Motivated by recent advances in rough volatility modeling, we introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semi-martingales, nor Markov processes in general. Nonetheless, their Fourier-Laplace functionals admit exponential-affine representations in terms of solutions of associated deterministic integral equations, extending the well-known Riccati equations for classical affine diffusions. Our findings generalize and simplify recent results in the literature on rough volatility.[-]

We study an optimal reinsurance problem under the criterion of maximizing the expected utility of terminal wealth when the loss process exhibits jump clustering features and the insurance company has restricted information about the claims arrival intensity. By solving the associated filtering problem we reduce the original problem to a stochastic control problem under full information. Since the classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite dimensionality of the filter, we choose an alternative approach based on Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize the value process and the optimal reinsurance strategy in terms of a BSDE driven by a marked point process. The talk is based on a joint work with M. Brachetta, G. Callegaro and C. Sgarra (arXiv:2207.05489, 2022).[-]

We study a financial market in which some assets, with prices adapted w.r.t. a reference filtration F are traded. In this presentation, we shall restrict our attention to the case where F is generated by a Brownian motion. One then assumes that an agent has some extra information, and may use strategies adapted to a larger filtration G. This extra information is modeled by the knowledge of some random time $\tau$, when this time occurs. We restrict our study to a progressive enlargement setting, and we pay particular attention to honest times. Our goal is to detect if the knowledge of $\tau$ allows for some arbitrage (classical arbitrages and arbitrages of the first kind), i.e., if using G-adapted strategies, one can make profit. The results presented here are based on two joint papers with Aksamit, Choulli and Deng, in which the authors study No Unbounded Profit with Bounded Risk (NUPBR) in a general filtration F and the case of classical arbitrages in the case of honest times, density framework and immersion setting. We shall also study the information drift and the growth of an optimal portfolio resulting from that model (forthcoming work with T. Schmidt).[-]

Non-convex random sets of admissible positions naturally arise in the setting of fixed transaction costs or when only a finite range of possible transactions is considered. The talk defines set-valued risk measures in such cases and explores the situations when they return convex result, namely, when Lyapunov's theorem applies. The case of fixed transaction costs is analysed in greater details.
Joint work with Andreas Haier (FINMA, Switzerland).[-]

For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. Here, we propose a new approach based on convex duality instead of martingale measures duality: our prices will be expressed using Fenchel conjugate and biconjugate.
This naturally leads to a weak condition of absence of arbitrage opportunity, called Absence of Immediate Profit (AIP), which asserts that the price of the zero claim should be zero. We study the link between (AIP), (NA) and the no-free lunch condition. We show in a one step model that, under (AIP), the super-hedging cost is just the payoff's concave envelop and that (AIP) is equivalent to the non-negativity of the super-hedging prices of some call option.
In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option. We also give some numerical illustrations.[-]