Marcel Nutz
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Linear Convergence of Gradient Descent for Quadratically Regularized Optimal Transport Open
In optimal transport, quadratic regularization is an alternative to entropic regularization when sparse couplings or small regularization parameters are desired. Here quadratic regularization means that transport couplings are penalized by…
Optimal Fees for Liquidity Provision in Automated Market Makers Open
Passive liquidity providers (LPs) in automated market makers (AMMs) face losses due to adverse selection (LVR), which static trading fees often fail to offset in practice. We study the key determinants of LP profitability in a dynamic redu…
Stability of Mean-Field Variational Inference Open
Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. This paper studies the stability properties of the mean-field approximation when the target d…
Sparse Regularized Optimal Transport without Curse of Dimensionality Open
Entropic optimal transport -- the optimal transport problem regularized by KL diver\-gence -- is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of di…
Randomization in Optimal Execution Games Open
We study optimal execution in markets with transient price impact in a competitive setting with $N$ traders. Motivated by prior negative results on the existence of pure Nash equilibria, we consider randomized strategies for the traders an…
Optimal Execution among $N$ Traders with Transient Price Impact Open
We study $N$-player optimal execution games in an Obizhaeva--Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists and we derive its closed form. Wherea…
Sparsity of Quadratically Regularized Optimal Transport: Scalar Case Open
The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling $π_{\varepsilon}$ has sparse support for small regularization parameter $\varepsilon$, in contrast to entropic regu…
Monotonicity in Quadratically Regularized Linear Programs Open
In optimal transport, quadratic regularization is a sparse alternative to entropic regularization: the solution measure tends to have small support. Computational experience suggests that the support decreases monotonically to the unregula…
Quantitative Convergence of Quadratically Regularized Linear Programs Open
Linear programs with quadratic regularization are attracting renewed interest due to their applications in optimal transport: unlike entropic regularization, the squared-norm penalty gives rise to sparse approximations of optimal transport…
Quadratically Regularized Optimal Transport: Existence and Multiplicity of Potentials Open
The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a solu…
On the Martingale Schrödinger Bridge between Two Distributions Open
We study a martingale Schrödinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schrödinger potentials for this coupling. Namely, under certain c…
View article: Estimating the Rate-Distortion Function by Wasserstein Gradient Descent
Estimating the Rate-Distortion Function by Wasserstein Gradient Descent Open
In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishe…
Unwinding Stochastic Order Flow: When to Warehouse Trades Open
We study how to unwind stochastic order flow with minimal transaction costs. Stochastic order flow arises, e.g., in the central risk book (CRB), a centralized trading desk that aggregates order flows within a financial institution. The des…
On the Guyon-Lekeufack Volatility Model Open
Guyon and Lekeufack recently proposed a path-dependent volatility model and documented its excellent performance in fitting market data and capturing stylized facts. The instantaneous volatility is modeled as a linear combination of two pr…
Martingale Schrodinger Bridges and Optimal Semistatic Portfolios Open
In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schrödinger bridge Q*; that is, the minimal-entropy martingale measure among all…
On the Convergence Rate of Sinkhorn's Algorithm Open
We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $π_{t}$ is shown to satisfy $H(π_{t}|π_{*})+H(π_{*}|π_{t})=O(t^{-1})$ where $H$ denotes relative entropy and $π_{*}$ the optimal …
Martingale Transports and Monge Maps Open
It is well known that martingale transport plans between marginals $μ\neqν$ are never given by Monge maps -- with the understanding that the map is over the first marginal $μ$, or forward in time. Here, we change the perspective, with surp…
Convergence Rates for Regularized Optimal Transport via Quantization Open
We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or $L^{p}$ regularization, general transport costs and multi-marg…
Limits of Semistatic Trading Strategies Open
We show that pointwise limits of semistatic trading strategies in discrete time are again semistatic strategies. The analysis is carried out in full generality for a two-period model, and under a probabilistic condition for multi-period, m…
Stability of Schrödinger Potentials and Convergence of Sinkhorn's Algorithm Open
We study the stability of entropically regularized optimal transport with respect to the marginals. Given marginals converging weakly, we establish a strong convergence for the Schrödinger potentials describing the density of the optimal c…
Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm Open
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general cond…
Quantitative Stability of Regularized Optimal Transport Open
We study the stability of entropically regularized optimal transport with
respect to the marginals. Lipschitz continuity of the value and H\older
continuity of the optimal coupling in $p$-Wasserstein distance are obtained
under general con…
Fine properties of the optimal Skorokhod embedding problem Open
We study the problem of stopping a Brownian motion at a given distribution \nu while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set \m…
Stability of Entropic Optimal Transport and Schrödinger Bridges Open
We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the u…
Mean Field Contest with Singularity Open
We formulate a mean field game where each player stops a privately observed Brownian motion with absorption. Players are ranked according to their level of stopping and rewarded as a function of their relative rank. There is a unique mean …
Entropic Optimal Transport: Geometry and Large Deviations Open
We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the …