arXiv subjects

Erhan Bayraktar

Publications and source records attributed to Erhan Bayraktar.

At least 127 records · Page 7Linked to original sources

arXiv2021

Prediction against a limited adversary

We study the problem of prediction with expert advice with adversarial corruption where the adversary can at most corrupt one expert. Using tools from viscosity theory, we characterize the long-time behavior of the value function of the game between the forecaster and the adversary. We provide lower and upper bounds for the growth rate of regret without relying on a comparison result. We show that depending on the description of regret, the limiting behavior of the game can significantly differ.

cs.LG↗

arXiv2021

Finite state Mean Field Games with Wright-Fisher common noise

We force uniqueness in finite state mean field games by adding a Wright-Fisher common noise. We achieve this by analyzing the master equation of this game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game; see Cardaliaguet et al. (2019). We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo (2013), has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. Among others, this requires a priori estimates of Hölder type for the corresponding Kimura operator when the drift therein is merely continuous.

math.PR↗

arXiv2020

Equilibrium concepts for time-inconsistent stopping problems in continuous time

A \emph{new} notion of equilibrium, which we call \emph{strong equilibrium}, is introduced for time-inconsistent stopping problems in continuous time. Compared to the existing notions introduced in ArXiv: 1502.03998 and ArXiv: 1709.05181, which in this paper are called \emph{mild equilibrium} and \emph{weak equilibrium} respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous-time Markov chain and the discount function is log sub-additive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.

q-fin.MF↗

arXiv2020

Extended Weak Convergence and Utility Maximization with Proportional Transaction Costs

In this paper we study utility maximization with proportional transaction costs. Assuming extended weak convergence of the underlying processes we prove the convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended weak convergence theory developed in [1] and the Meyer--Zheng topology introduced in [24].

q-fin.MF↗

arXiv2020

Asymptotics for Small Nonlinear Price Impact: a PDE Approach to the Multidimensional Case

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.

math.AP↗

arXiv2020

A Macroeconomic SIR Model for COVID-19

The current COVID-19 pandemic and subsequent lockdowns have highlighted the close and delicate relationship between a country's public health and economic health. Macroeconomic models that use preexisting epidemic models to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We develop an SIR model of the COVID-19 pandemic that explicitly considers herd immunity, behavior-dependent transmission rates, remote workers, and indirect externalities of lockdown. This model is presented as an exit time control problem where lockdown ends when the population achieves herd immunity, either naturally or via a vaccine. A social planner prescribes separate levels of lockdown for two separate sections of the adult population: low-risk (ages 20-64) and high-risk (ages 65 and over). These levels are determined via optimization of an objective function which assigns a macroeconomic cost to the level of lockdown and the number of deaths. We find that, by ending lockdowns once herd immunity is reached, high-risk individuals are able to leave lockdown significantly before the arrival of a vaccine without causing large increases in mortality. Moreover, if we incorporate a behavior-dependent transmission rate which represents increased personal caution in response to increased infection levels, both output loss and total mortality are lowered. Lockdown efficacy is further increased when there is less interaction between low- and high-risk individuals, and increased remote work decreases output losses. Overall, our model predicts that a lockdown which ends at the arrival of herd immunity, combined with individual actions to slow virus transmission, can reduce total mortality to one-third of the no-lockdown level, while allowing high-risk individuals to leave lockdown well before vaccine arrival.

physics.soc-ph↗

arXiv2020

Terminal Ranking Games

We analyze a mean field tournament: a mean field game in which the agents receive rewards according to the ranking of the terminal value of their projects and are subject to cost of effort. Using Schrödinger bridges we are able to explicitly calculate the equilibrium. This allows us to identify the reward functions which would yield a desired equilibrium and solve several related mechanism design problems. We are also able to identify the effect of reward inequality on the players' welfare as well as calculate the price of anarchy.

math.OC↗

arXiv2020

Continuity of Utility Maximization under Weak Convergence

In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the convergence in distribution of the underlying processes. We also establish a weak convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall risk) in the Heston model by building an appropriate lattice approximation.

q-fin.MF↗

arXiv2020

On the adversarial robustness of robust estimators

Motivated by recent data analytics applications, we study the adversarial robustness of robust estimators. Instead of assuming that only a fraction of the data points are outliers as considered in the classic robust estimation setup, in this paper, we consider an adversarial setup in which an attacker can observe the whole dataset and can modify all data samples in an adversarial manner so as to maximize the estimation error caused by his attack. We characterize the attacker's optimal attack strategy, and further introduce adversarial influence function (AIF) to quantify an estimator's sensitivity to such adversarial attacks. We provide an approach to characterize AIF for any given robust estimator, and then design optimal estimator that minimizes AIF, which implies it is least sensitive to adversarial attacks and hence is most robust against adversarial attacks. From this characterization, we identify a tradeoff between AIF (i.e., robustness against adversarial attack) and influence function, a quantity used in classic robust estimators to measure robustness against outliers, and design estimators that strike a desirable tradeoff between these two quantities.

