**Thursday 28 April 2022**

### 10.00 - 10.30:* welcome coffee*

10.30 - 11.15: Christa Cuchiero: *Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem*

11.15 - 12.00: Claudia Ceci: *Stochastic control problems under partial information and applications*

12.00 - 12.45: Rüdiger Frey: *Deep neural network algorithms for parabolic PIDEs and FBSDEJs: convergence and applications*

12.45 - 14.30*: lunch break*

14.30 - 15.15: Martin Schweizer: *New ramifications on absence of arbitrage*

15.15 - 16.00: Monique Jeanblanc: *A generalisation of Cox model in credit risk*

16.00 - 16.30: *coffee break*

16.30 - 17.15: Paolo Dai Pra: *Some remarks on non-uniqueness in mean-field games*

17.15 - 18.00: Andrea Pascucci: *Kolmogorov SPDEs and applications to stochastic filtering*

**Friday 29 April 2022**

### 9.00 - 9.45: Eckhard Platen: *Principles for modeling long-term market dynamics*

9.45 - 10.30: Zorana Grbac: *Term structure modelling with overnight rates beyond stochastic continuity*

10.30 - 11.00*: coffee break*

11.00 - 11.45: Thorsten Schmidt: *No Arbitrage in insurance and applications*

11.45 - 12.30: Yuri Kabanov: *Ruin problem with investments: solved and open problems*

12.30 - 14.30: *lunch break*

14.30 - 15.00: Matteo Brachetta: *A stochastic control approach to public debt management*

15.00 - 15.30: Alessandro Calvia: *On a class of partially observed systems arising in singular optimal control*

15.30 - 16.00: *coffee break*

16.00 - 16.30: Alekos Cecchin: *Finite state N-agent and mean field control problems*

16.30 - 17.00: Sara Svaluto-Ferro: *Signature-based models: theory and calibration*

**ABSTRACTS:**

### Christa Cuchiero (University of Vienna):

*Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem*

Abstract: We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash to a subset of the entities in order to limit defaults to a given proportion of entities. We prove that the value of the agent's control problem converges as the number of defaultable agents goes to infinity, and that this mean-field limit satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the agent's optimal strategy is to subsidize banks whose asset values lie in a non-trivial time-dependent region. We also study a linear-quadratic version of the model where instead of the terminal losses, the agent optimizes a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically. The talk is based on joint work with Christoph Reisinger and Stefan Rigger.

### Claudia Ceci (University of Chieti-Pescara):

*Stochastic control problems under partial information and applications*

Abstract: Stochastic control finds numerous and various applications in economics, finance and insurance. In many cases the controlled state process depends on some unobserved quantities and this leads to discuss stochastic control problems under partial information. By means of filtering techniques it is often possible to reduce the original problem to an equivalent problem under full information which involves the so-called filter process. We present three different applications where filtering can be applied. The first one concerns an optimal portfolio with intermediate consumption problem in the case where the risky asset return rate and its jump-local characteristics are affected by an unobserved stochastic factor and investors can only observe risky asset prices negotiated in the financial market. Next, we consider a singular stochastic control problem which arises in public debt management in the case where the growth rate of GDP is modulated by an unobserved continuous time Markov chain describing the state of the economy. The government aims at reducing the debt-to GDP ratio and faces partial observation of the regimes of the economy. In the last application we study an optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the claims arrival intensity. Due to the infinite dimensionality of the filter the first and the third problem are solved via a BSDEs approach.

### Rüdiger Frey (Vienna University of Economics and Business):

*Deep neural network algorithms for parabolic PIDEs and FBSDEJs: convergence and applications*

Abstract: High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given space-time point. This gave rise to a growing literature on deep learning based methods for solving partial differential equations; results for integro-differential equations on the other hand are scarce. In this talk we discuss a neural network algorithm for solving parabolic partial integro-differential equations based on their probabilistic representation in terms of backward stochastic differential equations with jumps. We extend convergence proofs for the no-jump case to our setup and we discuss applications in insurance and finance.

