arXiv subjects

Erhan Bayraktar

Publications and source records attributed to Erhan Bayraktar.

At least 109 records · Page 6Linked to original sources

arXiv2022

Data-Driven Nonparametric Robust Control under Dependence Uncertainty

We consider a multi-period stochastic control problem where the multivariate driving stochastic factor of the system has known marginal distributions but uncertain dependence structure. To solve the problem, we propose to implement the nonparametric adaptive robust control framework. We aim to find the optimal control against the worst-case copulae in a sequence of shrinking uncertainty sets which are generated from continuously observing the data. Then, we use a stochastic gradient descent ascent algorithm to numerically handle the corresponding high dimensional dynamic inf-sup optimization problem. We present the numerical results in the context of utility maximization and show that the controller benefits from knowing more information about the uncertain model.

math.OC↗

arXiv2022

A PDE approach for regret bounds under partial monitoring

In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior of the regret of the forecaster. Using a verification type argument, we show that the problem of obtaining regret bounds and efficient algorithms can be tackled by finding appropriate smooth sub/supersolutions of this parabolic PDE.

math.PR↗

arXiv2022

Supermartingale shadow couplings: the decreasing case

For two measures $μ$ and $ν$ that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab., 46(6):3351--3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling $π^D$ by establishing a Brenier-type result: (a generalised version of) $π^D$ concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale \textit{shadow} measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincaré Probab. Statist., 52(4):1823--1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer. Math. Soc., 374:4973--5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures $μ,ν$, introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the `martingale points' of each such coupling.

math.PR↗

arXiv2022

Stability of Equilibria in Time-inconsistent Stopping Problems

We investigate the stability of equilibrium-induced optimal values with respect to (w.r.t.) reward functions $f$ and transition kernels $Q$ for time-inconsistent stopping problems under nonexponential discounting in discrete time. First, with locally uniform convergence of $f$ and $Q$ equipped with total variation distance, we show that the optimal value is semi-continuous w.r.t. $(f,Q)$. We provide examples showing that continuity may fail in general, and the convergence for $Q$ in total variation cannot be replaced by weak convergence. Next we show that with the uniform convergence of $f$ and $Q$, the optimal value is continuous w.r.t. $(f,Q)$ when we consider a relaxed limit over $\varepsilon$-equilibria. We also provide an example showing that for such continuity the uniform convergence of $(f,Q)$ cannot be replaced by locally uniform convergence.

math.OC↗

arXiv2022

K-core in percolated dense graph sequences

We determine the size of $k$-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights $\{a^n_{i,j}\}_{i,j \in [n]}$ that converges to a kernel $W:[0,1]^2\to [0,+\infty)$ in the cut metric. Keeping an edge $(i,j)$ of $G_n$ with probability $\min \{ {a^n_{i,j}}/{n},1 \}$ independently, we obtain a sequence of random graphs $G_n(\frac{1}{n})$. Denote by $\mathcal{A}$ the property of a branching process that the initial particle has at least $k$ children, each of which has at least $k-1$ children, each of which has at least $k-1$ children, and so on. Using branching process and the theory of dense graph limits, under mild assumptions we obtain the size of $k$-core of random graphs $G_n(\frac{1}{n})$, \begin{align*} \text{size of $k$-core of } G_n\left(\frac{1}{n}\right) =n \mathbb{P}_{X^W}\left(\mathcal{A}\right) +o_p(n). \end{align*} Our result can also be used to obtain the threshold of appearance of a $k$-core of order $n$.

math.PR↗

arXiv2022

Concentration of measure for Graphon particle system

We study heterogeneously interacting diffusive particle systems with mean-field type interaction characterized by an underlying graphon and their finite particle approximations. Under suitable conditions, we obtain exponential concentration estimates over a finite time horizon for both 1 and 2 Wasserstein distances between the empirical measures of the finite particle systems and the averaged law of the graphon system, extending the work of Bayraktar-Wu.

math.PR↗

arXiv2022

Nonparametric Adaptive Robust Control Under Model Uncertainty

We consider a discrete time stochastic Markovian control problem under model uncertainty. Such uncertainty not only comes from the fact that the true probability law of the underlying stochastic process is unknown, but the parametric family of probability distributions which the true law belongs to is also unknown. We propose a nonparametric adaptive robust control methodology to deal with such problem. Our approach hinges on the following building concepts: first, using the adaptive robust paradigm to incorporate online learning and uncertainty reduction into the robust control problem; second, learning the unknown probability law through the empirical distribution, and representing uncertainty reduction in terms of a sequence of Wasserstein balls around the empirical distribution; third, using Lagrangian duality to convert the optimization over Wasserstein balls to a scalar optimization problem, and adopting a machine learning technique to achieve efficient computation of the optimal control. We illustrate our methodology by considering a utility maximization problem. Numerical comparisons show that the nonparametric adaptive robust control approach is preferable to the traditional robust frameworks.

math.OC↗

arXiv2022

A potential-based construction of the increasing supermartingale coupling

The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351--3398, 2018) is an extreme point of the set of `supermartingale' couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (The potential of the shadow measure, Electron. Commun. Probab., 27, paper no. 16, 1--12, 2022) obtained in the martingale setting.

math.PR↗

arXiv2022

Path-dependent Hamilton-Jacobi equations with super-quadratic growth in the gradient and the vanishing viscosity method

