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Register a new userA principal-agent problem with exit contract
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We study a principal-agent problem with one principal and multiple agents. The principal provides an exit contract which is identical to all agents, then each agent chooses her/his optimal exit time with the given contract. The principal looks for an optimal contract in order to maximize her/his reward value which depends on the agents choices. Under a technical monotone condition, and by using Bank-El Karouis representation of stochastic process, we are able to decouple the two optimization problems, and to reformulate the principals problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is obtained. We then show that the continuous time problem can be approximated by a sequence of discrete time ones, which would induce a natural numerical approximation method. We finally discuss the principal-agent problem if one restricts to the class of all Markovian and/or continuous contracts.
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This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $alpha$-stable processes, which help verify the solvability of the original HJB equation.
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In order to approximate the exit time of a one-dimensional diffusion process, we propose an algorithm based on a random walk. Such an algorithm so-called Walk on Moving Spheres was already introduced in the Brownian context. The aim is therefore to generalize this numerical approach to the Ornstein-Uhlenbeck process and to describe the efficiency of the method.
Crowdsourcing markets have emerged as a popular platform for matching available workers with tasks to complete. The payment for a particular task is typically set by the tasks requester, and may be adjusted based on the quality of the completed work, for example, through the use of bonus payments. In this paper, we study the requesters problem of dynamically adjusting quality-contingent payments for tasks. We consider a multi-round version of the well-known principal-agent model, whereby in each round a worker makes a strategic choice of the effort level which is not directly observable by the requester. In particular, our formulation significantly generalizes the budget-free online task pricing problems studied in prior work. We treat this problem as a multi-armed bandit problem, with each arm representing a potential contract. To cope with the large (and in fact, infinite) number of arms, we propose a new algorithm, AgnosticZooming, which discretizes the contract space into a finite number of regions, effectively treating each region as a single arm. This discretization is adaptively refined, so that more promising regions of the contract space are eventually discretized more finely. We analyze this algorithm, showing that it achieves regret sublinear in the time horizon and substantially improves over non-adaptive discretization (which is the only competing approach in the literature). Our results advance the state of art on several different topics: the theory of crowdsourcing markets, principal-agent problems, multi-armed bandits, and dynamic pricing.
Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the pull strength (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the nonlinear system at hand with that of a well-chosen process for which tight estimates of the containment probability are known or can be numerically obtained (e.g. an Ornstein-Uhlenbeck process). Specifically, the main technical contribution of this paper is to define a suitable dominance relationship between the pull strengths of two systems and to prove that this dominance relationship implies an order relationship between their containment probabilities. We also discuss the link with contraction theory and highlight some examples of applications.
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