Department of Mathematics

A point process is a mathematical model for random collections of points in a given space, often used to describe events occurring in time, locations in space, or other structured domains. Formally, i... More

This paper compares the risk management and payout policies of firms maximizing either assets’ average long-term growth or the present value of future dividends. When deleveraging incurs proportiona... More

It is well established that neural networks can effectively learn and replicate operators through such methods as DeepONet and the Fourier Neural Operator (FNO). However, significant research has yet... More

Langevin dynamics is Newton's law for the motion of N particles subject to friction, thermal fluctuations and potential forces. Aside from its relevance in statistical mechanics, its discretizations ... More

In recent years, the classical signatures have been very useful in machine learning and finance in the context of analyzing and predicting missing information from the data streams and pricing financi... More

The long-only minimum variance problem with the total investment constraints is a very important problem for portfolio managers and various practitioners, but an explicit solution for the problem does... More

This is the first part of a two-part series discussing partially observed Markov decision processes (POMDPs), a class of discrete-time stochastic control problems where the decision maker receives noi... More

We propose a new methodology to increase efficiency of numerical methods for optimal control in continuous-time. Our method leverages the solution to a coarsely discretized scheme to generate syntheti... More

This project investigates the real impacts of sustainable investing driven by both financial value and ethical values. Specifically, it examines how firms interact with investors who have a “warm gl... More

We explore the challenges of calibrating a linear combination of signature terms, expressed as polynomials using the Signature Method, to accurately model stochastic processes in financial application... More

In this article, we investigate three classes of equations: the McKean-Vlasov stochastic differential equation (MVSDE), the MVSDE with a subdifferential operator referred to as the McKean-Vlasov stoc... More

Automated Market Makers (AMMs) are a decentralized approach for creating financial markets by allowing investors to invest in liquidity pools of assets against which traders can transact. Liquidity pr... More

We study a stochastic control/stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We demonstrate that the stochastic contr... More

We consider systems of weakly interacting jump processes on heterogeneous random graphs and their large population limit. The interaction is of mean field type weighted by the underlying graphon. A la... More

We present a new dimension reduction method called the global active subspace (GAS) method, and new global sensitivity indices called global activity scores based on GAS. GAS uses expected values of ... More

Causal operators (CO), such as various solution operators to stochastic differential equations, play a central role in contemporary stochastic analysis; however, there is still no canonical framework ... More

Stochastic differential equations (SDEs) driven by fractional Brownian motion arise as natural non-Markovian models in financial mathematics. While upper estimates for the distribution of solution hav... More

Addressing the challenge of comparing unbalanced measures in distinct metric spaces, our work transforms the Partial Gromov-Wasserstein (PGW) problem into a more tractable variant, akin to the Gromov-... More

In this talk, I shall discuss the diffusion operator integral (DOI) method and its applications in options pricing and solving optimal investment problems. The DOI method is a useful tool to generate ... More

Stein's method is a collection of techniques for bounding probability distance between two distributions. While Stein's method has been developed for a wide variety of univariate targets, there is muc... More

Injecting heavy-tailed noise to the iterates of stochastic gradient descent (SGD) has received increasing attention over the past few years. While various theoretical properties of the resulting algor... More

The Markowitz Mean-Variance Optimization problem aims to provide an optimal portfolio with

minimum risk, but the most important parameter in the problem (the covariance

matrix) has to be... More

We will delve into multiple facets of operator learning for PDE. Specifically, I will address some of the key challenges in operator learning, including discretization invariance that the network rema... More

Algorithmic stability is an important notion that has proven powerful for deriving generalization bounds for practical algorithms. The last decade has witnessed an increasing number of stability bound... More

In this talk we study mean field games with possibly multiple mean field equilibria. Instead of focusing on the individual equilibria, we propose to study the set of values over all possible equilibri... More

Training modern deep neural networks (DNNs) can be cast as a non-convex Bayesian learning problem. A standard tool to address this challenge is Langevin Monte Carlo, but it can be arbitrarily slow and... More

Deep Operator Networks: Implementation and Applications (Ryan Bausback)

"In the traditional machine learning setting, the goal is to use a neural network 'f' to approximate the "true" function ... More

Langevin algorithms are core Markov Chain Monte Carlo methods for solving machine learning problems. These methods arise in several contexts in machine learning and data science including Bayesian (le... More

In this presentation, we consider the portfolio optimization problem in a financial market under a general utility function. Empirical analysis suggests that if a significant market fluctuation ("jump... More

