23 research outputs found

    Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition

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    This paper is devoted to present a method of proving verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term. The value function is assumed to be continuous in time and once differentiable in the space variable (C0,1C^{0,1}) instead of once differentiable in time and twice in space (C1,2C^{1,2}), like in the classical results. The results are obtained using a time dependent Fukushima - Dirichlet decomposition proved in a companion paper by the same authors using stochastic calculus via regularization. Applications, examples and comparison with other similar results are also given.Comment: 34 pages. To appear: Stochastic Processes and Their Application

    Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

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    The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process \X, taking values in a Hilbert space HH, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an It\^o type formula applied to f(t, \X(t)) where f:[0,T]×H→Rf:[0,T] \times H \rightarrow \R is a C0,1C^{0,1} function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of "mild solution" for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation

    Path-dependent equations and viscosity solutions in infinite dimension

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    Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.Comment: To appear in the Annals of Probabilit

    A regularization approach to functional It\^o calculus and strong-viscosity solutions to path-dependent PDEs

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    First, we revisit functional It\^o/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fourni\'e, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by C. Di Girolami and the second named author are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of classical solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness. The notion of strong-viscosity solution motivates the last part of the paper which is devoted to explore this new concept of solution for general semilinear PDEs in the finite dimensional case. We prove an equivalence result between the classical viscosity solution and the new one. The definition of strong-viscosity solution for semilinear PDEs is inspired by the notion of "good" solution, and it is based again on an approximating procedure

    Dynamic pricing with demand learning under competition

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 199-204).In this thesis, we focus on oligopolistic markets for a single perishable product, where firms compete by setting prices (Bertrand competition) or by allocating quantities (Cournot competition) dynamically over a finite selling horizon. The price-demand relationship is modeled as a parametric function, whose parameters are unknown, but learned through a data driven approach. The market can be either in disequilibrium or in equilibrium. In disequilibrium, we consider simultaneously two forms of learning for the firm: (i) learning of its optimal pricing (resp. allocation) strategy, given its belief regarding its competitors' strategy; (ii) learning the parameters in the price-demand relationship. In equilibrium, each firm seeks to learn the parameters in the price-demand relationship for itself and its competitors, given that prices (resp. quantities) are in equilibrium. In this thesis, we first study the dynamic pricing (resp. allocation) problem when the parameters in the price-demand relationship are known. We then address the dynamic pricing (resp. allocation) problem with learning of the parameters in the price-demand relationship. We show that the problem can be formulated as a bilevel program in disequilibrium and as a Mathematical Program with Equilibrium Constraints (MPECs) in equilibrium. Using results from variational inequalities, bilevel programming and MPECs, we prove that learning the optimal strategies as well as the parameters, is achieved. Furthermore, we design a solution method for efficiently solving the problem. We prove convergence of this method analytically and discuss various insights through a computational study.(cont.) Finally, we consider closed-loop strategies in a duopoly market when demand is stochastic. Unlike open-loop policies (such policies are computed once and for all at the beginning of the time horizon), closed loop policies are computed at each time period, so that the firm can take advantage of having observed the past random disturbances in the market. In a closed-loop setting, subgame perfect equilibrium is the relevant notion of equilibrium. We investigate the existence and uniqueness of a subgame perfect equilibrium strategy, as well as approximations of the problem in order to be able to compute such policies more efficiently.by Carine Simon.Ph.D

    Delayed Forward-Backward stochastic PDE's driven by non Gaussian LĂ©vy noise with application in finance

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    From the very first results, the mathematical theory of financial markets has undergone several changes, mostly due to financial crises who forced the mathematical-economical community to change the basic assumptions on which the whole theory is founded. Consequently a new mathematical foundation were needed. In particular, the 2007/2008 credit crunch showed the word that a new financial theoretical framework was necessary, since several empirical evidences emerged that aspects that were neglected prior to these years were in fact fundamental if one has to deal with financial markets. The goal of the present thesis goes in this direction; we aim at developing rigorous mathematical instruments that allow to treat fundamental problems in modern financial mathematics. In order to do so, the talk is thus divided into three main parts, which focus on three different topics of modern financial mathematics. The first part is concerned with delay equations. In particular, we will prove Feynman-Kac type result for BSDE's with time-delayed generator, as well as an ad hoc Ito formula for delay equations with jumps. The second part deal with infinite dimensional analysis and network models, focusing in particular on existence and uniqueness results for infinite dimensional SPDE's on networks with general non-local boundary conditions. The last part treats the topic of rigorous asymptotic expansions, providing a small noise asymptotic expansion for SDE with Lévy noise with several concrete application to financial models