math.ST↗

arXiv2020

On the asymptotic optimality of the comb strategy for prediction with expert advice

For the problem of prediction with expert advice in the adversarial setting with geometric stopping, we compute the exact leading order expansion for the long time behavior of the value function. Then, we use this expansion to prove that as conjectured in Gravin et al. [12], the comb strategies are indeed asymptotically optimal for the adversary in the case of 4 experts.

math.PR↗

arXiv2020

Malicious Experts versus the multiplicative weights algorithm in online prediction

We consider a prediction problem with two experts and a forecaster. We assume that one of the experts is honest and makes correct prediction with probability $μ$ at each round. The other one is malicious, who knows true outcomes at each round and makes predictions in order to maximize the loss of the forecaster. Assuming the forecaster adopts the classical multiplicative weights algorithm, we find upper and lower bounds for the value function of the malicious expert. Our results imply that the multiplicative weights algorithm cannot resist the corruption of malicious experts. We also show that an adaptive multiplicative weights algorithm is asymptotically optimal for the forecaster, and hence more resistant to the corruption of malicious experts.

math.OC↗

arXiv2020

Transport plans with domain constraints

This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain its Kantorovich duality. A corollary of this result is a monotonicity principle which gives a geometric way of identifying the optimizer.

math.PR↗

arXiv2020

On non-uniqueness in mean field games

We analyze an $N+1$-player game and the corresponding mean field game with state space $\{0,1\}$. The transition rate of $j$-th player is the sum of his control $α^j$ plus a minimum jumping rate $η$. Instead of working under monotonicity conditions, here we consider an anti-monotone running cost. We show that the mean field game equation may have multiple solutions if $η< \frac{1}{2}$. We also prove that that although multiple solutions exist, only the one coming from the entropy solution is charged (when $η=0$), and therefore resolve a conjecture of ArXiv: 1903.05788.

math.PR↗

arXiv2019

Finite-Time 4-Expert Prediction Problem

We explicitly solve the nonlinear PDE that is the continuous limit of dynamic programming of \emph{expert prediction problem} in finite horizon setting with $N=4$ experts. The \emph{expert prediction problem} is formulated as a zero sum game between a player and an adversary. By showing that the solution is $\mathcal{C}^2$, we are able to show that the strategies conjectured in arXiv:1409.3040G form an asymptotic Nash equilibrium. We also prove the "Finite vs Geometric regret" conjecture proposed in arXiv:1409.3040G for $N=4$, and and show that this conjecture in fact follows from the conjecture that the comb strategies are optimal.

math.PR↗

arXiv2019

On the quasi-sure superhedging duality with frictions

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchard, B., Deng, S. and Tan, X., Super-replication with proportional transaction cost under model uncertainty, Math. Fin., 29(3):837-860, 2019] and to show the duality under the more natural condition of No Strict Arbitrage. In addition, we extend the results to models with portfolio constraints.

q-fin.MF↗

arXiv2019

Embedding of Walsh Brownian Motion

Let $(Z,κ)$ be a Walsh Brownian motion with spinning measure $κ$. Suppose $μ$ is a probability measure on $\mathbb{R}^n$. We characterize all the $κ$ such that $μ$ is a stopping distribution of $(Z,κ)$. If we further restrict the solution to be integrable, we show that there would be only one choice of $κ$. We also generalize Vallois' embedding, and prove that it minimizes the expectation $\mathbb{E}[Ψ(L^Z_τ)]$ among all the admissible solutions $τ$, where $Ψ$ is a strictly convex function and $(L_t^Z)_{t \geq 0}$ is the local time of the Walsh Brownian motion at the origin.

math.PR↗

arXiv2019

Time Consistent Stopping For The Mean-Standard Deviation Problem --- The Discrete Time Case

Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among stopping times or randomized stopping times may not exist. This motivates us to consider the notion of liquidation strategies, which lets the stopping right to be divisible. We then argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. It turns out that an equilibrium liquidation strategy always exists. We then analyze whether optimal equilibrium liquidation strategies exist and whether they are unique and observe that neither may hold.

q-fin.MF↗

arXiv2019

On the Adversarial Robustness of Multivariate Robust Estimation

In this paper, we investigate the adversarial robustness of multivariate $M$-Estimators. In the considered model, after observing the whole dataset, an adversary can modify all data points with the goal of maximizing inference errors. We use adversarial influence function (AIF) to measure the asymptotic rate at which the adversary can change the inference result. We first characterize the adversary's optimal modification strategy and its corresponding AIF. From the defender's perspective, we would like to design an estimator that has a small AIF. For the case of joint location and scale estimation problem, we characterize the optimal $M$-estimator that has the smallest AIF. We further identify a tradeoff between robustness against adversarial modifications and robustness against outliers, and derive the optimal $M$-estimator that achieves the best tradeoff.

stat.ML↗