The non-exponential Schilder-type theorem in Backhoff-Veraguas, Lacker and Tangpi [Ann. Appl. Probab., 30 (2020), pp. 1321-1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton-Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton-Jacobi-Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semi-continuous solutions holds and state constraints are admitted.

math.PR↗

arXiv2022

$K_{r,s}$ graph bootstrap percolation

A graph $G$ percolates in the $K_{r,s}$-bootstrap process if we can add all missing edges of $G$ in some order such that each edge creates a new copy of $K_{r,s}$, where $K_{r,s}$ is the complete bipartite graph. We study $K_{r,s}$-bootstrap percolation on the Erdős-Rényi random graph, and determine the percolation threshold for balanced $K_{r,s}$ up to a logarithmic factor. This partially answers a question raised by Balogh, Bollobás, and Morris. We also establish a general lower bound of the percolation threshold for all $K_{r,s}$, with $r\geq s \geq 3$.

math.PR↗

arXiv2022

Propagation of Chaos of Forward-Backward Stochastic Differential Equations with Graphon Interactions

In this paper, we study graphon mean field games using a system of forward-backward stochastic differential equations. We establish the existence and uniqueness of solutions under two different assumptions and prove the stability with respect to the interacting graphons which are necessary to show propagation of chaos results. As an application of propagation of chaos, we prove the convergence of n-player game Nash equilibrium for a general model, which is new in the theory of graphon mean field games.

math.PR↗

arXiv2021

Optimal Investment and Consumption under a Habit-Formation Constraint

We formulate an infinite-horizon optimal investment and consumption problem, in which an individual forms a habit based on the exponentially weighted average of her past consumption rate, and in which she invests in a Black-Scholes market. The individual is constrained to consume at a rate higher than a certain proportion $α$ of her consumption habit. Our habit-formation model allows for both addictive ($α=1$) and nonaddictive ($0<α<1$) habits. The optimal investment and consumption policies are derived explicitly in terms of the solution of a system of differential equations with free boundaries, which is analyzed in detail. If the wealth-to-habit ratio is below (resp. above) a critical level $x^*$, the individual consumes at (resp. above) the minimum rate and invests more (resp. less) aggressively in the risky asset. Numerical results show that the addictive habit formation requires significantly more wealth to support the same consumption rate compared to a moderately nonaddictive habit. Furthermore, an individual with a more addictive habit invests less in the risky asset compared to an individual with a less addictive habit but with the same wealth-to-habit ratio and risk aversion, which provides an explanation for the equity-premium puzzle.

q-fin.MF↗

arXiv2021

Finite state mean field games with Wright Fisher common noise as limits of $N$-player weighted games

Forcing finite state mean field games by a relevant form of common noise is a subtle issue, which has been addressed only recently. Among others, one possible way is to subject the simplex valued dynamics of an equilibrium by a so-called Wright-Fisher noise, very much in the spirit of stochastic models in population genetics. A key feature is that such a random forcing preserves the structure of the simplex, which is nothing but, in this setting, the probability space over the state space of the game. The purpose of this article is hence to elucidate the finite player version and, accordingly, to prove that $N$-player equilibria indeed converge towards the solution of such a kind of Wright-Fisher mean field game. Whilst part of the analysis is made easier by the fact that the corresponding master equation has already been proved to be uniquely solvable under the presence of the common noise, it becomes however more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which hence may differ from $1/N$ and which, most of all, evolves with the common noise.

math.PR↗

arXiv2021

A Note on Utility Maximization with Proportional Transaction Costs and Stability of Optimal Portfolios

The aim of this short note is to establish a limit theorem for the optimal trading strategies in the setup of the utility maximization problem with proportional transaction costs. This limit theorem resolves the open question from [4]. The main idea of our proof is to establish a uniqueness result for the optimal strategy. The proof of the uniqueness is heavily based on the dual approach which was developed recently in [6,7,8].

q-fin.MF↗

arXiv2021

Mean field interaction on random graphs with dynamically changing multi-color edges

We consider weakly interacting jump processes on time-varying random graphs with dynamically changing multi-color edges. The system consists of a large number of nodes in which the node dynamics depends on the joint empirical distribution of all the other nodes and the edges connected to it, while the edge dynamics depends only on the corresponding nodes it connects. Asymptotic results, including law of large numbers, propagation of chaos, and central limit theorems, are established. In contrast to the classic McKean-Vlasov limit, the limiting system exhibits a path-dependent feature in that the evolution of a given particle depends on its own conditional distribution given its past trajectory. We also analyze the asymptotic behavior of the system when the edge dynamics is accelerated. A law of large number and a propagation of chaos result is established, and the limiting system is given as independent McKean-Vlasov processes. Error between the two limiting systems, with and without acceleration in edge dynamics, is also analyzed.

math.PR↗

arXiv2021

Disorder Detection with Costly Observations

We study the Wiener disorder detection problem where each observation is associated with a positive cost. In this setting, a strategy is a pair consisting of a sequence of observation times and a stopping time corresponding to the declaration of disorder. We characterize the minimal cost of the disorder problem with costly observations as the unique fix-point of a certain jump operator, and we determine the optimal strategy.

math.PR↗

arXiv2021

On the Continuity of the Root Barrier

We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.

math.PR↗

arXiv2021

Stationarity and uniform in time convergence for the graphon particle system

We consider the long time behavior of heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, including a certain convexity condition, we show the exponential ergodicity for both systems, establish the uniform-in-time law of large numbers for marginal distributions as the number of particles increases, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided.

math.PR↗