**Poster Session on Thursday, September 26, 2024:**

Iro René Kouarfate

Title: EXPLICIT SOLUTION SIMULATION METHOD FOR THE 3/2 MODEL

Abstract: An explicit weak solution for the 3/2 stochastic volatility model is obtained and used to develop a simulation algorithm for option pricing purposes. The 3/2 model is a non-a_ne stochastic volatility model whose variance process is the inverse of a CIR process. This property is exploited here to obtain an explicit weak solution, similarly, to Kouritzin (2018). A simulation algorithm based on this solution is proposed and tested via numerical examples. The performance of the resulting pricing algorithm is comparable to that of other popular simulation algorithms

Tak Wa Ng

Abstract: We introduce a new model for individual survivor fund account with bequest that allows the tontine’s participants to leave inheritances to their heirs. Two proposed designs, constant and variable participation, are examined through the lens of an individual account, addressing optimal investment and bequest proportions. Our formulation captures two types of bequest motives: the relative concern between terminal benefit and premature bequest and the intention to smooth the bequest plan. Our numerical illustrations show that the individual’s willingness to participate in the longevity risk pool will decrease with these two bequest motive levels and address the question of when and with what motive level the individual will join the pool.

Julie Bélanger

Title: Option Pricing with Quantum Computers

Abstract: Quantum Finance is a fast-growing field, promising algorithms that can provide a major computational speedup compared to classical calculations. In this project we explore the implementation of option pricing methods using quantum computers, and identify their respective advantages and challenges. For European option pricing, we implement techniques based on quantum amplitude estimation and Schrödingerisation as alternatives to classical simulations. Financial derivatives whose price is linked to an optimal stopping strategy (American options for instance) lead to challenging pricing problems since they cannot directly be obtained from Monte Carlo simulations. In this case, we exploit the similarity between the two-phase Stefan problem and the free boundary value problem formulation of the pricing problem to develop a hybrid classical-quantum framework using the Harrow-Hassidim-Lloyd quantum algorithm.

Kathleen Miao

Title: Robustifying elicitable functionals under Kullback-Leibler misspecification

Abstract: Elicitable functionals and (strict) consistent scoring functions are of interest due to their utility of determining (uniquely) optimal forecasts, and thus the ability to effectively backtest predictions. However, in practice, assuming that a distribution is correctly specified too strong a belief to reliably hold. To remediate this, we incorporate a notion of statistical robustness into the framework of elicitable functionals, meaning that our robust functional accounts for “small” misspecifications of a baseline distribution. Specifically, we propose a robustified version of elicitable functionals by using the Kullback-Leibler divergence to quantify potential misspecifications from a baseline distribution. We show that the robust elicitable functionals admit unique solutions lying at the boundary of the uncertainty region. Since every elicitable functional possesses infinitely many scoring functions, we propose the class of b-homogeneous strictly consistent scoring functions, for which the robust functionals maintain desirable statistical properties. We show the applicability of the REF in two examples: in a reinsurance setting and in robust regression problems.

Felipe José Pinto Antunes

Title: Picard Iteration Scheme for Principal Agent Mean Field Games

Abstract: We propose a numerical method for Principal Agent Mean Field Games based on Picard iterations over the McKean - Vlasov FBSDE formulation. At each iteration, the solutions for the Forward and Backward SDEs are approximated by deep neural networks (DNNs). Moreover, the solution for the BSDE component is formulated as a conditional expectation, and solved through elicitability methods.

Jonghwa Park

Title: Bounding adapted Wasserstein metrics

Abstract: The adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ defines the adapted weak topology in the space of laws of stochastic processes. In contrast with the usual Wasserstein distance $\mathcal{W}_p$, $\mathcal{A}\mathcal{W}_p$ accounts for the temporal dynamics of processes, making it particularly suited for analyzing time-dependent stochastic optimization problems.

While the topological differences between $\mathcal{A}\mathcal{W}_p$ and $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\le C\mathcal{A}\mathcal{W}_p$. This paper aims to answer the following question: can we find nice upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$? Through our investigation of the smooth adapted Wasserstein distance, we establish an explicit bound where the error term is controlled by a sum of Eder's modulus of continuity and the tail behaviours of measures. As a particular instance of our findings, we prove that $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the space of measures that have Lipschitz kernels.

Moreover, our works reveal how the smoothing affects the adapted weak topology. We find that the topology induced by the smooth adapted Wasserstein distance lies between the adapted weak topology and the weak topology, and the inclusion is governed by a decay of a bandwidth parameter.

