arXiv subjects

Thomas Mariotti

Publications and source records attributed to Thomas Mariotti.

5 recordsLinked to original sources

arXiv2026

Gradient extremals, talwegs, valleys, and directional alignment for generic gradient descent

Gradient extremals are loci along which the gradient is an eigenvector of the Hessian. These objects provide a natural geometric framework connecting several notions, notably valleys and talwegs, which we analyze from a variational viewpoint in the generic case. We then show that trajectories of the gradient flow and of its discrete counterpart exhibit directional alignment with the tangent spaces to gradient extremals, and generically to the talweg. Under non-resonance assumptions, and in contrast with the quadratic case, alignment rates are governed either by the first spectral gap or by the smallest eigenvalue of the Hessian at the limit point. Nonlinearities and the step length may both distort these rates in a complex manner. We further prove a volume concentration phenomenon emphasizing the structuring role of gradient extremals: for large times, the images of sets of initial conditions concentrate inside valleys and asymptotically around talwegs.

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arXiv2025

A class of singular control problems with tipping points

Tipping points characterize situations where a regulated system may experience a sudden and irreversible change and are generally associated with a random state of the system below which the change materializes. In this paper, we study a singular stochastic control problem in which the performance criterion depends on the hitting time of a random state that is not a stopping time for the reference filtration. We establish a connection between the value of this problem and that of a singular control problem involving a diffusion and its running minimum. We provide a verification lemma that we apply to explicitly solve a resource-extraction problem with an ex-ante unknown tipping point.

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arXiv2024

Mixed Markov-Perfect Equilibria in the Continuous-Time War of Attrition

We prove the existence of a Markov-perfect equilibrium in randomized stopping times for a model of the war of attrition in which the underlying state variable follows a homogenous linear diffusion. The proof uses the fact that the space of Markovian randomized stopping times can be topologized as a compact absolute retract, which in turn enables us to use a powerful fixed-point theorem by Eilenberg and Montgomery. We illustrate our results with an example of a war of attrition that admits a mixed-strategy Markov-perfect equilibrium but no pure-strategy Markov-perfect equilibrium.

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arXiv2022

Mixed-Strategy Equilibria in the War of Attrition under Uncertainty

We study a generic family of two-player continuous-time nonzero-sum stopping games modeling a war of attrition with symmetric information and stochastic payoffs that depend on an homogeneous linear diffusion. We first show that any Markovian mixed strategy for player $i$ can be represented by a pair $(μ^i,S^i)$, where $μ^i$ is a measure over the state space representing player $i$'s stopping intensity, and $S^i$ is a subset of the state space over which player $i$ stops with probability $1$. We then prove that, if players are asymmetric, then, in all mixed-strategy Markov-perfect equilibria, the measures $μ^i$ have to be essentially discrete, and we characterize any such equilibrium through a variational system satisfied by the players' equilibrium value functions. This result contrasts with the literature, which focuses on pure-strategy equilibria, or, in the case of symmetric players, on mixed-strategy equilibria with absolutely continuous stopping intensities. We illustrate this result by revisiting the model of exit in a duopoly under uncertainty, and exhibit a mixed-strategy equilibrium in which attrition takes place on the equilibrium path though firms have different liquidation values.

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arXiv2021

Investment Timing and Technological Breakthroughs

We study the optimal investment policy of a firm facing both technological and cash-flow uncertainty. At any point in time, the firm can decide to invest in a standalone technology or to wait for a technological breakthrough. Breakthroughs occur when market conditions become favorable enough, exceeding a certain threshold value that is ex-ante unknown to the firm. A microfoundation for this assumption is that a breakthrough occurs when the share of the surplus from the new technology accruing to its developer is high enough to cover her privately observed cost. We show that the relevant Markov state variables for the firm's optimal investment policy are the current market conditions and their current historic maximum, and that the firm optimally invests in the stand-alone technology only when market conditions deteriorate enough after reaching a maximum. Empirically, investments in new technologies requiring the active cooperation of developers should thus take place in booms, whereas investments in state-of-the-art technologies should take place in busts. Moreover, the required return for investing in the stand-alone technology is always higher than if this were the only available technology and can take arbitrarily large values following certain histories. Finally, a decrease in development costs, or an increase in the value of the new technology, makes the firm more prone to bear downside risk and to delay investment in the stand-alone technology.

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