Abstract: We consider the problem of estimating the expectation over a convex polyhedron specified by a set of linear inequalities. This problem encompasses a multitude of financial applications including systemic risk quantification, exotic option pricing, and portfolio management. We particularly focus on the case where the target event is rare, which corresponds to extreme systemic failures, deep out-of-the-money options, and high target returns in the aforementioned applications, respectively. This rare-event setting renders the naive Monte Carlo method inefficient and requires the use of variance reduction techniques. To address this issue, we develop a novel and strongly efficient method for the computation of the said expectation in a general rare-event setting by exploiting the geometry of the target polyhedron and concentrating the sampling density almost within the polyhedron. The proposed method significantly outperforms the existing approaches in various numerical experiments in terms of accuracy and computational costs.

Abstract: The classic optimal dividend problem aims to maximize the expected discounted dividend stream over the lifetime of a company. Since dividend payments are irreversible, this problem corresponds to a singular control problem with a risk-neutral utility function applied to the discounted dividend stream. In cases where the company's surplus process encounters model ambiguity under the Brownian filtration, we explore robust dividend payment strategies in worst-case scenarios. We establish a connection between ambiguity aversion in a robust singular control problem and risk aversion in Epstein-Zin preferences. To do so, we first formulate the dividend problem as a recursive utility function with the EZ aggregator within a singular control framework. We investigate the existence and uniqueness of the EZ dividend problem. By employing Backward Stochastic Differential Equation (BSDE) representations where singular controls are involved in the generators of BSDEs, we demonstrate that the EZ formulation is equivalent to the maximin problem involving risk-neutral utility on the discounted dividend stream, incorporating Meanhout's regularity that reflects investors' ambiguity aversion. Considering the equivalent Meanhout's preferences, we solve the robust dividend problem using a Hamilton-Jacobi-Bellman (HJB) approach combined with a variational inequality (VI). Our solution is obtained through a novel shooting method that simultaneously satisfies the VI and boundary conditions. This is a joint work with Kyunghyun Park and Hoi Ying Wong.

Abstract: We propose a two-time scale evolutionary game approach to solving multi-agent reinforcement learning (MARL) problems. Three key components underly the algorithm design. First, we use the perturbed best response to update agents’ belief. Second, we use the fictitious play rule to update the agents’ beliefs about their opponents. Third, policies and beliefs are updated at different learning rates from those used for Q-value updating. The new approach provably converges to epsilon-Nash equilibria of general-sum MARL problems without imposing restrictive assumptions that are typically needed in the literature.
  

 AI-powered algorithms are now widely adopted in marketplaces to price goods and services. However, serious concerns have been raised by the regulators and academics about the possibility that these algorithms may learn to collude through their strategic interactions. Researchers predominately use Q-learning to model the behavior of pricing algorithms, which lacks of convergence guarantees in multi-agent setup. Our approach provides an innovative framework for algorithmic collusion studies. Numerical experiments demonstrate how agents can learn collusive pricing policies and, more importantly, a punishment strategy to sustain collusion.

Abstract: We consider an optimal investment problem to maximize expected utility (CRRA) of the terminal wealth, in a market with search frictions and transaction costs. In the market model, an investor's attempt of transaction is successful only at arrival times of a Poisson process, and the investor pays proportional transaction costs. The optimal trading strategy is described by the no-trade region. We discuss asymptotic analysis of the value function and the no-trade boundaries, for small search frictions and transaction costs at the same time.

Abstract: This talk investigates factors causing different lifestyle choices and their long-run implications in a rational economic agent framework. To accomplish this, we establish a dynamic model of consumption, labor-leisure allocation, and risky investment decisions for an agent with recursive preference. We first characterize four different lifestyles including YOLO (You Only Live Once) based on the agent’s consumption and labor-leisure patterns, and then explore the long-term sustainability of these lifestyles. We discover that lifestyle choices are time-varying and can dramatically change according to the agent’s financial status. These findings provide profound policy implications. This is a joint work with Minsuk Kwak and Byung Hwa Lim.

Abstract: This study investigates the relationships for high frequency (HF) market-making strategies and the expected final wealth to diverse plausible market situations through simulation analysis. We employ the optimal trading strategy for HF market-making constructed on the Hawkes arrival model including self/mutually-exciting factors with synchronizing tendency existing in buy and sell order dynamics. Moreover, we adopt the optimal market-making strategies obtained by using a deep neural network (DNN) to estimate a solution for the relevant high dimensional partial differential equation. Accuracy and efficiency of the DNN-approximated solution are tested by verifying training loss and the final expected profit and loss to select the best hyperparameters including batch size and learning rate policy. With the solution we conduct sensitivity analysis on the optimal trading strategies and the maximal final wealth of a market-maker with respect to change in market stability, synchronizing effect, and market manipulation to assess the relationships. At this end, we discuss practical implications on HF market-making activities with the simulation results.

