Abstract : The mini-symposium we propose aims to feature the latest developments and promote research in the field of quantitative finance. The mini-symposium will enhance interaction and cooperation among researchers worldwide working on some specific topics in the field. In particular, we will focus on, but are not limited to, the following three topics:
• stochastic control in quantitative finance,
• dynamic game and mean-field game in quantitative finance, and
• machine learning and reinforcement learning in quantitative finance.
Consequently, we plan to have three sessions on the above three topics, respectively
Organizer(s) : Min Dai, Zuoquan Xu, Chao Zhou
Sponsor : This session is sponsored by the SIAM Activity Group on Financial Mathematics and Engineering.
Anders Max Reppen (Boston University Questrom School of Business)
Valentin Tissot-Daguette (Princeton University)
Abstract : A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several examples related to financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions. We also briefly show how this method applies tis the classical Stefan problem of solidification.
Abstract : We study a dynamic mean-variance portfolio optimization problem under the reinforcement learning framework,
where an entropy regularizer is introduced to induce exploration. Due to the time–inconsistency
involved in a mean-variance criterion, we aim to learn an equilibrium policy. Under an incomplete market
setting, we obtain a semi-analytical, exploratory, equilibrium mean-variance policy that turns out to follow a
Gaussian distribution. We then focus on a Gaussian mean return model and propose a reinforcement learning
algorithm to find the equilibrium policy. Thanks to a thoroughly designed policy iteration procedure
in our algorithm, we prove the convergence of our algorithm under mild conditions, despite that dynamic
programming principle and the usual policy improvement theorem failing to hold for an equilibrium policy.
Numerical experiments are given to demonstrate our algorithm. The design and implementation of our reinforcement
learning algorithm apply to a general market setup.
[05301] On Consistency of Selecting Signatures Using Lasso: A Tale of Ito and Stratonovich
Format : Talk at Waseda University
Author(s) :
Xin Guo (UC Berkeley)
Ruixun Zhang (Peking University)
Chaoyi Zhao (Peking University)
Abstract : We investigate the statistical consistency of using Lasso to select signatures in machine learning predictions. Signatures are defined as iterated path integrals of stochastic processes, and their universal nonlinearity warrants Lasso as a common tool to select sparse linear approximations. We study the consistency of Lasso for selecting signatures for the Brownian motion, the Ornstein--Uhlenbeck process, and the fractional Brownian motion, both theoretically and numerically. Our findings show that, for signatures defined by Ito integrals, Lasso is more consistent for processes that are closer to Brownian motion and have weaker inter-dimensional correlations. For signatures defined by Stratonovich integrals, we observe better Lasso consistency for mean-reverting processes than for mean-averting processes. Our results emphasize the importance of choosing appropriate definitions of signatures in statistical inference and machine learning, particularly for non-Brownian processes.
[02766] Portfolio choice with transaction costs and reinforcement learning
Format : Talk at Waseda University
Author(s) :
Min Dai (Hong Kong Polytechnic University)
Abstract : We provide a reinforcement learning approach for portfolio choice with transaction costs. Numerical results are provided to demonstrate the efficiency of our approach.
[05335] Dynamic programming for mean-variance portfolio selection
Format : Talk at Waseda University
Author(s) :
Martin Schweizer (ETH Zurich)
Abstract : We present a dynamic programming approach to solving the mean-variance portfolio selection problem in finite discrete time. This bypasses issues of time-inconsistency and hence does not need the introduction of an equilibrium or game-theoretic approach. The talk is based on joint work with Zhouyi Tan.
[05363] Non-Concave Utility Maximization with Transaction Costs
Format : Talk at Waseda University
Author(s) :
shuaijie qian (Hong Kong University of Science and Technology)
Chen Yang (The Chinese University of Hong Kong)
Abstract : We consider the non-concave utility maximization problem, which appears in plenty of areas in finance, with transaction costs. Technically, we propose a proper terminal condition and lay the corresponding theoretical foundation of viscosity solutions. This terminal condition implies that any transaction close to maturity provides a marginal contribution to the target. We find that the introduction of transaction costs into non-concave utility problems can prevent the portfolio from unbounded leverage and also result in richer action regions than classical transaction costs problems with concave utilities.
