| 书目名称 | Stochastic Analysis, Filtering, and Stochastic Optimization | | 副标题 | A Commemorative Volu | | 编辑 | George Yin,Thaleia Zariphopoulou | | 视频video | | | 概述 | Consists of contributions from leading experts in these fields, former colleagues, graduate students, graduate advisor, and research associates of Professor Davis.Compiles a body of research to honor | | 图书封面 |  | | 描述 | .This volume is a collection of research works to honor the late Professor Mark H.A. Davis, whose pioneering work in the areas of Stochastic Processes, Filtering, and Stochastic Optimization spans more than five decades. Invited authors include his dissertation advisor, past collaborators, colleagues, mentees, and graduate students of Professor Davis, as well as scholars who have worked in the above areas. Their contributions may expand upon topics in piecewise deterministic processes, pathwise stochastic calculus, martingale methods in stochastic optimization, filtering, mean-field games, time-inconsistency, as well as impulse, singular, risk-sensitive and robust stochastic control. . | | 出版日期 | Book 2022 | | 关键词 | Piecewise deterministic processes; Pathwise stochastic calculus; Martingale methods; Impulse and singul | | 版次 | 1 | | doi | https://doi.org/10.1007/978-3-030-98519-6 | | isbn_softcover | 978-3-030-98521-9 | | isbn_ebook | 978-3-030-98519-6 | | copyright | The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl |
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Front Matter |
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Abstract
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,Control in Hilbert Space and First-Order Mean Field Type Problem, |
Alain Bensoussan,Hang Cheung,Sheung Chi Phillip Yam |
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Abstract
We extend the work [9] by two of the coauthors, which dealt with a deterministic control problem for which the Hilbert space could be generic and investigated a novel form of the ‘lifting’ technique proposed by P. L. Lions. In [9], we only showed the local existence and uniqueness of solutions to the FBODEs in the Hilbert space which were associated to the control problems with drift function consisting of the control only. In this article, we establish the global existence and uniqueness of the solutions to the FBODEs in Hilbert space corresponding to control problems with separable drift function which is nonlinear in state and linear in control.We shall also prove the sufficiency of the Pontryagin Maximum Principle and derive the corresponding Bellman equation. Finally, by using the ‘lifting’ idea as in [6, 7], we shall apply the result to solve the linear quadratic mean field type control problems, and to show the global existence of the corresponding Bellman equations.
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,Risk-Sensitive Markov Decision Problems under Model Uncertainty: Finite Time Horizon Case, |
Tomasz R. Bielecki,Tao Chen,Igor Cialenco |
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Abstract
In this paper we study a class of risk-sensitive Markovian control problems in discrete time subject to model uncertainty. We consider a risk-sensitive discounted cost criterion with finite time horizon. The used methodology is the one of adaptive robust control combined with machine learning.
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| 4 |
,Optimal Control of Piecewise Deterministic Markov Processes, |
O. L. V. Costa,F. Dufour |
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Abstract
This chapter studies the infinite-horizon continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure . of the process. Two optimality criteria are considered, the discounted cost case and the long run average cost case. We provide conditions for the existence of a solution to an integro-differential optimality equality, the so called Hamilton-Jacobi-Bellman HJB) equation, for the discounted cost case, and a solution to an HJB inequality for the long run average cost case, aswell as conditions for the existence of a deterministic stationary optimal policy. From the results for the discounted cost case and under some continuity and compactness hypothesis on the parameters and non-explosive assumptions for the process, we derive the conditions for the long run average cost case by employing the so-called vanishing discount approach.
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| 5 |
,Pathwise Approximations for the Solution of the Non-Linear Filtering Problem, |
Dan Crisan,Alexander Lobbe,Salvador Ortiz-Latorre |
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Abstract
We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark [2] and Davis [10, 11] and prove their robustness property. In particular, we show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. Moreover, the pathwise viewpointwill be a stepping stone into the rigorous development ofmachine learning methods for the filtering problem. This work is a cotinuation of [5] where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the work in [5] by showing that robust approximations can be derived from the discretisations therein.
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,Discrete-Time Portfolio Optimization under Maximum Drawdown Constraint with Partial Information and |
Carmine de Franco,Johann Nicolle,Huyên Pham |
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Abstract
We study a discrete-time portfolio selection problem with partial information and maximum drawdown constraint. Drift uncertainty in the multidimensional framework is modeled by a prior probability distribution. In this Bayesian framework, we derive the dynamic programming equation using an appropriate change of measure, and obtain semi-explicit results in the Gaussian case. The latter case, with a CRRA utility function is completely solved numerically using recent deep learning techniques for stochastic optimal control problems. We emphasize the informative value of the learning strategy versus the non-learning one by providing empirical performance and sensitivity analysis with respect to the uncertainty of the drift. Furthermore, we show numerical evidence of the close relationship between the non-learning strategy and a no short-sale constrained Merton problem, by illustrating the convergence of the former towards the latter as the maximum drawdown constraint vanishes.