Shukun Long

Title: Markovian projections for It\^o semimartingales with jumps and the inversion problems for pure jump processes

Abstract: Given a general It\^o semimartingale, its Markovian projection is an It\^o process, with Markovian differential characteristics, that matches the one-dimensional marginal laws of the original process. We construct Markovian projections for It\^o semimartingales with jumps, whose flows of one-dimensional marginal laws are solutions to non-local Fokker--Planck--Kolmogorov equations (FPKEs). As an application, we show how Markovian projections appear in building calibrated diffusion/jump models with both local and stochastic features, leading to the study of the inversion problems. We invert the Markovian projections for pure jump processes, which can be used to construct calibrated local stochastic intensity (LSI) models for credit risk applications. Such models are jump process analogues of the notoriously hard to construct local stochastic volatility (LSV) models used in equity modeling.

Emma Kroell

Title: Optimal reinsurance in a monotone mean-variance framework

Abstract: Optimizing criteria based on static risk measures often leads to time-inconsistent strategies. In the optimal reinsurance literature, mean-variance preferences are a common time-inconsistent decision criterion. To mitigate the time-consistency issue, we study the optimal behaviour of an insurer with monotone mean-variance preferences (Maccheroni et al. (2009)) who purchases reinsurance over a finite continuous-time horizon. Leveraging a dual representation of the monotone mean-variance criterion as a min-max problem, we solve for the insurer's optimal time-consistent ceded loss function. Assuming a Cramér-Lundberg loss model and the expected value premium principle, we explicitly obtain the optimal contract and conclude with some numerical illustrations.

Licheng Zhang

Title: Time-varying equity universe in stochastic portfolio theory: an empirical analysis

Abstract: Stochastic portfolio theory (SPT) aims to study the dynamics of large equity markets, especially over long time horizons. Rank-based modeling is a basic paradigm in this theory, where the growth rates and volatility of any given stock depend on its rank within the universe of all stocks. It is commonly assumed that the universe is fixed over time, and while it is broadly recognized that this assumption is not satisfied in practice, the extent of its impact is not well understood. In this work we perform a comprehensive empirical analysis of the effect that a time-varying equity universe has on the estimation and analysis of rank-based models. We find that the impact is substantial. Listing and delisting events are critical drivers of the well-documented stability of the distribution of capital and are able to recover this stability without the need for an "Atlas" stock to prop up the entire market, which is an undesirable, but necessary, feature in many SPT models. Furthermore, ignoring them introduces significant biases in the estimation of model parameters, especially so-called collision rates which determine rates of return. We develop new estimators that correct for these biases, and find that the corrected estimates are remarkably consistent with the relationship between collision rates, volatility, and particle density predicted by simple diffusion models.

Lu Vy

Title: Insider Trading and Stochastic Liquidity in Higher Dimensions

Abstract: We consider a Kyle-Back model where the noise trading volatility is stochastic and there are multiple traded assets. This may be viewed as a multidimensional generalization of the 2016 Econometrica paper by Dufresne and Fos. Our approach differs from theirs in our emphasis on variational methods. We start with the causal optimal coupling between the fundamental price of the assets and the Wiener process which drives the noise trades. This leads to a finite fuel type optimization problem, but its dual turns out to be of greater interest. From the dual problem, we discover the equilibrium minimizes an average of the initial market depth and the noise traders’ slippage costs. The dual problem admits a solution in one dimension thanks to a result from Ekren, Mostowski, and Zitkovic (2022), and we extend this solution to the multidimensional problem via eigen-decomposition.

Liam Welsh

Title: Nash Equilibria in Reinforcement Learning of Optimal Stopping

Abstract: Stochastic finite player games are often times unable to be solved analytically. In these cases, practitioners often turn to machine learning techniques, including reinforcement learning (RL). RL provides many benefits due to its versatility to numerous problem types. In our work, we pose a finite player discrete-time optimal stopping game and solve for the Nash equilibria using a deep Q-learning algorithm. The game we solve is based off of Canada’s recently updated greenhouse gas offset credit market, and our results demonstrate the profitability of actively participating in this market and characterizes the interactions between agents.

Hang Cheung

Title: Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space

Abstract: I will present a comprehensive theory of viscosity solutions for a class of second-order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, specifically addressing mean field control problems that incorporate common noise. The inherent lack of local compactness in the Wasserstein space complicates the development of viscosity solutions. I will discuss how the introduction of a well-chosen gauge function can mitigate these challenges, providing a clearer framework for addressing these complex equations.

