**Thursday 13 April 2023**

### 10.00 - 10.30:* welcome coffee (room 701)*

10.30 - 11.15: Athena PICARELLI

11.15 - 12.00: Sergio PULIDO

12.00 - 12.45: Yuliya MISHURA

12.45 - 14.30*: lunch break*

14.30 - 15.15: Caroline HILLAIRET

15.15 - 16.00: Katia COLANERI

16.00 - 16.45: Cosimo Andrea MUNARI

16.45 - 17.30:* afternoon tea (room 701)*

**ABSTRACTS:**

### KATIA COLANERI (University of Rome - Tor Vergata, Italy)

A filtering approach for parameter estimation in the stochastic SIR model and an application to Austrian Covid19 data

Abstrct: We consider a discrete time stochastic SIR model under partial information where the transmission rate is random and the true number of infectious people at any observation time is not directly observable. In this way we account for asymptomatic and non-tested people. We develop a nested particle filtering approach to estimate the reproduction rate and the model parameters. We apply our methodology to Austrian Covid-19 infection data in the period from May 2020 to June 2022. Finally, we discuss forecasts and model tests. This presentation is based on a joint work with Ruediger Frey and Camilla Damian.

### CAROLINE HILLAIRET (ENSAE CREST, France)

Hawkes processes, Malliavin calculus, and application to cyber-insurance derivatives

Abstrct: In this talk, we provide an expansion formula for the valuation of reinsurance contracts (such that Stop-Loss contracts) whose payoff depends on a cumulative loss indexed by a Hawkes process. It can be applied to cyber-insurance contracts, as the times of occurrence of cyber-claims exhibit self-exciting behavior. The methodology relies on the Poisson imbedding representation and Malliavin calculus. The expansion formula involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the integration by parts formula for Poisson functional. From the actuarial point of view, these processes can be seen as "stressed" scenarios. From a theoretical point of view, Malliavin calculus is a useful and original tool to provide new results on Hawkes processes. This talk is based on joint works with Anthony Réveillac and Mathieu Rosenbaum.

### YULIYA MISHURA (National University of Kyiv, Ukraine)

High-frequency trading and option pricing with fractional Brownian motion

Abstrct: We consider two problems connected to finance in the conditions of long- or short-range dependence modeled by fractional Brownian motion with all spectra of the values of Hurst index. First, we find an explicit formula for locally mean-variance optimal strategies and their performance for an asset price that follows fBm. Without trading costs, risk-adjusted profits are linear in the trading horizon and riseasymmetrically as the Hurst exponent departs from standard Bm, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalize numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit. Second, we consider option pricing for possibly discontinuous payoffs and stochastic volatility involving fBm. The rate of convergence of discretization is estimated.

### COSIMO ANDREA MUNARI (University of Zurich, Switzerland)

Fundamental theorem of asset pricing with acceptable risk in markets with frictions

Abstrct: We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk-reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume. The talk is based on joint work with Maria Arduca.

### ATHENA PICARELLI (University of Verona, Italy)

A deep solver for BSDEs with jumps

Abstrct: The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned solver, starting from a discretized version of the FBSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the FBSDE is viewed as model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation with finitely many jumps of the forward process. We successfully apply our algorithm to option pricing problems in low and high dimension and discuss the applicability in the context of counterparty credit risk.

### SERGIO PULIDO (ENSIIE - LaMME, France)

Affine Volterra processes with jumps

Abstrct: The theory of affine processes has been recently extended to the framework of stochastic Volterra equations with continuous trajectories. These so–called affine Volterra processes overcome modeling shortcomings of affine processes because they can have trajectories whose regularity is different from the regularity of the paths of Brownian motion. More specifically, singular kernels yield rough affine processes. We extend the theory by considering affine stochastic Volterra equations with jumps. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories. We illustrate possible applications to model volatility in financial markets. This talk is based on joint works with Alessandro Bondi, Giulia Livieri and Simone Scotti.