### Martin Schweizer (ETH Zurich):

*New ramifications on absence of arbitrage*

Abstract: Consider a general financial market which may have no tradable numeraire. What is then "absence of arbitrage"? After motivating the need for answering this question, we present a new definition and characterisation of absence of arbitrage (AOA) for such general markets. This new AOA property is invariant under fully general changes of numeraire, and it is characterised in dual terms by certain martingale-like properties. If time permits, we also briefly comment on how this could link up with robust absence of arbitrage. The talk is based on joint work with Daniel Balint.

### Monique Jeanblanc (University of Evry):

*A generalisation of Cox model in credit risk*

Abstract: We study a model where $\tau$ is the first time where an increasing process (not absolutely continuous with respect to Lebesgue's measure) and adapted with respect to a reference filtration hits a level independent of the reference filtration. We determine the characteristics of this time and show how the F-stopping times not avoided by $\tau$ play a role in pricing defaultable claims. Joint work with D. Gueye.

### Paolo Dai Pra (University of Verona):

*Some remarks on non-uniqueness in mean-field games*

Abstract: Symmetric N-player games whose running costs are convex in the control have a unique Nash equilibrium for every N. However, their mean-field limit may admit multiple equilibria. After reviewing some examples of this phenomenon, use discuss its stability under time discretisation.

### Andrea Pascucci (University of Bologna):

*Kolmogorov SPDEs and applications to stochastic filtering*

Abstract: We study existence, regularity in Hölder classes and estimates from above and below of the fundamental solution of a degenerate SPDE satisfying the weak Hörmander condition. Our method is based on a Wentzell's reduction of the SPDE to a PDE with random coeffcients to which we apply the parametrix technique to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for the SPDE and the related measurability issues that appear in the stochastic framework. Applications to stochastic filtering are also discussed.

### Eckhard Platen (University of Technology Sydney):

*Principles for modeling long-term market dynamics*

Abstract: The paper derives eight principles that allow predicting the long-term dynamics of large stock markets, the typical distribution of the market capitalization of stocks, the risk premia for stock portfolios, the key role of optimal portfolios at the growth efficient frontier, the least expensive pricing and hedging of long-term payoffs, and other market features. By applying the concepts of entropy maximization and energy conservation in a richer modeling world than typically considered, most of these market features follow rather directly. Furthermore, popular fundamental tools for portfolio and risk management, including the intertemporal capital asset pricing model and the preferred pricing rule, become revised.

### Zorana Grbac (University of Paris Cité):

*Term structure modelling with overnight rates beyond stochastic continuity*

Abstract: In the current reform of interest rate benchmarks, a central role is played by risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A key feature of RFRs is the presence of

jumps and spikes at periodic time intervals as a result of regulatory and liquidity constraints. This corresponds to stochastic discontinuities (i.e., jumps occurring at predetermined dates) in the dynamics of RFRs. In this work, we propose a general modeling framework where RFRs and term rates can have stochastic discontinuities and characterize absence of arbitrage in an extended HJM setup. When the term rate is generated by the RFR itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. In general, this BSDE may admit multiple solutions and we provide sufficient conditions ensuring uniqueness. We develop a tractable specification driven by affine semimartingales, also extending the classical short rate approach to the case of stochastic discontinuities. In this context, we show that a simple specification allows to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Finally, we study hedging in the sense of local risk-minimization when the underlying term structures have stochastic discontinuities. This is joint work with C. Fontana and T. Schmidt.

### Thorsten Schmidt (University of Freiburg):

*No arbitrage in insurance and applications*

Abstract: This work is an attempt to study fundamentally the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it via interest rates, or – as in hybrid products, equity-linked life insurance and variable annuities – directly to stocks or indices. By defining portfolio strategies on an insurance portfolio and combining them with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). A fundamental theorem provides two sufficient conditions for presence or absence of IFA, respectively. For the first one it utilizes the conditional law of large numbers and risk-neutral valuation. As a key result we obtain a simple valuation rule, called QP-rule, which is market consistent and excludes IFA.

Utilizing the theory of enlargements of filtrations we construct a tractable framework for general valuation results, working under weak assumptions. The generality of the approach allows to incorporate many important aspects, like mortality risk or dependence of mortality and stock markets which is of utmost importance in the recent corona crisis. For practical applications, we provide an affine formulation which leads to explicit valuation formulas for a large class of hybrid products. This is joint work with Karl-Theodor Eisele, Philippe Artzner and Raquel Gaspar.

