Minicourses
Xin Guo:
A control-theoretical perspective of continuous time reinforcement learning
Reinforcement Learning (RL) is a one of the fundamental machine learning paradigms, where an agent learns to make a sequence of decisions by interacting with an environment and possibly with other agents. Though primarily developed for discrete environments, RLs are intrinsically continuous. This minicourse presents an overview of recent progress in continuous-time reinforcement learning (RL) from a control-theoretic perspective. Time permitting, the discussion will also cover how techniques from large language models can be integrated into the RL framework to enhance algorithmic performance and facilitate convergence analysis
Christain Bayer:
Rough volatility modelling
Rough volatility models are stochastic volatility models with a "rough" stochastic volatility process. In this context, rough means that the process behaves like fractional Brownian motion with Hurst index $0 < H< 1/2$. Rough volatility models have become popular in the last few years, because they allow to take into account two consistent empirical observations: [1] Realized variance on short-scales is much rougher than the asset price time-series itself. When estimating Hölder/scaling coefficients, we typically see values significantly smaller than $1/2$. [2] The implied volatility surface exhibits a singularity for very short maturity. Specifically, the derivative w.r.t. the log-moneyness variable (a.k.a. "skew") shows a power-law explosion in terms of time-to-maturity going to $0$. As a consequence, rough volatility models are neither semi-martingales nor Markov processes, leading to considerable challenges in theoretical and numerical analysis. In this mini-course, we will explore the path-regularity of realized variance time series of asset prices, and the empirical behavior of the implied volatility skew. This motivates the introduction of rough volatility models, specifically the rough Bergomi and rough Heston models. Using tools of the theory of large deviations, we show that rough volatility models exhibit power law behavior of the implied volatility skew. Finally, we study Markovian approximations of rough volatility models from a theoretical and a numerical point of view. Special invited lectures
Eduardo Abi Jaber:
Path-Signatures: Memory and Stationarity
We explore the interplay between path-signatures, memory, and stationarity, highlighting their implications for machine learning, representation of stochastic processes and applications in mathematical finance. In a first part, we provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H in (0, 1). Our expressions allow to disentangle an infinite dimensional Markovian structure. In addition they open the door to: (i) straightforward and simple approximation schemes that we illustrate numerically, (ii) representations of certain Fourier-Laplace transforms in terms of a non-standard infinite dimensional Riccati equation with important applications for pricing and hedging in quantitative finance. In a second part, we introduce a time-invariant version of the signature: the fading-memory signature, and establish powerful algebraic, analytic and probabilistic properties with applications to learning stationary relationships in time series. This is based on joint works with Paul Gassiat, Louis-Amand Gérard, Yuxing Huang, Dimitri Sotnikov.
Giorgia Callegaro:
A stochastic Gordon-Loeb model for optimal cybersecurity investment under clustered attacks
We develop a continuous-time stochastic model for optimal cybersecurity investment under the threat of cyberattacks. The arrival of attacks is modeled using a Hawkes process, capturing the empirically relevant feature of clustering in cyberattacks. Extending the Gordon-Loeb model, each attack may result in a breach, with breach probability depending on the system's vulnerability. We aim at determining the optimal cybersecurity investment to reduce vulnerability. The problem is cast as a two-dimensional Markovian stochastic optimal control problem and solved using dynamic programming methods. Numerical results illustrate how accounting for attack clustering leads to more responsive and effective investment policies, offering significant improvements over static and Poisson-based benchmark strategies. Our findings underscore the value of incorporating realistic threat dynamics into cybersecurity risk management. This is a joint work with: C. Fontana, C. Hillairet and B. Ongarato.
