# STAR - STochastics And Risk Online Seminars

This series of seminars brings together all interested people in probability, stochastic analysis, control, risk evaluation, statistics, with a view towards applications, in particular to renewable energy markets and production. This series brings together the major research themes of the projects STORM, SCROLLER, and SPATUS. If you want to take part to the seminars, please register here.

The seminars will take place on Zoom and a link with the invitation to the virtual room will be sent out in advance, before each event, to all members of the Section Risk and Stochastics as well as to all those who registered at the registration page.

Below you will find time and date for the upcoming seminars:

**Friday 23rd. October, Time 11.00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Samy Tindel - University of Perdue, Indiana

**Title: A coupling between Sinai’s random walk and Brox diffusion**

**Abstract: ** Sinai’s random walk is a standard model of 1-dimensional random walk in random environment. Brox diffusion is its continuous counterpart, that is a Brownian diffusion in a Brownian environment. The convergence in law of a properly rescaled version of Sinai’s walk to Brox diffusion has been established 20 years ago. In this talk, I will explain a strategy which yields the convergence of Sinai’s walk to Brox diffusion thanks to an explicit coupling. This method, based on rough paths techniques, opens the way to rates of convergence in this demanding context. Notice that I’ll try to give a maximum of background about the objects I’m manipulating, and will keep technical considerations to a minimum.

**Friday 6th. November, Time 11.00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Yaozhong Hu - University of Alberta

**Title: TBA**

**Abstract: ** TBA

Here is a list of the past seminars:

**Friday 18th. September, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Andreas Petersson

**Title: Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs**

**Abstract: ** We consider the computation of quadratic functionals of the solution to a linear parabolic stochastic partial differential equation (SPDE) with multiplicative Gaussian noise on a bounded domain. The functionals are allowed to be path dependent and the noise is white in time and may be white in space. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional of the SPDE solution, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. We also use these estimates to derive weak error rates for a fully discrete approximation of the SPDE itself. In the setting of finite element approximations, a computational complexity comparison reveals that approximating the Lyapunov equation allows us to compute quadratic functionals more cheaply compared to applying Monte Carlo or covariance-based methods directly to the discretized SPDE. We illustrate the theoretical results with numerical simulations.

This is joint work with Adam Andersson, Annika Lang and Leander Schroer.

**Friday 25th. September, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Emel Savku

**Title: Optimal investment strategies in a Markov Regime-Switching Market**

**Abstract: ** We discuss two optimal investment problems by using zero-sum and nonzerosum stochastic game approaches in a continuous-time Markov regimeswitching jump-diffusion environment. We represent different states of an economy by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzerosum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains.We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies.We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results.

Joint work with Gerhard Wilhem Weber.

**Friday 2nd. October, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Jasmina Djordjevic

**Title: Perturbation effect on Reflected Backward Stochastic Differential Equations**

**Abstract: ** Perturbed stochastic differential equations, in general, are the topic of permanent interest of many authors, both theoretically and in applications. Stochastic models of complex phenomena under perturbations in analytical mechanics, control theory and population dynamics, for example, can be sometimes compared and approximated by appropriate unperturbed models of a simpler structure. In this way, the problems can be translated into more simple and familiar cases which are easier to solve and investigate. Problems of perturbed backward stochastic differential equations (BSDEs) are very interesting because of their applications in economy and finance. The most interesting problem in this field of perturbations of BSDEs deals with a large class of reflected backward stochastic differential equations whose generator, barrier process and final condition are arbitrarily dependent on a small parameter. The solution of perturbed equation, is compared in the L p -sense, with the solutions of the appropriate unperturbed equations. Conditions under which the solution of the unperturbed equation is L p -stable are given. It is shown that for an arbitrary η > 0 there exists an interval [t(η), T] ⊂ [0, T] on which the L p -difference between the solutions of both the perturbed and unperturbed equations is less than η.

**Friday 9th. October, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Leonardo Rydin Gorjão - Institute of Theoretical Physics, University of Cologne

**Title: Applications and developments of stochastic processes in power-grid frequency measurements: A data-driven study.**

**Abstract: ** Power-grid frequency is a key measurement of stability of power-grid systems. It comprises the balance of power generation and consumption, electricity market exchanges, and control mechanism. Power-grid frequency, as stochastic process, has been scarcely studied. We will present the developments in power-grid frequency data collection, the design of a N-dimensional non-parametric estimator for time-continuous Markov processed, and the design of a computationally efficient Multifractal Detrended Fluctuation Analysis (MFDFA) algorithm. Lastly, we will report on the design of a surrogate stochastic model for power-grid frequency via a fractional Ornstein–Uhlenbeck process, the application of a Hurst index and a volatility estimator, and the limitations due to multifractional and time-and-space coloured noise.

**Friday 16th. October, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Marta Sanz-Sole - University of Barcelona

**Title: Stochastic wave equations with super-linear coefficients**

**Abstract: ** We consider a stochastic wave equation on R^d , d ∈ {1, 2, 3}, driven by a Gaussian noise in (t, x), white in time. We assume that the free terms b and σ are such that, for |x| → ∞,

|σ(x)| ≤ σ_1 + σ2_|x| (ln_+(|x|))^a , |b(x)| ≤ θ_1 + θ_2|x| (ln_+(|x|))^δ , (1)

where θ_2, σ_2 > 0, δ, a > 0, with b dominating over σ. For any fixed time horizon T > 0 and with a suitable constraints on the parameters a, δ, σ_2 and θ_2, we prove existence of a random field solution to the equation and that this solution is unique, and bounded in time and in space a.s. The research is motivated by the article [R. Dalang, D. Khoshnevisan, T. Zhang, AoP, 2019] on a 1-d reaction-diffusion equation with coefficients satisfying conditions similar to (1). We see that the L^∞- method used by these authors can be successfully implemented in the case of wave equations. This is joint work with A. Millet (U. Paris 1, Panthéon-Sorbonne)