Abstract: We consider a model with producers making decisions on how much to produce and how much to invest in expansion of capacity of future production. With demand functions exogenously given, we study a multi-agent setting where prices are formed within equilibrium. Depending on the form of the production function, this leads to either a singular or standard control problem. The solutions to the latter are either given explicitly, or characterised via a second-order non-linear ODE. (Based on works with Junchao Jia, Alexander Pavlis and Michael Zervos.)

Abstract: This paper considers the execution of transactions in a blockchain system as a priority queueing game, more specifically, M/G^K/1 queue. We aim to provide new insights into the system dynamics and its impact on user behaviors. For this purpose, we estimate the waiting cost structure of users and analyze optimal bidding strategies.

Abstract: In this talk, we study the optimization problem of an economic agent who chooses the best time for retirement as well as consumption and investment in the presence of a mandatory retirement date. Moreover, the agent faces the borrowing constraint which is constrained in the ability to borrow against future income during working. By utilizing the dual-martingale method for the borrowing constraint, we derive a dual two-person zero-sum game between a singular-controller and a stopper over finite-time horizon. The value of the game satisfies a minmax type of parabolic variational inequality involving both obstacle and gradient constraints, which gives rise to two time-varying free boundaries that correspond to the optimal retirement and the wealth binding, respectively. Using partial differential equation (PDE) techniques, including many technical and non-standard arguments, we establish the uniqueness and existence of a strong solution to the variational inequality, as well as the monotonicity and smoothness of the two free boundaries. Furthermore, the value of game is shown to be the solution to the variational inequality, and we establish a duality theorem to characterize the optimal strategy. To the best our knowledge, this paper is the first to study the zero-sum games between a singular-controller and a stopper over finite-time horizon in the mathematical finance literature. This is a joint work with Junkee Jeon and Zhou Yang.

Abstract: We assume a robust, in general not dominated, probabilistic framework and pro- vides necessary and sufficient conditions for a bipolar representation of subsets of the set of all quasi-sure equivalence classes of non-negative random variables without any further conditions on the underlying measure space. This generalises and unifies existing bipolar theorems proved under stronger assumptions on the robust framework. It enables applications in areas of robust financial modeling. This talk is based on joint work with Gregor Svindland.

Abstract: We explore an optimal token holding and staking problem for cryptocurrency investors. Our investigation revolves around understanding the tradeoff between staking rewards and the consequent illiquidity that emerges from the distinct structure of blockchain platforms or Decentralized Autonomous Organizations (DAOs). We present comprehensive analytic solutions, which enable us to examine the novel implications stemming from the staking mechanism for trading and staking policies and the dynamics of risk-taking behaviors. Our model provides the insights distinguishing between token investments with staking rewards and conventional investment avenues, such as stocks and commodities. This is a joint joint work with Kyung Jin Choi, Junkee Jeon, and Minsuk Kwak.

Abstract: Multi-objective learning is crucial in various real-world problems where the simultaneous optimization of multiple criteria is necessary. It is widely applied in fields such as machine learning, engineering, economics, and finance. However, optimizing multiple tasks simultaneously presents more challenges than solving a standard single-objective optimization. In this work, we identify pathological behaviors in multi-objective learning when the gradients of each loss function exhibit significant imbalances in their magnitudes, along with the presence of negative inner product values. To address these issues, we propose a novel optimization framework named Dual Cone Gradient Descent (DCGD), which adjusts the direction of the updated gradient to ensure it falls within a dual cone region. This region is defined as a set of vectors where the inner products among all loss functions are nonnegative. Theoretically, we analyze the convergence properties of the DCGD algorithms in a nonconvex setting. We then demonstrate the performance of DCGD through comparisons against existing optimization algorithms on a series of benchmark examples. Moreover, we discuss its applications in finance, including multi-asset option valuation and multi-objective portfolio optimization.

Abstract: We present an unsupervised statistical learning approach that investigates all martingale couplings with one variable and one fixed marginal. In this framework, we propose that the variable marginal distribution, which maximizes variance, serves as a solution to the learning objective. We illustrate the applicability of this approach in a variety of unsupervised learning contexts, such as data clustering and principal curve and surface inference. We establish the existence of solutions to our optimization scheme and provide consistency result. Additionally, we show that a specific instance of our method is equivalent to classical principal component analysis (PCA), implying that our approach generalizes PCA.