Sigrid Källblad Nordin:
Tractable infinite dimensional models: theory and applications
In this talk we will focus on a particular class of stochastic processes referred to as measure-valued martingales. They are probability-measure-valued processes featuring a certain martingale property. Such processes arise naturally for many problems such as stochastic control under partial information, enlargement of filtrations, and robust option pricing. Both old and new results will be presented. In particular, we will discuss how the classical theory of stochastic control theory can be modified to encompass controlled processes of this type. Numerous examples and applications from both filtering theory and robust option pricing will then be discussed. Based on joint work with A. Cox, M. Larsson, S. Svaluto and C. Wang Short lectures
Josha Dekker:
Stochastic optimal control with randomly arriving control moments
Control problems with randomly arriving control moments occur naturally. Financial situations in which control moments may arrive randomly are e.g., asset-liquidity spirals or optimal hedging in illiquid markets. We develop methods and algorithms to analyze such problems in a continuous time finite horizon setting, under mild conditions on the arrival process of control moments. Operating on the random timescale implied by the control moments, we obtain a discrete time, infinite-horizon problem. This problem may be solved accordingly or suitably truncated to a finite-horizon problem. We develop a stochastic primal-dual simulation-and-regression algorithm that does not require knowledge of the transition probabilities, as these may not be readily available for such problems. To this end, we present a corresponding dual representation result. We also discuss some insights regarding choices of regression functions and sampling methods and illustrate their effect on the duality gaps. We then apply our methods to several examples, where we explore in particular the effect of randomly arriving control moments on the optimal control policies. Joint work with Roger J.A. Laeven, John G.M. Schoenmakers and Michel H. Vellekoop
Guanyu Jin:
Constructing Uncertainty Sets for Robust Risk Measures: A Composition of $\phi$-Divergences Approach to Combat Tail Uncertainty
Risk measures, which typically evaluate the impact of large losses, are highly sensitive to model misspecification in the tails. In this talk, we discuss a robust optimization approach to combat tail uncertainty by proposing a unifying framework to construct uncertainty sets for a broad class of risk measures, given a specified nominal model. Our framework is based on a parametrization of robust risk measures using two (or multiple) $\phi$-divergence functions, which enables us to provide uncertainty sets that are tailored to both the sensitivity of each risk measure to tail losses and the tail behavior of the nominal distribution. In addition, our formulation allows for a tractable computation of robust risk measures, and elicitation of the $\phi$-divergences that describe a decision maker's risk and ambiguity preferences. We illustrate and implement our results in several examples, including a newsvendor problem and a financial hedging problem.
Gijs Mast :
A COS-tensor Framework for Credit Exposure Calculations
Monte Carlo (MC) simulation methods remain the predominant approach for computing credit exposures in the pricing and risk management practices of the financial industry, owing to their flexibility, implementation simplicity, and transparency. However, accurate computation of high-quantile exposure metrics for large portfolios remains time consuming due to the intrinsically slow convergence of MC simulation. This paper introduces a novel ``COS-tensor'' framework that transcends MC simulation. It has the potential to serve as a computationally efficient alternative, particularly for large, liquid portfolios, while maintaining sufficient flexibility and transparency. Our key insight is that the problem can be transformed and solved in the Fourier domain through two steps: First, rather than generating total exposure samples as in MC methods, we numerically compute the characteristic function (ch.f.) of the total exposure and subsequently recover the cumulative distribution function via the one-dimensional COS method, see Fang and Oosterlee (2009). Second, to circumvent the curse of dimensionality in ch.f. computation, we apply tensor decomposition to the Fourier-cosine coefficient tensor of the joint density function. This ``COS-tensor'' approach constitutes a general framework that generates distinct dimensionally reduced cosine expansions for different tensor decomposition techniques, effectively shifting the curse of dimensionality to the offline training phase for tensor decomposition. The main part of this paper builds and studies the COS-CPD method, resulted from inserting low-rank Canonical Polyadic (CP) decomposition into the COS-tensor framework. A secondary innovation herewith is our Fourier-domain training algorithm for offline CP decomposition, which demonstrates over 100-fold improvements in speed and accuracy compared to physical-domain backpropagation with gradient descent. Extensive testing on real-sized portfolios shows that achieving 0.1\% error for portfolios of thousands of trades under seven risk factors requires only a fraction of the computation time of MC simulation. Results confirm our theoretical error analysis: exponential convergence with respect to Fourier-cosine terms and quadrature points at netting-set level, versus algebraic convergence at counterparty level. Notably, the computational performance remains largely unaffected as portfolio size increases, which is a stark contrast to MC methods. The computational bottleneck lies in the offline training, where the dimensionality curse for risk factors persists. This limitation, combined with diverse extension avenues, indicates substantial potential for further research. For instance, as we will demonstrate in a companion paper, using Tensor Train decomposition can markedly alleviate high-dimensional training constraints. We further note that existing instrument-level acceleration methods remain compatible with our framework, and for portfolios with numerous risk factors, the COS-tensor methods can serve as effective variance reduction techniques for MC simulation.
Marco Zullino:
Dynamic star-shaped risk measures via BSDEs
In this talk, we present characterization results for dynamic return and star-shaped risk measures induced by backward stochastic differential equations (BSDEs). We begin by characterizing a broad family of static star-shaped functionals in a locally convex Fréchet lattice. Then, using the Pasch-Hausdorff envelope, we construct a suitable family of convex drivers of BSDEs, which induce a corresponding family of dynamic convex risk measures. The dynamic return and star-shaped risk measures arise as their essential minimum. Moreover, we prove that if the set of star-shaped supersolutions of a BSDE is non-empty, then for each terminal condition there exists at least one convex BSDE with a non-empty set of supersolutions, yielding the minimal star-shaped supersolution. Finally, we illustrate our theoretical results with a few examples |