/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Exploring and identifying structure is even more important for multivariate data than univariate data, given the difficulties in graphically presenting multivariate data and the comparative lack of parametric models to represent it. Unfortunately, such exploration is also inherently more difficult.

Multivariate Density Estimation

Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

Recent years have witnessed significant advances in reinforcement learning (RL), which has registered tremendous success in solving various sequential decision-making problems in machine learning. Most of the successful RL applications, e.g., the games of Go and Poker, robotics, and autonomous driving, involve the participation of more than one single agent, which naturally fall into the realm of multi-agent RL (MARL), a domain with a relatively long history, and has recently re-emerged due to advances in single-agent RL techniques. Though empirically successful, theoretical foundations for MARL are relatively lacking in the literature. In this chapter, we provide a selective overview of MARL, with focus on algorithms backed by theoretical analysis. More specifically, we review the theoretical results of MARL algorithms mainly within two representative frameworks, Markov/stochastic games and extensive-form games, in accordance with the types of tasks they address, i.e., fully cooperative, fully competitive, and a mix of the two. We also introduce several significant but challenging applications of these algorithms. Orthogonal to the existing reviews on MARL, we highlight several new angles and taxonomies of MARL theory, including learning in extensive-form games, decentralized MARL with networked agents, MARL in the mean-field regime, (non-)convergence of policy-based methods for learning in games, etc. Some of the new angles extrapolate from our own research endeavors and interests. Our overall goal with this chapter is, beyond providing an assessment of the current state of the field on the mark, to identify fruitful future research directions on theoretical studies of MARL. We expect this chapter to serve as continuing stimulus for researchers interested in working on this exciting while challenging topic.

Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms

https://hal.archives-ouvertes.fr/hal-00015486/document

Discrete time approximation of decoupled Forward-Backward SDE with jumps

We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e., finding the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike in the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics and are shown to recover the explicit solution of Follmer and Leukert [Finance Stoch., 3 (1999), pp. 251-273].

/pdf/stochastic-target-problems-with-controlled-loss-w8fqljlb19.pdf

Stochastic Target Problems with Controlled Loss

We consider the infinite-horizon optimal consumption-investment problem under a drawdown constraint, i.e., when the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the with constant coefficients. For a general class of utility functions, we provide the value function in explicit form and derive closed-form expressions for the optimal consumption and investment strategy.

/pdf/optimal-lifetime-consumption-and-investment-under-a-drawdown-4b0buj779x.pdf

Optimal lifetime consumption and investment under a drawdown constraint

https://basepub.dauphine.fr/bitstream/handle/123456789/9750/be12_BSDEquaddelay.pdf;sequence=2

A simple constructive approach to quadratic BSDEs with or without delay

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^{\nu}$ is constrained to satisfy an almost sure constraint $Z^{\nu}(T)\in G\subset\mathbb{R}^{d+1}$ $\mathbb{P}$-a.s. at some final time $T>0$. When the set is of the form $G:=\{(x,y)\in\mathbb{R}^d\times\mathbb{R}:g(x,y)\geq0\}$, with $g$ nondecreasing in $y$, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a state constraint problem, where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^{\nu}(t))\in[0,T]\times\mathbb{R}^{d+1}:Z^{\nu}(T)\in G$ a.s. for some $\nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$, which itself is a viscosity solution of a nonlinear parabolic PDE. Applying ideas recently developed in Bouchard, Elie, and Touzi [SIAM J. Control Optim., 48 (2009), pp. 3123-3150], our general result also allows us to consider optimal control problems with moment constraints of the form $\mathbb{E}[g(Z^{\nu}(T))]\geq0$ or $\mathbb{P}[g(Z^{\nu}(T))\geq0]\geq p$.

/pdf/optimal-control-under-stochastic-target-constraints-4uvpgorzaw.pdf

Romuald Elie

Papers

Discrete time approximation of decoupled Forward-Backward SDE with jumps

Stochastic Target Problems with Controlled Loss

Optimal lifetime consumption and investment under a drawdown constraint

A simple constructive approach to quadratic BSDEs with or without delay

Optimal Control under Stochastic Target Constraints