# STAR seminars - STochastics And Risk

This series of webinars addresses 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.

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

**Upcoming**

**Friday 4th. December, Time 11.00-12.00 (45 min seminar, Q&A and coffee break).**

**Speaker: **David Ruiz Baños

**Title: ***Policies with cash flows subject to interest rate regimes: a path dependent Thiele's partial differential equation *

**Abstract:** A life or pension insurance is a contract between an insurance company and a person, where the insurer promises to pay a sum of money, either at once or periodically, to the insured or a beneficiary (e.g. family member) under some specified events. Actuaries must assess the value of such contracts and their risk. For example, how much is it worth today a pension agreement for a 30 year old Norwegian citizen consisting of a yearly pension of NOK200 000 from a retirement age of 70 years? This question, although it may seem easy to answer, is not. There are two main risks for such contract from the insurance company perspective. First, interest rate risk (too low/high interest) and logenvity or mortality risk (wrong forecast of mortality). In this talk we will discuss interest rate risk and derive a formula for the value of insurance contracts where the cash flow (e.g. NOK200 000) is also random, and not fixed. For example: a pension which pays NOK200 000 in high interest rate regimes and NOK150 000 in low interest rate regimes. Also, it is appealing to consider path dependent conditions on the contract, which requires the use of a functional Itô calculus in order to derive the corresponding Thiele's PDE for reserving. Outline: We will introduce the main and basic definitions and examples. Then we will derive the so-called Thiele's partial differential equation for computing prospective reserves for interest rate (path) dependent policies and finally we will look at specific examples under the Vasicek model by either solving the problem explicitly (tedious but worth it) or numerically (implicit and explicit finite difference method). Finally, we will also overview some possible open questions and future research plans.

**Friday 18th. December, Time 11.00-12.00 (45 min seminar, Q&A and coffee break).**

**Speaker:** Federica Masiero - Università degli Studi di Milano-Bicocca

**Title: TBA**

**Abstract: TBA**

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Past seminars Fall 2020

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

**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).**

**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).**

**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).**

**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-Solé - 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)

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

**Speaker: **Samy Tindel - University of Purdue, 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).**

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

**Title: Functional central limit theorems for stick-breaking priors**

**Abstract: ** We obtain the empirical strong law of large numbers, empirical Glivenko-Cantelli theorem, central limit theorems, functional central limit theorems for various nonparametric Bayesian priors which include the Dirichlet process with general stick-breaking weights, the Poisson-Dirichlet process, the normalized inverse Gaussian process, the normalized generalized gamma process, and the generalized Dirichlet process.

For the Dirichlet process with general stick-breaking weights, we introduce two general conditions such that the central limit theorem holds. Except in the case of generalized Dirichlet process, since the finite dimensional distributions of these processes are either hard to obtain or are complicated to use even they are available, we use the general moment method to obtain the convergence results.

For the generalized Dirichlet process we use its finite dimensional marginal distributions to obtain the asymptotics although the computations are highly technical.

**Friday 13th. November, Time 11.00-12.00 (45 min seminar, Q&A and coffee break).**

**Speaker:** Rama Cont - University of Oxford

**Title: Excursion risk**

**Abstract: ** A broad class of dynamic trading strategies may be characterized in terms of *excursions* of the market price of a portfolio away from a reference level. We propose a mathematical framework for the risk analysis of such strategies, based on a description in terms of price excursions, first in a pathwise setting, without probabilistic assumptions, then in a probabilistic setting, when the price is modelled as a Markov process.

We introduce the notion of δ-excursion, defined as a path which deviates by δ from a reference level before returning to this level. We show that every continuous path has a unique decomposition into such δ-excursions, which turn out to be useful for the scenario analysis of dynamic trading strategies, leading to simple expressions for the number of trades, realized profit, maximum loss and drawdown.

When the underlying asset follows a Markov process, we combine these results with Ito's excursion theory to obtain a tractable decomposition of the process as a concatenation of independent δ-excursions, whose distribution is described in terms of Ito's excursion measure. We provide analytical results for linear diffusions and give new examples of stochastic processes for flexible and tractable modeling of excursions. Finally, we describe a non-parametric scenario simulation method for generating paths whose excursions match those observed in a data set.

This is joint work with: Anna Ananova and RenYuan Xu.

The paper can be found here.

**Friday 20th. November, Time 11.00-12.00 (45 min seminar, Q&A and coffee break).**

**Speaker:** Tusheng Zhang - University of Manchester

**Title: Reflected Brownian motion with measure-valued drifts**

**Abstract:**In this talk, I will present some recent results on the uniqueness and existence of weak solution to the reflected Brownian motion with measure-valued drifts. Furthermore, we obtain some Gaussian type estimates of the transition density function of the solution and we also provide solutions to the associated Neumann boundary value problems.