Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark H. A. Davis's Contributions

Preface
Table of Contents
Bibliography of Mark H. A. Davis
	Books
	Published Papers
	Unpublished Works
	Obituaries and Legacy
Control in Hilbert Space and First-Order Mean Field Type Problem
	1 Introduction
	2 The Model
		2.1 Assumptions
		2.2 The Problem
	3 Necessary Conditions of Optimality
		3.1 The System
		3.2 Decoupling
	4 A Priori Estimates
		4.1 First Estimate
		4.2 Second Estimate
	5 Local Time Solution
		5.1 Fixed Point Approach
		5.2 Choice of Functions μt and ρt
		5.3 Contraction Mapping
	6 Global Solution
		6.1 Statement of Results
		6.2 Optimal Control
		6.3 Bellman Equation
	7 Application to Mean Field Type Control Theory
		7.1 Wasserstein Space
		7.2 Functional Derivatives
		7.3 Mean Field Type Control Problems
		7.4 The Hilbert Space Hm and the Push-Forward Map
			7.4.1 Settings
			7.4.2 Extending the Domain of Functions to Hm
		7.5 Control Problem in the Hilbert Space Hm
		7.6 Necessary and Sufficient Condition for Optimality
		7.7 Properties of the Value Function
		7.8 Bellman Equation
	8 Acknowledgements
	References
Risk-Sensitive Markov Decision Problems under Model Uncertainty: Finite Time Horizon Case
	1 Introduction
	2 Risk-sensitive Markovian discounted control problems with model uncertainty
	3 The adaptive robust risk sensitive discounted control problem
		3.1 Adaptive robust Bellman equation
	4 Exponential Discounted Tamed Quadratic Criterion Example
	5 Machine Learning Algorithm and Numerical Results
	Acknowledgements
	References
Optimal Control of Piecewise Deterministic Markov Processes
	1 Introduction
	2 Notation and definition
	3 Problem formulation for the controlled PDMP
		3.1 Parameters of the model
		3.2 Construction of the controlled process ξt
		3.3 Admissible strategies
		3.4 Problems formulation
	4 Main assumptions and auxiliary results
		4.1 Main assumptions
		4.2 Auxiliary results
	5 The discounted control problem
	6 The average control problem
	Appendix
	References
Pathwise Approximations for the Solution of the Non-Linear Filtering Problem
	1 Introduction
	2 Preliminaries
		2.1 The filtering problem
		2.2 High order time discretisation of the filter
			2.2.1 Stochastic Taylor expansions
			2.2.2 Discretisation of the log-likelihood process
			2.2.3 Discretisation of the filter
			2.2.4 Order of approximation for the filtering functionals
	3 Robustness of the approximation
	4 Proof of the robustness of the approximation
	References
Discrete-Time Portfolio Optimization under Maximum Drawdown Constraint with Partial Information and Deep Learning Resolution
	1 Introduction
	2 Problem setup
	3 Dynamic programming system
		3.1 Change of measure and Bayesian filtering
		3.2 The static set of admissible controls
		3.3 Derivation of the dynamic programming equation
		3.4 Special case: CRRA utility function
	4 The Gaussian case
		4.1 Bayesian Kalman filtering
		4.2 Finite-dimensional dynamic programming equation
	5 Deep learning numerical resolution
		5.1 Architectures of the deep neural networks
		5.2 Hybrid-Now algorithm
		5.3 Numerical results
			5.3.1 Learning and non-learning strategies
			5.3.2 Learning, non-learning and constrained equally-weighted strategies
			5.3.3 Non-learning and Merton strategies
		5.4 Sensitivities analysis
	6 Conclusion
	Appendix
		6.1 Proof of Proposition 1
		6.2 Proof of Proposition 2
		6.3 Proof of Lemma 1
		6.4 Proof of Lemma 2
		6.5 Proof of Lemma 3
		6.6 Proof of Lemma 4
	References
Estimating the Matthew Effects: Switching Pareto Dynamics
	1 Introduction
	2 Generating Pareto Random Variables
