# Spring with STORM - Online Seminars 2020

During these strange corona times, the need for continuous interaction in science is coming stronger.

We now launch this new Spring Seminar Series Online in probability, stochastic analysis and applications.

These online seminars are both meant to give a scientific update of our doings, as well as being a social event that we can attend once a week from our home-office. Prepare yourself with a good cup of coffee or tea!

If you want to take part to the seminars, please register at this link.

The plan is to arrange one seminar per week. To begin with, the seminars will be given by members of the STORM group as well as members from the stochastic analysis group at UiO to give an update on their recent work. With time we also hope to invite international speakers.

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 interested in participating.

If you are interested in attending these seminars, please email Fabian A. Harang at fabianah<at>math.uio.no for more details.

Below you will find time and date for the seminars:

**Thursday 7. May, 11:00-12.00. (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Fabian A. Harang

**Title: ***Deterministic regularization by noise*

**Abstract: ** In this talk we will discuss the concept of regularization by noise in SDEs from a pathwise point of view. Together with Prof. Nicolas Perkowski at Humboldt University we recently proved that solutions to ODE's perturbed by a particular irregular (but continuous) path exists uniquely, even when the drift vector field of the ODE is only a Scwartz distribution. Moreover, the flow associated to such perturbed ODE is infinitely differentiable. This gives insight into the powerful effect that noise may have on certain equations. We will also discuss an ongoing extension of these results to the regularization by noise towards PDE/SPDEs. By this we mean that we consider an a-priori ill-posed non-linear PDE/SPDE, and show that by perturbation of a sufficiently irregular path, one obtains well posedness of such equations.

**Friday 15. May 11:00-12.00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Michele Giardono

**Title: ***Lifting of Volterra processes: optimal control and HJB equations*

**Abstract: **In this talk we present a new approach to solve an optimization problem with Volterra dynamics driven by a Brownian motion. Thanks to a lift of the original problem to an infinite dimension Banach space, we are able to recover some Markovian properties, which in turn allow us to recover the HJB equations for the lifted problem.

With this approach we are thus able to solve the original Volterra optimization problem with a "classical" HJB approach.

Joint work with Giulia di Nunno

**Wednesday 20. May 11:00-12:00 (45 min seminar+Q&A and "coffee break" after). **

**Speaker: **Bernt Øksendal

**Title: ***Optimal Control of SPDEs with Space Interactions*

**Abstract: **We consider optimal control of a new type of stochastic partial differential equations (SPDEs), in which the dynamics of the system state at a point also depends on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology.

Both sufficient and necessary maximum principles for the optimal control of such systems are proved. We also prove the existence and uniqueness of solutions of such equations. As an illustration, we apply the results to an optimal harvesting problem from a population whose density is modelled as a space-mean stochastic reaction-diffusion equation.

The talk is based on joint works with Nacira Agram and Astrid Hilbert, Linnaeus University (LNU), Växjö, Sweden

**Friday 29. May 11:00-12:00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Silvia Lavagnini

**Title: ***Accuracy of Deep Learning in Calibrating HJM Forward Curves*

**Abstract: **We price European-style options written on forward contracts in the commodity market, which we model with a state-dependent infinite-dimensional Heath-Jarrow-Morton (HJM) approach. We introduce a new volatility operator which maps the square integrable noise into the Filipovi{\'{c}} space of forward curves, and we specify a deterministic parametrized version of it. We train a neural network to approximate the option price as a function of the model parameters. We then use it to calibrate the HJM parameters starting from (simulated) options market data. Finally we introduce a new loss function taking into account bid and ask prices, providing a solution to the liquidity problem. A key issue discovered is that the trained neural network might be non-injective, which could potentially lead to poor accuracy in calibrating the forward curve parameters, even when showing high degree of accuracy in recovering the prices. This implies that the original meaning of the model parameters gets somehow lost in the approximation step.

This is a joint work with Fred Espen Benth (UiO) and Nils Detering (UCSB).

**Wednesday 03. June 11:00-12:00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Emanuela Rosazza-Gianin (University of Milano Bicocca, Italy)

**Title: ***Capital allocation rules and acceptance sets*

**Abstract: **In the literature, capital allocation problems are classically studied and associated to risk measures. The aim of this talk is to introduce a new approach to face capital allocation problems from the perspective of acceptance sets, by defining the family of sub-acceptance sets.

We study the relations between the notions of sub-acceptability and acceptability of a risky position as well as their impact on capital allocation rules; in this context, indeed, capital allocation rules are interpretable as tools for assessing the contribution of a sub-portfolio to a given portfolio in terms of acceptability instead of necessarily involving a risk measure.

Furthermore, we investigate under which conditions on a capital allocation rule a representation of an acceptance set holds in terms of the capital allocation rule itself in quite a general (convex, quasiconvex, S-additive) framework.

Finally, we investigate the correspondence between properties at the level of capital allocation rules and those at the level of sub-acceptance families.

This talk is based on a joint work with Gabriele Canna and Francesca Centrone.

**Friday 12. June 11:00-12:00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Anton Yurchenko-Tytarenko

**Title: ***Sandwiched processes driven by Hölder noises*

**Abstract: **We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. Additionally, numeric schemes for the solution are considered. As illustrations of our approach, generalised CIR and CEV processes will be discussed.

This talk is based on joint work with Giulia Di Nunno and Yuliya Mishura.

**Wednesday 17. June 11:00-12:00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **José Manuel Corcuera (University of Barcelona, Spain)

**Title: ***Path-dependent Kyle equilibrium model*

**Abstract: **Kyle model is a classic model to explain the formation of prices in a financial market. It is assumed that prices increase with the total demand of the stock and market makers provide liquidity in the market and fix the prices in a competitive way. The demand comes from noise traders and its is assumed the presence of an informed trader who knows the liquidation value of the stock. This informed trader tries to optimise their strategy. When all the actors are satisfied then we say that we have an equilibrium.

Different kind of equilibriums have been obtained under the assumption that prices depend on the spot value of the total demand or a particular path-dependence. In this work, inspired by the functional Itô calculus, we study the equilibrium when prices are a functional of the path of the aggregate demand in a very general form. We consider the case when the informed trader is risk neutral as well as the risk-averse case.

This is joint work with Giulia di Nunno and José Fajardo who sadly passed away this third of May.

**Friday 26. June 11:00-12:00 (45 min seminar+Q&A and "coffee break" after).**

**Speaker: **Dennis Schroers

**Title: ***Copulas and Sklar's theorem in infinite dimensions*

**Abstract: **Copulas describe statistical dependence between the components of multivariate random variables in full generality by virtue of Sklar’s theorem. Although they are used and defined for certain infinite dimensional objects (e.g. Gaussian processes, Markov processes or infinite dimensional Archimedean copulas) there is no prevalent notion of a copula as an infinite dimensional law that unifies these concepts. To this end we define copulas as probability measures on product spaces and prove Sklar’s theorem in this general setting.

Afterwards we use this result on Banach spaces to construct cylindrical probability measures with predefined marginals and underlying copula. This induces the functional analytic problem of finding criteria in which cases the obtained cylindrical law induces a real probability measure, which is in general difficult to decide. We solve this problem in the p-Wasserstein space on the space of p-summable sequences (including separable Hilbert spaces) and show that copulas effectively solve a restricted optimal coupling problem.

This is joint work with Fred Espen Benth and Giulia Di Nunno