[05320] Optimal stopping without time consistency
Format : Talk at Waseda University
Author(s) :
Hanqing Jin (University of Oxford)
Yanzhao Yang (University of Oxford)
Abstract : We study a continuous time dynamic optimal stopping problem with a flow of preferences, which can be in non-expectation form and can depend on both the current time and state of the system in general. We will define a solution to the problem by the rationality of the agent, and compare it with other solutions appeared in literature.
[03404] Optimal dividend payout with non-decreasing constraint
Format : Talk at Waseda University
Author(s) :
Zuo Quan Xu (The Hong Kong Polytechnic University )
Chonghu Guan (Jiaying University)
Abstract : We study a dividend payout problem under the classical Cram ́er-Lundberg 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.
[05500] Extended mean-field control problems with multi-dimensional singular controls
Format : Talk at Waseda University
Author(s) :
Ulrich Horst (Humboldt University Berlin)
Robert Denkert (Humboldt University Berlin)
Abstract : We consider extended mean-field control problems with multi-dimensional singular controls. A key challenge when analysing singular controls are jump costs. When controls are one-dimensional, jump costs are most naturally computed by linear interpolation. When the controls are multi-dimensional the situation is more complex, especially when the model parameters depend on an additional mean-field interaction term, in which case one needs to ``jointly'' and ``consistently'' interpolate jumps both on a distributional and a pathwise level. This is achieved by introducing the novel concept of two-layer parametrisations of stochastic processes. Two-layer parametrisations allow us to equivalently rewrite rewards in terms of continuous functions of parametrisations of control process and to derive an explicit representation of rewards in terms of minimal jump costs. From this we derive a DPP for extended mean-field control problems with multi-dimensional singular controls. Under the additional assumption that the value function is continuous we characterise the value function as the minimal super-solution to a certain quasi-variational inequality in the Wasserstein space.
[05172] Lightning Network Economics: Channels and Topology
Format : Talk at Waseda University
Author(s) :
Paolo Guasoni (Dublin City University)
Gur Huberman (Columbia Business School)
Clara Shikhelman (Chaincode Labs)
Abstract : Designed to address Bitcoin’s scalability challenge, the Lightning Network (LN) is a protocol allowing two parties to secure bitcoin payments and escrow holdings between them. In a lightning channel, each party commits collateral towards future payments to the counterparty and payments are cryptographically secured updates of collaterals. First, we identify conditions for two parties to optimally establish a channel, find explicit formulas for channel costs and optimal collaterals, and derive the implied reduction in congestion of the blockchain. Then we obtain necessary conditions for cost-minimizing topologies and bounds on the cost of the optimal topology, showing the unusual circumstances in which it is a hub that connects all other nodes.
[05309] Non-monotone linear-quadratic mean field games with a major player
Format : Talk at Waseda University
Author(s) :
Chenchen Mou (City University of Hong Kong)
Min Li (Shandong University)
Zhen Wu (Shandong University)
Chao Zhou (National University of Singapore)
Abstract : In this talk, we consider a class of linear-quadratic mean field games with a major player. The game has the feature that the major player can have an impact on all the minor players while the miner players as a whole influence the major player. The value functions corresponding to the major and minor players satisfy the corresponding master equations, which generate the so-called master system. To the best of our knowledge, the global wellposedness of the master system remains open. The main focus of the paper is to tackle the open problem in the linear-quadratic setting without any monotonicity conditions. The key idea is to use the non-degeneracy of the common noise to show the global wellposedness of the Nash certainty equivalence system for the minor players. Meanwhile, we also study the corresponding $N$-minor player and a major player game. More precisely, we show the quantitative convergence from the $N$-minor player and a major player game to the mean field game and the propagation of chaos property for the related optimal trajectories for both the minor players and the major player. This is based on a joint work with M. Li, Z. Wu and C. Zhou.
[05501] Large ranking games with diffusion control
Format : Talk at Waseda University
Author(s) :
Stefan Ankirchner (University of Jena)
Nabil Kazi-Tani (Université de Lorraine)
Julian Wendt (University of Jena)
Chao Zhou (National University of Singapore)
Abstract : We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best α ∈ (0, 1) of all states receive a fixed prize. Within the mean field limit version of the game we compute an explicit equilibrium, a threshold strategy that consists in choosing the maximal fluctuation intensity when the state is below a given threshold, and the minimal intensity else. We show that for large n the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two player case.