	3 Switching Parameter Values
	4 Estimation
		A Filter
	5 Parameter Estimation
	6 Recursive Estimates
	7 Implemention
	References
Optimal Couplings onWiener Space and An Extension of Talagrand’s Transport Inequality
	1 Introduction
	2 Preliminaries
	3 Intrinsic drift and optimal coupling in the absolutely continuous case
	4 Specific Relative Entropy
	5 Intrinsic Wiener Process and Optimal Coupling for Semimartingale Measures
	References
Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation
	1 Introduction
	2 Extended HJB Equation
		2.1 Notations
		2.2 Time-Inconsistent Stochastic Control Problems
		2.3 Intra-Personal Equilibrium
		2.4 Sufficient and Necessary Condition
		2.5 Extended HJB
	3 Discussions
		3.1 Intra-Personal Equilibria with Fixed Initial Data
		3.2 Set of Alternative Strategies
		3.3 Regular and Strong Intra-Personal Equilibrium
		3.4 Existence and Uniqueness
		3.5 Non-Markovian Strategies
	4 Closed-Loop versus Open-Loop Intra-Personal Equilibria
	5 Optimal Stopping
	6 Discretization Approach
	7 Applications
		7.1 Present-bias Preferences
		7.2 Mean-Variance
		7.3 Non-EUT Preferences
	8 Dynamically Consistent Preferences
	References
N-Player and Mean-Field Games in Itô-Diffusion Markets with Competitive or Homophilous Interaction
	1 Introduction
	2 Incomplete Itô-diffusion common market and CARA utilities
		2.1 The N-player game
			2.1.1 The Markovian case
			2.1.2 A fully solvable example
		2.2 The common-noise MFG
			2.2.1 The Itô-diffusion common-noise MFG
			2.2.2 The Markovian case
	3 Complete Itô-diffusion common market and CARA utilities with random risk tolerance coefficients
		3.1 The Itô-diffusion market and random risk tolerance coefficients
		3.2 The single-player problem
			3.2.1 The Markovian case
		3.3 N-player game
		3.4 The Itô-diffusion common-noise MFG
	4 Conclusions and future research directions
	References
A Variational Characterization of Langevin–Smoluchowski Diffusions
	1 Introduction
		1.1 Preview
	2 The setting
		2.1 Invariant measure, likelihood ratio, and relative entropy
		2.2 The probabilistic setting
	3 Reversal of time
	4 A stochastic control problem
		4.1 Entropic interpretation of the expected cost when Q(Rn) < ∞
	5 From local to square-integrable martingales
		5.1 Reversing time once again
		5.2 The dynamics of the relative entropy process
		5.3 Relative entropy dissipation
	6 From backwards dynamics “back” to forward dynamics
		6.1 Entropic interpretation of the expected cost when Q(Rn) < ∞
	7 The case of finite invariant measure, and an iterative procedure
		Appendix 1: The decrease of the relative entropy without convexity assumption
		Appendix 2: The triviality of the tail σ-algebra H(∞)
	References
Incomplete Stochastic Equilibria with Exponential Utilities Close to Pareto Optimality
	Introduction
		The equilibrium problem
		The “fast-and-slow” model
		The representative-agent approach, and its failure in incomplete markets
		Our probabilistic-analytic approach
		Some notational conventions
	1 The Equilibrium Problem and its BSDE Reformulation
		1.1 The financial market, its agents, and equilibria
		1.2 A simple risk-aware reparametrization
		1.3 A solution of the single-agent utility-maximization problem
		1.4 A BSDE characterization of equilibria
	2 Main Results
		2.1 Equilibria close to Pareto optimality
	3 Proofs
		3.1 Proof of Proposition 1
		3.2 Proof of Lemma 1
		3.3 Proof of Theorem 1
		3.4 Proof of Theorem 2
		3.5 Proof of Corollary 1
		3.6 Proof of Corollary 2
		3.7 An a-priori bmo-estimate
	References
Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability I
	1 Introduction