In this presentation we will explore an approach for estimating static parameters in state-space models. To illustrate this approach, our focus is directed to the specific task of estimating permeabil... More

This talk will present the deep signature algorithms for solving path-dependent options. We extend the backward scheme in [Huré-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-de... More

We develop a fitted value iteration (FVI) method to compute bicausal optimal transport (OT) where couplings have an adapted structure. Based on the dynamic programming formulation, FVI adopts a functi... More

Portfolio managers need to estimate risk for many assets simultaneously with a limited number of useful observations. The standard approach is to do this using factor models, which reduce the number ... More

In this talk, we provide a review of numerical methods for stochastic control problems and the proposed numerical solutions with emphasis on Hamilton-Jacobi-Bellman (HJB) equations. Then, we present m... More

In this talk, we provide a review of numerical methods for stochastic control problems and the proposed numerical solutions with emphasis on Hamilton-Jacobi-Bellman (HJB) equations. Then, we present m... More

We investigate the statistical evidence for the use of `rough' fractional processes with Hurst exponent $H< 0.5$ for modeling the volatility of financial assets, using a model-free approach. We intro... More

We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, whil... More

The Whittaker-Shannon-Kotel′nikov sampling theorem and the Paley-Wiener theorem state that an entire function of exponential type (band-limited function) can be reconstructed exactly from its values... More

Consider the Monge-Kantorovich optimal transport problem where the cost function is given by a Bregman divergence. The associated transport cost, termed the Bregman-Wasserstein divergence here, presen... More

We study the properties of value functions in joint optimal control and stopping problems where (i) the state variable may be multi-dimensional, (ii) the domain may be unbounded, and (iii) the primiti... More

The idea of regularization that combines a simply structured target with classical estimators is popular in high-dimensional data analysis. We propose a new regularization method and its fast machine ... More

Noh: Solvability of the Gaussian Kyle model with imperfect information and risk aversion



Abstract: We investigate a Kyle model with imperfect information and a risk-averse informed tra... More

We report on recent works (with Fausto Gozzi, Idris Kharroubi, Huyên Pham, Mauro Rosestolato) on mean field control. In particular, we focus on the law invariance property of the value function. We a... More

For a general entropy-regularized stochastic control problem on an infinite horizon, we prove that a policy improvement algorithm (PIA) converges to an optimal relaxed control. Contrary to the standar... More

Fully nonlinear second-order partial differential equations (PDEs) play an important role in many areas. For example, value functions in stochastic optimal control and option prices in mathematical fi... More

The now classical Malliavin-Stein method, a combination of Stein's method with Malliavin calculus, has been very successful in deriving quantitative limit theorems for non-linear approximation. One im... More

TBAWe study the long-time regime of the prediction with expert advice problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to... More

Organizational Meeting

Market risks are growing due to volatility, securitization and increased trading. In this talk we will introduce the concept of market risk, focusing on value at risk (VaR) measure. VaR is a statistic... More

The transition from LIBOR to new alternate risk-free rates will bring considerable challenges for financial institutions in many areas. One typical challenge is how to handle this transition properly ... More

A Risk Participation Swap (RPS) is an agreement by which a bank hedges/offsets a portion or all of the credit risk arising from an over-the-counter (OTC) derivatives transaction with one of its counte... More

Researchers from different areas have independently defined extensions of the usual weak topology between laws of stochastic processes. This includes Aldous' extended weak convergence, Hellwig's info... More

We explore the extent to which using ambiguous communication in persuasion can be beneficial for the sender when facing a receiver who behaves dynamic-consistently under ambiguity. For a family of dec... More

For a general entropy-regularized stochastic control problem on an infinite horizon, we prove that a policy improvement algorithm (PIA) converges to an optimal relaxed control. Contrary to the standar... More

Yue: The Global Active Subspace Method



Abstract: The active subspace (AS) method is a popular dimension reduction method used in problems from sciences and engineering. The method uses... More

Nguyen: Anchored Langevin Dynamics in Sampling and Optimization



Abstract: In this talk, we introduce our newly developed algorithm, the anchored Langevin dynamics. The Langevin dynamic... More

In this talk, we introduce a notion of viscosity solutions for second order elliptic Hamilton-Jacobi-Bellman (HJB) equations with infinite delay associated with infinite-horizon optimal control probl... More