    Essays on modeling and analysis of dynamic sociotechnical systems

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    A sociotechnical system is a collection of humans and algorithms that interact under the partial supervision of a decentralized controller. These systems often display in- tricate dynamics and can be characterized by their unique emergent behavior. In this work, we describe, analyze, and model aspects of three distinct classes of sociotech- nical systems: financial markets, social media platforms, and elections. Though our work is diverse in subject matter content, it is unified though the study of evolution- and adaptation-driven change in social systems and the development of methods used to infer this change. We first analyze evolutionary financial market microstructure dynamics in the context of an agent-based model (ABM). The ABM’s matching engine implements a frequent batch auction, a recently-developed type of price-discovery mechanism. We subject simple agents to evolutionary pressure using a variety of selection mech- anisms, demonstrating that quantile-based selection mechanisms are associated with lower market-wide volatility. We then evolve deep neural networks in the ABM and demonstrate that elite individuals are profitable in backtesting on real foreign ex- change data, even though their fitness had never been evaluated on any real financial data during evolution. We then turn to the extraction of multi-timescale functional signals from large panels of timeseries generated by sociotechnical systems. We introduce the discrete shocklet transform (DST) and associated similarity search algorithm, the shocklet transform and ranking (STAR) algorithm, to accomplish this task. We empirically demonstrate the STAR algorithm’s invariance to quantitative functional parameteri- zation and provide use case examples. The STAR algorithm compares favorably with Twitter’s anomaly detection algorithm on a feature extraction task. We close by using STAR to automatically construct a narrative timeline of societally-significant events using a panel of Twitter word usage timeseries. Finally, we model strategic interactions between the foreign intelligence service (Red team) of a country that is attempting to interfere with an election occurring in another country, and the domestic intelligence service of the country in which the election is taking place (Blue team). We derive subgame-perfect Nash equilibrium strategies for both Red and Blue and demonstrate the emergence of arms race inter- ference dynamics when either player has “all-or-nothing” attitudes about the result of the interference episode. We then confront our model with data from the 2016 U.S. presidential election contest, in which Russian military intelligence interfered. We demonstrate that our model captures the qualitative dynamics of this interference for most of the time under stud

    Topics in stochastic calculus in infinite dimension for financial applications

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    This thesis is devoted to study delay/path-dependent stochastic differential equations and their connection with partial differential equations in infinite dimensional spaces, possibly path-dependent. We address mathematical problems arising in hedging a derivative product for which the volatility of the underlying assets as well as the claim may depend on the past history of the assets themselves. The starting point is to provide a robust framework for working with mild solutions to path-dependent SDEs: well-posedness, continuity with respect to the data, regularity with respect to the initial condition. This is done in Chapter 1. In Chapter 2, under Lipschitz conditions on the data, we prove the directional regularity needed in order to write the hedging strategy. In Chapter 3 we introduce a new notion of viscosity solution to semilinear path-dependent PDEs in Hilbert spaces (PPDEs), we prove well-posedness and show that the solution is given by the Fyenman-Kac formula. In Chapter 4 we extend to Hilbert spaces the functional It\uafo calculus and, under smooth assumptions on the data, we prove a path-dependent It\uafo\u2019s formula, show existence of classical solutions to PPDEs, and obtain a Clark-Ocone type formula. In Chapter 5 we introduce a new notion of C0-semigroup suitable to be applied to Markov transition semigroups, hence to mild solutions to Kolmogorov PDEs, and we prove all the basic results analogous to those available for C0-semigroups in Banach spaces. Additional theoretical results for stochastic analysis in Hilbert spaces, regarding stochastic convolutions, are given in Appendix A. Our methodology varies among different chapters. Path-dependent models can be studied in their original path-dependent form or by representing them as non-pathdependent models in infinite dimension. We exploit both approaches. We treat pathdependent Kolmogorov equations in infinite dimension with two notions of solution: classical and viscosity solutions. Each approach leads to original results in each chapter

    Generalized averaged Gaussian quadrature and applications

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    A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

    MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

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    Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
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