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,Estimating the Matthew Effects: Switching Pareto Dynamics, |
Robert J. Elliott |
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Abstract
Pareto distributions can describe the clustering of observations and give rise to sayings such as ‘The rich gets richer and the poor gets poorer’. They are sometimes generated by counting processes whose rate depends on external factors. In turn, these factors are modelled by a finite state Markov chain .. New filters are derived which estimate . together with other parameters of the model.
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,Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality, |
Hans Föllmer |
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Abstract
For a probability measure . on Wiener space, Talagrand’s transport inequality takes the form.ϰ (.,.)2 ≤2.(.|.), where theWasserstein distance.ϰ is defined in terms of the Cameron-Martin norm, and where .(.|.) denotes the relative entropy with respect to Wiener measure .. Talagrand’s original proof takes a bottom-up approach, using finite-dimensional approximations. As shown by Feyel and Üstünel in [3] and Lehec in [10], the inequality can also be proved directly on Wiener space, using a suitable coupling of . and .. We show how this top-down approach can be extended beyond the absolutely continuous case .<<.. Here the Wasserstein distance is defined in terms of quadratic variation, and .(.|.) is replaced by the specific relative entropy .(.|.) on Wiener space that was introduced by N. Gantert in [7].
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,Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation, |
Xue Dong He,Xun Yu Zhou |
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Abstract
Time inconsistency is prevalent in dynamic choice problems: a plan of actions to be taken in the future that is optimal for an agent today may not be optimal for the same agent in the future. If the agent is aware of this intra-personal conflict but unable to commit herself in the future to following the optimal plan today, the rational strategy for her today is to reconcile with her future selves, namely to correctly anticipate her actions in the future and then act today accordingly. Such a strategy is named intra-personal equilibrium and has been studied since as early as in the 1950s. A rigorous treatment in continuous-time settings, however, had not been available until a decade ago. Since then, the study on intra-personal equilibrium for time-inconsistent problems in continuous time has grown rapidly. In this chapter, we review the classical results and some recent development in this literature.
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,N-Player and Mean-Field Games in Itˆo-Diffusion Markets with Competitive or Homophilous Interaction |
Ruimeng Hu,Thaleia Zariphopoulou |
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Abstract
In Itô-diffusion environments, we introduce and analyze .-player and common-noise mean-field games in the context of optimal portfolio choice in a common market. The players invest in a finite horizon and also interact, driven either by competition or homophily. We study an incomplete market model in which the players have constant individual risk tolerance coefficients (CARA utilities). We also consider the general case of random individual risk tolerances and analyze the related games in a complete market setting. This randomness makes the problem substantially more complex as it leads to (. or a continuum of) auxiliary “individual” Itô-diffusion markets. For all cases, we derive explicit or closed-form solutions for the equilibrium stochastic processes, the optimal state processes, and the values of the games.
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,A Variational Characterization of Langevin-Smoluchowski Diffusions, |
Ioannis Karatzas,Bertram Tschiderer |
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Abstract
We show that Langevin–Smoluchowski measure on path space is invariant under time-reversal, followed by stochastic control of the drift with a novel entropic-type criterion. Repeated application of these forward-backward steps leads to a sequence of stochastic control problems, whose initial/terminal distributions converge to the Gibbs probability measure of the diffusion, and whose values decrease to zero along the relative entropy of the Langevin–Smoluchowski flow with respect to this Gibbs measure.
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,Incomplete Stochastic Equilibria with Exponential Utilities Close to Pareto Optimality, |
Constantinos Kardaras,Hao Xing,Gordan Žitković |
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Abstract
We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete markets model. An assumption of “smallness” type— imposed through the new notion of “closeness to Pareto optimality”—is shown to be sufficient for existence and uniqueness. Central role in our analysis is played by a fully-coupled nonlinear system of quadratic BSDEs.
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,Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability I, |
Thomas G. Kurtz,Jason Swanson |
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Abstract
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like aMarkov chain on the local minima.By thework ofBovier and collaborators, the local minimacan be associated with the small eigenvalues of the diffusion generator. Applying a Markov mapping theorem, we use the eigenfunctions of the generator to couple this diffusion to a Markov chain whose generator has eigenvalues equal to the eigenvalues of the diffusion generator that are associated with the local minima and establish explicit formulas for conditional probabilities associatedwith this coupling. The fundamental question then becomes to relate the coupled Markov chain to the approximateMarkov chain suggested by the results of Freidlin and Wentzel.