	2 The general coupling
		2.1 Assumptions and definitions
		2.2 Construction of the coupling
	3 Reversible diffusions
		3.1 Assumptions on the potential function
		3.2 Spectral properties of the generator
		3.3 The coupled process
	Appendix
	References
Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability II
	1 Introduction
	2 Ordering the local minima
	3 Structure of the second eigenfunction
		3.1 Tools and preliminary results
		3.2 Location of the nodal point
		3.3 Behavior near the minima
		3.4 Behavior near the nodal point
	4 Asymptotic behavior of the coupled process
	Appendix 1
	Appendix 2
	References
Maximally Distributed Random Fields under Sublinear Expectation
	1 Introduction
	2 Preliminaries
	3 Maximally distributed random fields
	4 Maximally distributed white noise
	5 Spatial and temporal maximally distributed white noise and related stochastic integral
		5.1 Stochastic integral with respect to the spatial maximally distributed white noise
		5.2 Maximally distributed random fields of temporal-spatial types and related stochastic integral
	References
Pairs Trading under Geometric Brownian Motion Models
	1 Introduction
	2 Pairs Trading under a GBM
	3 Pairs Trading with Cutting Losses
	4 A Pairs Selling Rule with Regime Switching
	5 Conclusions
	References
Equilibrium Model of Limit Order Books: A Mean-Field Game View
	1 Introduction
	2 Preliminaries
		2.1 Mean-field SDEs with reflecting boundary conditions
		2.2 An Itô’s formula
	3 A Bertrand game among the sellers (static case)
		3.1 The Bertrand game and its Nash equilibrium
		3.2 A linear mean-field case
	4 Mean-field type liquidity dynamics in continuous time
		4.1 A general description
		4.2 Problem formulation
	5 Dynamic programming principle
	6 HJB equation and its viscosity solutions
	References
Bounded Regret for Finitely Parameterized Multi-Armed Bandits
	1 Introduction
	2 Problem Formulation
	3 UCB Algorithm for Finitely Parameterized Multi-Armed Bandits
	4 Analysis of the FP-UCB Algorithm
	5 Simulations
	6 Conclusion and Future Work
	References
	Appendix
		6.1 Proof of Lemma 1
		6.2 Proof of Lemma 2
Developing the Path Signature Methodology and Its Application to Landmark-Based Human Action Recognition
	1 Introduction
	2 Related work
		2.1 Path signature feature (PSF)
		2.2 Landmark-based human action recognition
	3 Path Signature
		3.1 Definition and geometric interpretation
		3.2 Calculation of the signature for a discrete path
		3.3 Properties of the path signature
			3.3.1 Uniqueness
			3.3.2 Invariance under translation
			3.3.3 Invariance under time reparameterization
			3.3.4 Nonlinearity of the signature
			3.3.5 Fixed dimension under length variations
	4 Path disintegrations and transformations
		4.1 Path disintegrations
			4.1.1 Pose Disintegration
			4.1.2 Temporal Disintegration
		4.2 Path transformations
			4.2.1 Time-incorporated transformation
			4.2.2 Invisibility-reset transformation
			4.2.3 Multi-delayed lead-lag transformation
	5 Feature extraction for human action recognition
		5.1 Spatial structural features
		5.2 Temporal dynamical features
	6 Experimental results and analysis
		6.1 Datasets
		6.2 Network configurations
		6.3 Data preprocessing and benchmark
		6.4 Investigation of the spatial features
		6.5 Investigation of the temporal features
			6.5.1 Investigation of T-J-PSF
			6.5.2 Investigation of T-S-PSF
		6.6 Ablation study
		6.7 Comparison with the state-of-the-art methods
			6.7.1 Comparison over small datasets
			6.7.2 Comparison over large-scale datasets
		6.8 Toward understanding of human actions
	7 Conclusions
	References
Index