Duan: Implementations on LIBOR Market Model



Abstract: In this talk, we will give a brief introduction to LIBOR Market Model(LMM) and SOFR Market Model frameworks. We will focus on the d... More

Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model has found an amazing range of applications from neuroscience to Josephson junctions in superconduc... More

Duan: Implementations on LIBOR Market Model



Abstract: In this talk, we will give a brief introduction to LIBOR Market Model(LMM) and SOFR Market Model frameworks. We will focus on the ... More

Bose: Backward-looking SABR model for pricing RFR caplets

Abstract: In this talk, we will study the effects of transition from LIBOR to SOFR on interest rate derivatives (caps and floors). I w... More

We study the topic of cost-efficiency in incomplete markets. A portfolio payoff is called cost-efficient if it achieves a given probability distribution at some given investment horizon with a minimum... More

Portfolio risk forecasts require an estimate of the covariance matrix of asset returns, often for a large number of assets. When only a small number of observations are available, we are in the high-d... More

We show that the problem of existence of equilibrium in Kyle's model with imperfect information can be studied by considering a system of forward-bakward equations involving random measures and singul... More

Motivated by activism trading, we solve a generalized Kyle's model type problem using the theory of optimal transport and backward stochastic partial differential equations. Our problem can be recast ... More

Motivated by problems in unsupervised learning, we study gradient flow in the space of probability densities using the notion of linear functional derivative. Gradient flow in this space gives rise to... More

The multi-armed bandit is a classic sequential prediction problem. At each round, the predictor (player) selects a probability distribution from a finite collection of distributions (arms) with the go... More

We investigate optimal execution problems in discrete time with instantaneous price impact and stochastic resilience. First, in the setting of linear transient price impact we derive a closed-form rec... More

We propose a general framework of mass transport between vector-valued measures, which will be called simultaneous mass transport. The new framework is motivated by the need to transport resources of ... More

We introduce a continuous time model for stock prices in a general factor representation with the noise driven by a geometric Brownian motion process. We derive the theoretical hitting probability dis... More

We establish existence and uniqueness of equilibrium in a generalized one-period Kyle (1985) model where insider trading can be subject to a penalty cost that is non-decreasing in the trade size.
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The very high liquidity of S&P 500 (SPX) and VIX derivatives requires that financial institutions price, hedge, and risk-manage their SPX and VIX options portfolios using models that perfectly fit mar... More

Efficient inventory management in the face of product variety is an important part of retail operations management. We analyze the optimal stocking policy for a retailer, in a setup with a single prod... More

We examine the optimal capital structure of a firm with stochastic revenues, stochastic variable costs, and fixed costs. In this two-state variable setting with stochastic operating leverage, we estab... More

We study a one-shot Nash competition between an arbitrary number of identical dealers that compete for the order flow of a client. The client trades either because of proprietary information, exposure... More

The financial math quant symposium at the Florida State University is a twenty three years old tradition to provide our students with guidance on the new trends in the quantitative finance and the suc... More

We consider an infinite horizon optimal investment and consumption problem for an agent who invests in a Black-Scholes-Samuelson market and is unwilling to consume below a fixed proportion her consump... More

We consider a symmetric stochastic game with weak interactions between many players. Time is continuous, the number of states is finite, and costs are ergodic. We prove the existence of a unique Nash ... More

In this talk, I will discuss an equilibrium model with transaction costs and price impact where two agents are incentivized to trade towards a target. The two types of frictions -- price impact and tr... More

In this talk we generalize the Kyle-Back model for insider trading by allowing the volatility to vary stochastically over time. To motivate this extension, we review the case where volatility is deter... More

The objective of this paper is to prove the existence of equilibrium in Kyle’s model with multiple assets. We begin with the single stock version of the Kyle-Back model where the informed trader is ... More

Portfolio managers faced with limited sample sizes must use factor models to estimate the covariance matrix of a high-dimensional returns vector. For the simplest one-factor market model, success rest... More

In this work, we developed a Gaussian Mixture Model (GMM) algorithm to implement the recursive Bayesian filter for nonlinear filtering problem. In the prediction step, we use GMM to approximate the so... More

A novel algorithm for solving stochastic control problems is constructed by combining the SGD technique and the stochastic maximum principle. Under the strong convexity assumption, a rigorous analytic... More

Estimation of the covariance of a high-dimensional returns vector is well-known by practitioners to be impeded by the lack of long data history. We extend the work of Goldberg, Papanicolaou, and Shkol... More