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,Finite Markov Chains Coupled to General Markov Processes and An Application to Metastability II, |
Thomas G. Kurtz,Jason Swanson |
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Abstract
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like aMarkov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. In Part I of this work [10], by applying a Markov mapping theorem, we used the eigenfunctions of the generator to couple this diffusion to a Markov chain whose generator has eigenvalues equal to the eigenvalues of the diffusion generator that are associated with the local minima and established explicit formulas for conditional probabilities associated with this coupling. The fundamental question now becomes to relate the coupled Markov chain to the approximateMarkov chain suggested by the results of Freidlin andWentzel. In this paper, we take up this question and provide a complete analysis of this relationship in the special cas
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,Maximally Distributed Random Fields under Sublinear Expectation, |
Xinpeng Li,Shige Peng |
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Abstract
This paper focuses on the maximal distribution on sublinear expectation space and introduces a new type of random fields with the maximally distributed finite-dimensional distribution. The corresponding spatial maximally distributed white noise is constructed, which includes the temporal-spatial situation as a special case due to the symmetrical independence property of maximal distribution. In addition, the stochastic integrals with respect to the spatial or temporal-spatial maximally distributed white noises are established in a quite direct way without the usual assumption of adaptability for integrand.
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,Pairs Trading under Geometric Brownian Motion Models, |
Phong Luu,Jingzhi Tie,Qing Zhang |
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Abstract
This survey paper is concerned with pairs trading strategies under geometric Brownian motion models. Pairs trading is about trading simultaneously a pair of securities, typically stocks. The idea is to monitor the spread of their price movements over time. A pairs trade is triggered by their price divergence (e.g., one stock moves up a significant amount relative to the other) and consists of a short position in the strong stock and a long position in the weak one. Such a strategy bets on the reversal of their price strengths and the eventual convergence of the price spread. Pairs trading is popular among trading institutions because its risk neutral nature. In practice, the trader needs to decide when to initiate a pairs position (how much divergence is enough) and when to close the position (how to take profits or cut losses). It is the main goals of this paper to address these issues and theoretical findings along with related practical considerations.
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,Equilibrium Model of Limit Order Books: A Mean-Field Game View, |
Jin Ma,Eunjung Noh |
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Abstract
In this paper, we propose a continuous time equilibrium model of the (sellside) limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with state-dependent intensity. To motivate the model we first study an .-seller static mean-field type Bertrand game among the liquidity providers. We shall then formulate the continuous time model as the limiting mean-field dynamics of the representative seller, and argue that the frontier of the LOB (e.g., the best ask price) is the value function of a mean-field stochastic control problem by the representative seller. Using a dynamic programming approach, we show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which can be used to determine the equilibrium density function of the LOB, in the spirit of [32].
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,Bounded Regret for Finitely Parameterized Multi-Armed Bandits, |
Kishan Panaganti,Dileep Kalathil,Pravin Varaiya |
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Abstract
We consider multi-armed bandits where the model of the underlying stochastic environment is characterized by a common unknown parameter. The true parameter is unknown to the learning agent. However, the set of possible parameters, which is finite, is known a priori. We propose an algorithm that is simple and easy to implement, which we call Finitely Parameterized Upper Confidence Bound (FP-UCB) algorithm, which uses the information about the underlying parameter set for faster learning. In particular, we show that the FP-UCB algorithm achieves a bounded regret under a structural condition on the underlying parameter set.We also show that, if the underlying parameter set does not satisfy this structural condition, the FP-UCB algorithm achieves a logarithmic regret, but with a smaller preceding constant compared to the standard UCB algorithm. We also validate the superior performance of the FP-UCB algorithm through extensive numerical simulations.
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,Developing the Path Signature Methodology and Its Application to Landmark- Based Human Action Recog |
Weixin Yang,Terry Lyons,Hao Ni,Cordelia Schmid,Lianwen Jin |
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Abstract
Landmark-based human action recognition in videos is a challenging task in computer vision. One key step is to design a generic approach that generates discriminative features for the spatial structure and temporal dynamics. To this end, we regard the evolving landmark data as a high-dimensional path and apply path signature techniques to provide an expressive, robust, non-linear, and interpretable representation for the sequential events. We do not extract signature features from the raw path, rather we propose path disintegrations and path transformations as preprocessing steps. Path disintegrations turn a high-dimensional path linearly into a collection of lower-dimensional paths; some of these paths are in pose space while others are defined over a multi-scale collection of temporal intervals. Path transformations decorate the paths with additional coordinates in standard ways to allow the truncated signatures of transformed paths to expose additional features. For spatial representation, we apply the non-linear signature transform to vectorize the paths that arise out of pose disintegration, and for temporal representation, we apply it again to describe this evolving vectoriza
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| 20 |
,Correction to: Stochastic Analysis, Filtering, and Stochastic Optimization, |
George Yin,Thaleia Zariphopoulou |
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Abstract
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书目名称Stochastic Analysis, Filtering, and Stochastic Optimization影响因子(影响力)
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发表于 2025-3-21 16:51:54