Reza Arabpour

Title: Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach

Abstract: Predicting the conditional evolution of Volterra processes with stochastic volatility is a crucial challenge in mathematical finance. While deep neural network models offer promise in approximating the conditional law of such processes, their effectiveness is hindered by the curse of dimensionality caused by the infinite dimensionality and non-smooth nature of these problems. To address this, we propose a two-step solution. Firstly, we develop a stable dimension reduction technique, projecting the law of a reasonably broad class of Volterra process onto a low-dimensional statistical manifold of non-positive sectional curvature. Next, we introduce a sequentially deep learning model tailored to the manifold’s geometry, which we show can approximate the projected conditional law of the Volterra process. Our model leverages an auxiliary hypernetwork to dynamically update its internal parameters, allowing it to encode non-stationary dynamics of the Volterra process, and it can be interpreted as a gating mechanism in a mixture of expert models where each expert is specialized at a specific point in time. Our hypernetwork further allows us to achieve approximation rates that would seemingly only be possible with very large networks. For this talk, we will first review the underlying problem, followed by an introduction to our hypernetwork approach. Explaining the challenges alongside the solutions and techniques we used to conquer these computation difficulties would be the next part. In the end, we will see how effective our work was in approximating the conditional law of such processes supported by an ablation study of each parameter in our setting.

Gabriela Kováčová

Title: Robust multi-objective stochastic control

Abstract: Model uncertainty is relevant for various dynamic optimization problems within the field of financial mathematics. As one example, let us mention the portfolio selection problem – an investor does not know the true distribution of asset return on the market. There has been a significant body of work dedicated to the study of uncertain stochastic control problem within the literature. Various approaches to handling the uncertainty have been developed, among them the robust approach with the aim of optimizing under the worst-case scenario. While model uncertainty and robust optimization are relatively well understood for stan-dard control problems with a single (scalar) objective, this is much less the case for problems with multiple objectives. In recent years, several (dynamic) problems of financial math-ematics have been approached through methods of multi-objective and set optimization. Set-valued Bellman’s principle, a version of the well known Bellman’s principle for problems with multiple or set-valued objectives, has been derived across different problems. In this work we explore the robust approach to model uncertainty for multi-objective stochastic control problems. Robust multi-objective optimization has been explored in the static but not in the dynamic setting. We are particularly interested in the application of dynamic programming and the impact model uncertainty has on the set-valued Bellman’s principle. We show how the set-valued Bellman’s principle is replaced by certain set rela-tions (or inclusions) under robustness and present assumptions under which equality can be obtained. These results are the first step to extending dynamic programming also to multi-objective problems in the context of model uncertainty.

Pertiny Wilfried Nkuize Ketchiekmen

Title: Optimal Investment and Entropy-Regularized Learning Under Stochastic Volatility Models with Portfolio Constraints

*Abstract: *Nous étudions le problème classique de maximisation de l’utilité espérée de Merton dans un marché incomplet où le portefeuille est soumis à des contraintes. La volatilité est stochastique, ajoutant ainsi une complexité supplémentaire. Plutôt que d’utiliser la dynamique exploratoire classique, nous adoptons une autre version pour surmonter les difficultés rencontrées. En utilisant une fonction d’utilité puissance et une approche d’apprentissage par renforcement, nous apprenons directement des politiques de portefeuille optimales sans avoir besoin d’estimer explicitement les paramètres du modèle. Nous prouvons également un théorème de vérification spécifique à notre fonction d’utilité puissance, montrant que la distribution optimale est une loi normale tronquée. Notre nouvelle dynamique exploratoire, adaptée à la volatilité stochastique et aux contraintes du portefeuille, met en lumière l’interaction complexe entre ces facteurs et démontre l’efficacité des techniques de RL dans ce contexte.

**Ruiyang Hong **

*Title:* Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity

*Abstract: *The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes which generalize but may be too small/constrained to be universal approximators. Motivated by real-world deep learning implementations which are both expressive and statistically reliable, we ask: ``Is there a class of neural networks that are both large enough to be universal but structured enough to generalize?''

This paper constructively provides a positive answer to this question by identifying a highly structured class of ReLU multilayer perceptions (MLPs), which are optimal function approximators and are statistically well-behaved. We first show that any L-Lipschitz function from [0,1]^d to [-n,n] can be approximated to a uniform 1/n error on [0,1]^d with an L-Lipschitz ReLU MLP of depth log_2(d)+4, width O(n^d), and whose weights and biases have magnitude less than 1 in absolute value, except in the first and final layers which have magnitude at-most n. We show that, unlike known classes of universal ReLU MLPs, the empirical Rademacher complexity of our class of ReLU MLPs remains bounded even when its depth and width are allowed to be arbitrarily large. Moreover, our class of MLPs achieves a near-optimal sample complexity of O(\log(N)/\sqrt{N}), on training sets consisting of N i.i.d.\ normalized sub-Gaussian samples.

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