Blockchain-based token platform economy is a new branch of digital platform economics. Constructing a continuous time dynamic model of token platform economy, this paper analyzes what kind of ESG policy is appropriate...
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Blockchain-based token platform economy is a new branch of digital platform economics. Constructing a continuous time dynamic model of token platform economy, this paper analyzes what kind of ESG policy is appropriate for the government, meanwhile the token platform participants (developers, users and speculators) make optimal investments and decisions under ESG policy. Simulation result shows neutral ESG policy is optimal. Based on the given neutral ESG policy, we have done the research on ESG investment and decision strategies for platform participants. Our research shows that the tokens selling rate and efforts of green platform (ESG score greater than 0) developers are lower than the ones of brown platform (ESG score less than 0). Consequently, when developers' token retention is about half of the initial amount, users should invest more brown tokens. Speculators should invest brown tokens for developers' high token retention. Green token investments of speculators and users are needed in other cases. Next, the impact of the government's three ESG policies on the maturity or termination of the platform also been analyzed. An important conclusion occurred: the government's aggressive or conservative ESG policy cannot make the development of the green platform better;Therefore, we suggest a neutral ESG policy which means that the government could adopt high tax incentive and high tax burden on the green and brown platform while it is not necessary to implement the extra subsidy and punishment policy on the green and brown platform.
We introduce a new class of strongly degenerate nonlinear parabolic PDEs ((p - 2)Delta(N)(infinity,X) + Delta(X))u(X, Y, t) + (m +p)(X . del(Yu)(X, Y, t) - partial derivative(t)u(X, Y, t)) = 0, (X, Y, t) is an element...
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We introduce a new class of strongly degenerate nonlinear parabolic PDEs ((p - 2)Delta(N)(infinity,X) + Delta(X))u(X, Y, t) + (m +p)(X . del(Yu)(X, Y, t) - partial derivative(t)u(X, Y, t)) = 0, (X, Y, t) is an element of R-m x R-m x R, p is an element of (1, infinity), combining the classical PDE of Kolmogorov and the normalized p-Laplace operator. We characterize solutions in terms of an asymptotic mean value property and the results are connected to the analysis of certain tug-of-war games with noise. The value functions for the games in-troduced approximate solutions to the stated PDE when the parameter that controls the size of the possible steps goes to zero. Existence and uniqueness of viscosity solutions to the Dirichlet problem is established. The asymptotic mean value property, the associated games and the geometry underlying the Dirichlet prob-lem, all reflect the family of dilation and the Lie group underlying operators of Kolmogorov type, and this makes our setting different from the context of standard parabolic dilations and Euclidean translations applicable in the context of the heat operator and the normalized parabolic infinity Laplace operator. (C) 2022 The Author(s). Published by Elsevier Inc.
We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, na...
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We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one time-step robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the S&P500$S\&P\nobreakspace 500$. We present two different types of ambiguity sets;one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.
By introducing a new type of minimality condition, this paper gives a novel approach to the reflected backward stochastic differential equations (RBSDEs) with cadlag obstacles. Our first step is to prove the dynamic p...
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By introducing a new type of minimality condition, this paper gives a novel approach to the reflected backward stochastic differential equations (RBSDEs) with cadlag obstacles. Our first step is to prove the dynamic programming principles for nonlinear optimal stopping problems with g-expectations. We then use the nonlinear DoobMeyer decomposition theorem for g-supermartingales to get the existence of the solution. With a new type of minimality condition, we prove a representation formula of solutions to RBSDEs, in an efficient way. Finally, we derive some a priori estimates and stability results.
In this survey work, we introduce Stochastic Differential Delay Equations and their impacts on Stochastic Optimal Control problems. We observe time delay in the dynamics of a state process that may correspond to inert...
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In this survey work, we introduce Stochastic Differential Delay Equations and their impacts on Stochastic Optimal Control problems. We observe time delay in the dynamics of a state process that may correspond to inertia or memory in a financial system. For such systems, we demonstrate two special approaches to handle delayed control problems by applying the dynamic programming principle. Moreover, we clarify the technical challenges rising as a consequence of the conflict between the path-dependent, infinite-dimensional nature of the problem and the necessity of the Markov property. Furthermore, we present two different Deep Learning algorithms to solve targeted delayed control tasks and illustrate the results for a complete memory portfolio optimization problem.
In this paper we introduce a new approach to discrete-time semi-Markov decision processes based on the sojourn time process. Different characterizations of discrete-time semi-Markov processes are exploited and decisio...
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In this paper we introduce a new approach to discrete-time semi-Markov decision processes based on the sojourn time process. Different characterizations of discrete-time semi-Markov processes are exploited and decision processes are constructed by their means. With this new approach, the agent is allowed to consider different actions depending also on the sojourn time of the process in the current state. A numerical method based on Q-learning algorithms for finite horizon reinforcement learning and stochastic recursive relations is investigated. Finally, we consider two toy examples: one in which the reward depends on the sojourn-time, according to the gambler's fallacy;the other in which the environment is semi-Markov even if the reward function does not depend on the sojourn time. These are used to carry on some numerical evaluations on the previously presented Q-learning algorithm and on a different naive method based on deep reinforcement learning.
We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Diri...
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We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.
We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman's dynamic programming principle. Th...
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We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman's dynamic programming principle. This also corresponds to solving first-order Hamilton-Jacobi-Bellman (HJB) equations. Our work builds upon the research conducted by Hur & eacute;et al. (SIAM J Numer Anal 59(1):525-557, 2021), which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions.
Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individu...
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Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value 0). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls);and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game.
In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller-Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman...
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In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller-Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.
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