arXiv subjects

Georgy Gaitsgori

Publications and source records attributed to Georgy Gaitsgori.

6 recordsLinked to original sources

arXiv2026

The Game of Marginal Utilities

We study a noncooperative resource-allocation game in which $m$ players distribute fixed resources among $n$ projects and the payoff of player $j$ is given by \[ F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}, \] where $a_i$ and $b_i$ are project parameters, while $x_i^j$ is the amount of resources player $j$ allocates to project $i$. This specification combines diminishing returns with congestion generated by competitors. We prove that the game has a unique Nash equilibrium and characterize it by an equimarginal principle. We show that, after the projects are ordered by $a_i/b_i$, each player invests in an initial segment of projects, and these segments are nested across players, so the equilibrium decomposes into consecutive activity zones; players with larger resources invest weakly more in every project. In the fully active regime, where every player invests in every project, we reduce the equilibrium to a single scalar nonlinear equation for the aggregate marginal-utility rate; all individual marginal rates, investments, and payoffs then follow from explicit formulas. We also provide a projected marginal-utility algorithm with global linear convergence under an explicit step-size condition, together with a structure-exploiting \textit{Block Pandora} algorithm that reconstructs and certifies the equilibrium conditional on a proposed nested cutoff structure.

cs.GT↗

arXiv2026

Optimal Stopping for a Diffusion with Unobserved Bernoulli Drift

We solve fairly explicitly an optimal stopping problem for a Wiener process with unobserved Bernoulli drift, in the presence of a cost on terminal position which is symmetric and increases with distance from the origin, and of a fixed positive cost per unit time $c > 0$. After filtering, the problem reduces to Markovian optimal stopping with complete observations for the state process ``centered'' by its starting position $x \in \mathbb R$. However, the solution becomes possible only after foliating by an additional state-parameter $y \in \mathbb{R}$, representing the displacement from the initial position; this foliation ``lifts'' the problem from the real line to the plane, solves the augmented problem for each fixed initial position $x$, characterizes fairly explicitly the optimal stopping region in $(x,y)$-space, and finally obtains the solution of the original problem by ``slicing'' along $y=0$. Following this procedure, we show that, under suitable structural assumptions on the terminal cost, each fixed-$x$ continuation section is either empty or a single bounded interval, whose endpoints are determined uniquely by a balancing condition; the corresponding value function is then given in semi-explicit form. The two-dimensional continuation region is obtained by gluing these fixed-$x$ intervals over $x$; its two free boundaries satisfy natural monotonicity properties and, at regular points, can be described by a coupled system of ordinary differential equations. The resulting description yields a threshold-type solution of the original one-dimensional problem whenever the horizontal slice $y=0$ enters the two-dimensional continuation region.

math.PR↗

arXiv2025

A Sequential Testing Problem with Signal Control

We study a controlled version of the Bayesian sequential testing problem for the drift of a Wiener process, in which the observer exercises discretion over the signal intensity. This control incurs a running cost that reflects the resource demands of information acquisition. The objective is to minimize the total expected cost, combining both the expenditure on control and the loss from misclassifying the unknown drift. By allowing for a general class of loss functions and any measurable cost of control, our analysis captures a broad range of sequential inference problems. We show that when a function, determined by the cost structure, admits a global minimizer, the optimal control is constant and explicitly computable, thereby reducing our setting to a solvable optimal stopping problem. If no such minimizer exists, an optimal control does not exist either, yet the value function remains explicit. Our results thus demonstrate that full tractability can be retained even when extending sequential inference to include endogenous control over the information flow.

math.OC↗

arXiv2025

Grab It Before It's Gone: Testing Uncertain Rewards under a Stochastic Deadline

We study a sequential estimation problem for an unknown reward in the presence of a random horizon. The reward takes one of two predetermined values that can be inferred from the drift of a Wiener process, which serves as a signal. The objective is to use the information in the signal to estimate the reward which is made available until a stochastic deadline that \textit{depends} on its value. The observer must therefore work quickly to determine if the reward is favorable and claim it before the deadline passes. Under general assumptions on the stochastic deadline, we provide a full characterization of the solution that includes an identification with the unique solution to a free-boundary problem. Our analysis derives regularity properties of the solution that imply its ``smooth fit'' with the boundary data, and shows that the free-boundary solves a particular integral equation. The continuity of the free-boundary is also established under additional structural assumptions that lead to its representation in terms of a continuous transformation of a monotone function. We provide illustrations for several examples of interest.

math.PR↗

arXiv2024

Drift Control with Discretionary Stopping for a Diffusion

We consider stochastic control with discretionary stopping for the drift of a diffusion process over an infinite time horizon. The objective is to choose a control process and a stopping time to minimize the expectation of a convex terminal cost in the presence of a fixed operating cost and a control-dependent running cost per unit of elapsed time. Under appropriate conditions on the coefficients of the controlled diffusion, an optimal pair of control and stopping rules is shown to exist. Moreover, under the same assumptions, it is shown that the optimal control is a constant which can be computed fairly explicitly; and that it is optimal to stop the first time an appropriate interval is visited. We consider also a constrained version of the above problem, in which an upper bound on the expectation of available stopping times is imposed; we show that this constrained problem can be reduced to an unconstrained problem with some appropriate change of parameters and, as a result, solved by similar arguments.

math.OC↗

arXiv2023

A Dynkin Game with Independent Processes and Incomplete Information

We analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete information. We assume that each player observes his own Brownian motion, which is not only independent of the other player's Brownian motion but also not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. Under appropriate growth conditions on the reward function, we show that there are infinitely many Nash equilibria in which both players attain infinite expected payoffs. In contrast, the only equilibrium with finite expected payoffs mandates immediate stopping by at least one of the players. Our results hold in the settings of both pure and mixed strategies.

math.PR↗