Erhan Bayraktar

Prof. Erhan Bayraktar

Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA
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Academic Career Length

24Years (2003-Present)

(Calculated once Scilit publication data is synced: current year − year of first publication + 1)

Representative Publications

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14 June 2012 in Mathematical Finance
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EBErhan Bayraktar
Erhan Bayraktar
MLMichael Ludkovski
Michael Ludkovski
15 November 2017 in Transactions of the American Mathematical Society
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EBErhan Bayraktar
Erhan Bayraktar
ACAndrea Cosso
Andrea Cosso
Huyên PHAMHuyên PHAM
Huyên PHAM
We analyze a stochastic optimal control problem, where the state process follows a McKean-Vlasov dynamics and the diffusion coefficient can be degenerate. We prove that its value function V V admits a nonlinear Feynman-Kac representation in terms of a class of forward-backward stochastic differential equations, with an autonomous forward process. We exploit this probabilistic representation to rigorously prove the dynamic programming principle (DPP) for V V . The Feynman-Kac representation we obtain has an important role beyond its intermediary role in obtaining our main result: in fact it would be useful in developing probabilistic numerical schemes for V V . The DPP is important in obtaining a characterization of the value function as a solution of a nonlinear partial differential equation (the so-called Hamilton-Jacobi-Belman equation), in this case on the Wasserstein space of measures. We should note that the usual way of solving these equations is through the Pontryagin maximum principle, which requires some convexity assumptions. There were attempts in using the dynamic programming approach before, but these works assumed a priori that the controls were of Markovian feedback type, which helps write the problem only in terms of the distribution of the state process (and the control problem becomes a deterministic problem). In this paper, we will consider open-loop controls and derive the dynamic programming principle in this most general case. In order to obtain the Feynman-Kac representation and the randomized dynamic programming principle, we implement the so-called randomization method, which consists of formulating a new McKean-Vlasov control problem, expressed in weak form taking the supremum over a family of equivalent probability measures. One of the main results of the paper is the proof that this latter control problem has the same value function V V of the original control problem.
10 July 2013 in ASTIN Bulletin
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EBErhan Bayraktar
Erhan Bayraktar
AKAndreas E. Kyprianou
Andreas E. Kyprianou
KYKazutoshi Yamazaki
Kazutoshi Yamazaki
We revisit the dividend payment problem in the dual model of Avanzi et al. ([2–4]). Using the fluctuation theory of spectrally positive Lévy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Lévy processes. Moreover, we characterize the optimal barrier using the functional inverse of a scale function. We also consider the capital injection problem of [4] and show that its value function has a very similar form to the one in which the horizon is the time of ruin.
1 October 2012 in Proceedings of the American Mathematical Society
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EBErhan Bayraktar
Erhan Bayraktar
MSMihai Sîrbu
Mihai Sîrbu
We introduce a stochastic version of the classical Perron's method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub- and super-) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a martingale along the solutions of the stochastic differential equation. The unique viscosity solution is actually equal to the expected payoff. This amounts to a verification result (Itô's Lemma) for non-smooth viscosity solutions of the linear parabolic equation.
1 October 2023 in The Annals of Applied Probability
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EBErhan Bayraktar
Erhan Bayraktar
SCSuman Chakraborty
Suman Chakraborty
RWRuoyu Wu
Ruoyu Wu
We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system.
1 January 2018 in SIAM Journal on Control and Optimization
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EBErhan Bayraktar
Erhan Bayraktar
ACAsaf Cohen
Asaf Cohen
We consider an $n$-player symmetric stochastic game with weak interactions between the players. Time is continuous, and the horizon and the number of states are finite. We show that the value function of each of the players can be approximated by the solution of a partial differential equation called the master equation. Moreover, we analyze the fluctuations of the empirical measure of the states of the players in the game and show that it is governed by a solution to a stochastic differential equation. Finally, we prove the regularity of the master equation, which is required for the above results.