moreover, the uniqueness is established without the divergence of the drift vector field bounded from below.

Joint work with Christian Olivera (Universidade Estadual de Campinas, Brazil).

Friday, July 16, 15:35 ~ 16:10 UTC-3

## Stochastic Evolution Equations with Lévy Noise in Spaces of Distributions

### Christian Fonseca-Mora

#### University of Costa Rica, Costa Rica - This email address is being protected from spambots. You need JavaScript enabled to view it.

In the first part of this talk, I will give sufficient and necessary conditions for the existence of a weak and mild solution to a time-dependent linear stochastic evolution equation with linear Lévy noise taking values in the dual of a nuclear space (of which spaces of distributions are a particular example). We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. Later we provide sufficient conditions for the weak convergence of the mild solutions to a sequence of linear stochastic evolution equations with Lévy noise.

We will end this talk by giving sufficient conditions for existence and uniqueness of weak and mild solutions to non-linear stochastic evolution equations driven by general multiplicative Lévy noise.