In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popul... More

We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems from an iid sample. This procedure is the first one that exhibits the optimal statistical p... More

We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with e... More

We solve in closed-form a continuous-time Nash equilibrium model in which a finite number of investors with exponential utilities continuously consume and trade strategically with price-impact. Compar... More

The GPS (Goldberg, Papanicolaou, Shkolnik) method shrinks the leading eigenvector of the sample covariance matrix towards the vector of all 1’s by a data driven amount in the low sample-high dimensi... More

In this talk we present a contracting problem between anvinsurance buyer and the seller, subject to prevention efforts in thevform of self-insurance and self-protection. We start with a static formula... More

In the analysis of the financial crises 2008, risk caused by financial modelling was identified as one of the main challenges. In order to reduce the model risk, we discuss an economically justified a... More

Motivated by the connection between univariate dynamic risk measures and backward stochastic differential equations, we start building a theory for set-valued backward stochastic differential equation... More

Consider the stochastic optimal control of systems driven by stochastic differential equations with irregular drift coefficient. In this talk, we will present a necessary and sufficient stochastic max... More

The Graphon Mean Field Game equations consist of a collection of parameterized Hamilton-Jacobi-Bellman equations, and a collection of parameterized Fokker-Planck-Kolmogorov equations coupled through a... More

In this talk, we show that the existence of equilibrium in the Kyle-Back models can be characterized by considering a system of forward Fokker-Planck equation and a system of backward quasilinear para... More

We propose a mean field game model to study the question of how centralization of reward and computational power occur in the Bitcoin-like cryptocurrencies. Miners compete against each other for mini... More

We model sequential auctions in financial markets during a given time period receiving orders of market participants. A clearing price of the auction is determined as the price maximizing the exchange... More

Rough volatility is a new paradigm in finance. We shall talk about the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option... More

Harry Markowitz transformed finance by framing the portfolio construction as a trade-off between the mean and variance of return. The classic Markowitz problem, as solved by every investor in the Capi... More

Backward stochastic differential equations (BSDEs) are probabilistic analogues of semi-linear partial differential equations (PDEs). In particular, BSDEs are used to describe the solutions of stochast... More

We investigate the numerical convergence of a deep learning method of a solution to a PDE to that of a BSDE. We transform the BSDE to a coupled PDE, and find the relationship between the solutions to ... More

We consider an optimal investment and risk control problem for an insurer under the mean-variance (MV) criterion. By introducing a deterministic auxiliary process defined forward in time, we formulate... More

The neural networks have become an extremely useful tool in various applications such as statistical learning and sampling. The empirical success urges a theoretical investigation based on mathematica... More

We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. T... More

A hedge fund manager invests the fund in a constant investment opportunity, and receives high-water mark fees when the fund reaches a new maximum relative to a stochastic benchmark, aiming to maximize... More

We consider a variety of semi-parametric models for a risky asset S = Log X and show how to robustly price and replicate a variety of path dependent claims. The semi-parametric models we consider may... More

The past few years have seen unusual shapes of the implied volatility in equities markets: under certain conditions the implied volatility can deviate from the familiar U-shaped smile, and is better d... More

The classical models for asset processes in math finance are SDEs driven by Brownian motion, and the option price solves a parabolic PDE. In this talk, we will consider that the asset process follows ... More

This paper develops a theory of dynamic pessimism and its impact on asset prices. Notions of time-varying pessimism arise endogenously in our setting as a consequence of agents’ concern for model mi... More

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering... More

We solve a continuous time dynamic Stackelberg game, where a large uninformed seller executes optimally, fully cognizant of the response of Cournot-competitive market makers. The game therefore endoge... More

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In this talk, we present a flexible technique to solve a continuous-time multi-asset/multioption Kyle’s model under general assumptions, including on the (possibly time-varying)

distribution ... More

We introduce a dynamic model of default waterfall of derivatives CCPs and propose a risk sensitive method for sizing the initial margin (IM), and the default fund (DF) and its allocation among clearin... More

The online telemedicine platforms represent a rapidly growing segment of healthcare delivery markets. In this paper, we develop a model of telemedicine platform operations that focuses on managing mul... More

To fulfill the interest of economics community, it is important to establish sensitivity analysis for the stochastic control problems we solve. In the case of principal-agent problem, we benefit from... More

The Black-Scholes option pricing model is well known to be a limit of binomial tree models. What happens if the branching times of the binomial tree are given by a random point process, such as the s... More

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