Abstract: In this talk we first introduce some basic notions of quantum computing. Then we provide a quantum Monte Carlo algorithm to solve high-dimensional  Black-Scholes PDEs with correlation for high-dimensional option pricing. The payoff function of the option is of general form and is only required to be continuous and piece-wise affine (CPWA), which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension d of the PDE and the reciprocal of the prescribed accuracy ε.  Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. This talk is based on joint work with Jianjun Chen and Yongming Li

Abstract: How do firms respond to changes in recession risk? We study a rich dynamic model with time-varying recession risk and heterogeneous firm size. In recessions, cash flows decrease, cash-flow volatility increases, external financing becomes unavailable, and liquidation costs increase. Recession risk leads to preemptive equity issuances by low-cash firms, investment cuts by intermediate-cash firms, and payout cuts by high-cash firms. Interestingly, large firms’ policies and values co-vary more with changes in recession risk because small firms prepare more when recession risk is low. We provide empirical support for these predictions.

Abstract:  We study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting. The talk is based on joint work with Michael Kupper and Max Nendel.

Abstract: When the reinforcement learning (RL) is applied to optimize portfolio strategy in a continuous-time economy, the learning algorithm is often designed based on the Bellman or dynamic programming principle (DPP). Unlike stochastic optimal control problems, time- consistent mean-variance (TCMV) strategies are formulated as subgame perfect equilibrium problems. If the extended HJB equation enforcing DPP is applied to the problem, there are subtle issues concerning about existence and uniqueness of an equilibrium strategy. Spike variation method becomes a promising alternative to derive the equilibrium strategy and uniqueness results to the equilibrium. Uniqueness is important for RL algorithm to obtain equilibrium policy through iterations. We propose a novel RL to distributional (or relaxed) equilibrium control based on spike variation. The policy is then updated through an equilibrium equation so that the computation of value function is not needed. For the case with a constant risk version, our algorithm numerically obtains an exploration control policy close to the estimate from RL algorithm based on DPP in the literature. This shows the consistency of our algorithm because the equilibrium strategies obtained from DPP and spike variation coincide with each other under a constant risk version. For the case of state-dependent risk aversion, the classical solutions from DPP and spike variation differ from each other, but the latter solution is unique. Our algorithm still performs well in a numerical example with a state-dependent risk aversion.

Abstract: This paper investigates the continuous-time entropy-regularized reinforcement learning (RL) for mean-field control problems with common noise. We study the continuous-time counterpart of the Q-function in the mean-field model, coined as q-function in Jia and Zhou (2023) in the single agent's model. It is shown that the controlled common noise gives rise to a nonlocal term of the policy in the exploratory HJB equation, rendering the policy improvement iteration intricate. To devise the model-free RL algorithm, we introduce the integrated q-function (Iq-function) on distributions of both state and action, and an optimal policy can be identified as a two-layer fixed point to the argmax operator of the Iq-function. The martingale characterization of the value function and Iq-function is established by exhausting all test policies. This allows us to propose several algorithms including the Actor-Critic learning algorithm, in which the policy is updated in the Actor-step based on the policy improvement rule induced by the linear derivative of the Iq-function with respect to the action distribution, and the value function and Iq-function are updated in the Critic-step based on the orthogonal martingale loss function involving all test policies. In two examples, within and beyond LQ-control framework, we implement and compare some simulation experiments of our RL algorithms with satisfactory performance.  

Abstract: We study a dividend payout problem under the classical Brownian motion model. The dividend payout must be non-decreasing over time and is subject to an upper bound constraint. Finding the optimal dividend payout strategy in this model is a long-standing open problem in risk theory. To overcome the difficulty, we first introduce a regime-switching problem --- a sequence of single-obstacle problems in ODE --- to approximate the original two-dimensional HJB equation and then take limit. We find a smooth switching boundary and the optimal strategy is given by the boundary.

Abstract: We analyze dynamic imperfectly competitive markets based on Cournot competition among traders who have private information and hold inventory. We propose a recursive and a nonrecursive method to characterize equilibrium in settings where price process is endogenously non-Markovian, and we discuss the advantages of each characterization. The model accommodates heterogeneity in traders’ risk preferences. Equilibrium strategies in any round depend on the entire history of past prices. When multiple equilibia with Markovian versus non-Markovian prices exist (namely, with symmetric traders), we show that the non-Markovian equilibrium is robust to infinitesimal changes in preferences and shock distributions, while the Markovian equilibrium is not.