### Yuri Kabanov (University of Besancon):

*Ruin problem with investments: solved and open problems*

### Matteo Brachetta (Politecnico di Milano):

*A stochastic control approach to public debt management*

Abstract: We discuss a class of debt management problems in a stochastic environment model. We propose a model for the debt-to-GDP (Gross Domestic Product) ratio where the government interventions via fiscal policies affect the public debt and the GDP growth rate at the same time. We allow for stochastic interest rate and possible correlation with the GDP growth rate through the dependence of both the processes (interest rate and GDP growth rate) on a stochastic factor which may represent any relevant macroeconomic variable, such as the state of economy. We tackle the problem of a government whose goal is to determine the fiscal policy in order to minimize a general functional cost. We prove that the value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation and provide a Verification Theorem based on classical solutions. We investigate the form of the candidate optimal fiscal policy in many cases of interest, providing interesting policy insights. Finally, we discuss two applications to the debt reduction problem and debt smoothing, providing explicit expressions of the value function and the optimal policy in some special cases.

### Alessandro Calvia (LUISS University):

*On a class of partially observed systems arising in singular optimal control*

Abstract: Partially observed systems model phenomena that appear in various disciplines, such as engineering, economics, and finance, where some quantity of interest, described by a stochastic process called signal, is not directly measurable or observable. The signal process affects another quantity, the observed process, through which one can obtain probabilistic estimates of the state of the unobserved signal. The estimate that one seeks is provided by the filtering process, defined as the conditional distribution of the signal at each time t ≥ 0, given the observation available at time t. This estimate is required, for instance, in optimal control problems with partial observation, where an agent (or controller) aims at optimizing some functional, depending on the stochastic processes previously introduced, by means of a control process. In continuous time, these problems have been deeply studied in the literature. However, to the best of our knowledge, a particularly relevant case for applications has not yet received proper attention: the singular control case. Indeed, few papers study singular control problems for partially observed systems and they do so only (apart from the linear-Gaussian setting) in the case where the control process acts on the observation. Instead, the case where the control acts on the signal process is more delicate, from a technical point of view, and requires a careful novel analysis. In this talk, we will introduce a class of singular control problems with partial information, underline their relevance in applications, and provide the explicit filtering equation (i.e., the SPDE satisfied by the filtering process), together with a uniqueness result. These results lay the ground to solve the corresponding singular optimal control problem under partial observation, that I will introduce and discuss. This is joint work with Giorgio Ferrari, Bielefeld University.

### Alekos Cecchin (University of Padova):

*Finite state N-agent and mean field control problems*

Abstract: Mean field control problems, also called control of McKean-Vlasov SDEs, are control problems involving also the law of the process. They can be seen as limit models for a centralized optimization with a large number of individuals; equivalently, in the language of game theory, such optimization is given by Pareto equilibria for N cooperative players. We examine mean field control problems in which agent's dynamics belong to a finite state space, in continuous time and over a finite time horizon. We characterize the value function of the mean field control problem as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation writted in the (finite-dimensional) simplex of probability measures. In absence of any convexity assumption, we exploit this characterization to prove convergence, as N grows, of the value functions of the centralized N-agent optimal control problem to the limit mean field control problem value function, the main result being to provide a convergence rate of order $1/\sqrt{N}$. Then, assuming convexity, we show that the limit value function is smooth and establish propagation of chaos, i.e. convergence of the N-agent optimal trajectories to the unique limiting optimal trajectory, with an explicit rate.

### Sara Svaluto-Ferro (University of Verona):

*Signature-based models: theory and calibration*

Abstract: Universal classes of dynamic processes based on neural networks and signature methods have recently entered the area of stochastic modeling and Mathematical Finance. This has opened the door to robust and more data-driven model selection mechanisms, while first principles like no arbitrage still apply. Here we focus on signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process. The framework is universal in the sense that any classical model can be approximated arbitrarily well and that the model characteristics can be learned from all sources of available data by simple methods. Indeed, we derive formulas for the expected signature in terms of the expected signature of the primary underlying process. These formulas enter directly in the calibration procedure to option prices, while time series data calibration just reduces to a simple regression.