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Thibaut Mastrolia

Assistant Professor, University of California Berkeley

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Since July 2021, I have been an Assistant Professor in the Department of Industrial Engineering and Operations Research (IEOR) at the University of California, Berkeley.

e-mail: mastrolia(AT)berkeley(DOT)edu
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My research develops stochastic control and game-theoretic tools and contract theory to design resilient financial and cyber systems under risk and uncertainty.

It lies at the intersection of stochastic control, financial engineering, and game theory, with applications to market design and regulation, energy systems, and cyber risk. I focus in particular on backward stochastic differential equations, high-frequency trading and market design withmodel and volatility uncertainty, principal–agent problems, mean field games, and dynamic cyber-risk management.

I have been awarded by the France-Berkeley Fund 2023.

Key words: Cyber risk and stochastic control [Cyber]; FinTech, incentives and market microstructure [Fintech]; Contract theory and games [Contract]; Stochastic Analysis and BSDE [BSDE]

Recent works/ArXiv preprints 

  • [Fintech] Julius Graf and Thibaut Mastrolia.
    Learning Market Making with Closing Auctions.
    In this work, we investigate the market-making problem on a trading session in which a continuous phase on a limit order book is followed by a closing auction. Whereas standard optimal market-making models typically rely on terminal inventory penalties to manage end-of-day risk, ignoring the significant liquidity events available in closing auctions, we propose a Deep Q-Learning framework that explicitly incorporates this mechanism. We introduce a market-making framework designed to explicitly anticipate the closing auction, continuously refining the projected clearing price as the trading session evolves. We develop a generative stochastic market model to simulate the trading session and to emulate the market. Our theoretical model and Deep Q-Learning method is applied on the generator in two settings: (1) when the mid price follows a rough Heston model with generative data from this stochastic model; and (2) when the mid price corresponds to historical data of assets from the S&P 500 index and the performance of our algorithm is compared with classical benchmarks from optimal market making.

 

  • [Fintech][Contract] Thibaut Mastrolia, Hao Wang.
    Regulation or Competition: Major-Minor Optimal Liquidation across Dark and Lit Pools.
    We study the optimal liquidation problem in both lit and dark pools for investors facing execution uncertainty in a continuous-time setting with market impact. First, we design an optimal make–take fee policy for a large investor liquidating her position across both pools, interacting with small investors who pay trading fees. We explicitly characterize the large investor’s optimal liquidation strategies in both lit and dark pools using BSDEs under a compensation scheme proposed by an exchange to mitigate market impact in the lit venue. Second, we consider a purely competitive model with major–minor traders in the absence of regulation. We provide explicit solutions to the associated HJB–Fokker–Planck system. Finally, we illustrate our results through numerical experiments, comparing market impact under a regulated market with a strategic large investor to that in a purely competitive market with both small and large investors.

 

  • [BSDE] Alberto Gennaro, Thibaut Mastrolia.
    2BSDE with uncertain horizon and application to stochastic control in erratic environments.
    We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a Markovian setting. Our theoretical results are applied to non-Markovian stochastic control problems in two settings: (1) when an agent seeks to maximize utility from a payoff received at an uncertain terminal time by controlling both the drift and volatility of a diffusion process; and (2) when the agent contends with volatility uncertainty stemming from an external source, referred to as Nature, and optimizes the drift in a worst-case scenario for the ambiguous volatility. We term this class of problems erratic stochastic control, reflecting the dual uncertainty in both model parameters and the timing of the terminal horizon.

 

  • [Cyber][Contract] Thibaut Mastrolia, William Yan.
    Agency Problems and Adversarial Bilevel Optimization under Uncertainty and Cyber Threats.
    We study an agency problem between a holding company and its subsidiary, exposed to cyber threats that affect the overall value of the subsidiary. The holding company seeks to design an optimal incentive scheme to mitigate these losses. In response, the subsidiary selects an optimal cybersecurity investment strategy, modeled through a stochastic epidemiological SIR (Susceptible-Infected-Recovered) framework. The cyber threat landscape is captured through an L-hop risk framework with two primary sources of risk: (i) internal risk propagation via the contagion parameters in the SIR model, and (ii) external cyberattacks from a malicious external hacker. The uncertainty and adversarial nature of the hacking lead to consider a robust stochastic control approach that allows for increased volatility and ambiguity induced by cyber incidents. The agency problem is formulated as a max-min bilevel stochastic control problem with accidents. First, we derive the incentive compatibility condition by reducing the subsidiary’s optimal response to the solution of a second-order backward stochastic differential equation with jumps. Next, we demonstrate that the principal’s problem can be equivalently reformulated as an integro-partial Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation. By extending the stochastic Perron’s method to our setting, we show that the value function of the problem is the unique viscosity solution to the resulting integro-partial HJBI equation.

 

  • [Fintech][Contract] Thibaut Mastrolia, Tianrui Xu.
    Optimal Rebate Design: Incentives, Competition and Efficiency in Auction Markets.
    This study explores the design of an efficient rebate policy in auction markets, focusing on a continuous-time setting with competition among market participants. In this model, a stock exchange collects transaction fees from auction investors executing block trades to buy or sell a risky asset, then redistributes these fees as rebates to competing market makers submitting limit orders. Market makers influence both the price at which the asset trades and their arrival intensity in the auction. We frame this problem as a principal-multi-agent problem and provide necessary and sufficient conditions to characterize the Nash equilibrium among market makers. The exchange’s optimization problem is formulated as a high-dimensional Hamilton-Jacobi-Bellman equation with Poisson jump processes, which is solved using a verification result. To numerically compute the optimal rebate and transaction fee policies, we apply the Deep BSDE method. Our results show that optimal transaction fees and rebate structures improve market efficiency by narrowing the spread between the auction clearing price and the asset’s fundamental value, while ensuring a minimal gain for both market makers indexed on the price of the asset on a coexisting limit order book.

 

  • [Cyber] Caroline Hillairet, Thibaut Mastrolia, Wissal Sabbagh.
    Optimal Impulse Control for Cyber Risk Management.
    We explore an optimal impulse control problem wherein an electronic device owner strategically calibrates protection levels against cyber attacks. Utilizing epidemiological compartment models, we qualitatively characterize the dynamics of cyber attacks within the network. We determine the optimal protective measures against effective hacking by formulating and solving a stochastic control problem with optimal switching. We demonstrate that the value function for the cluster owner constitutes a viscosity solution to a system of coupled variational inequalities associated with a fully coupled reflected backward stochastic differential equation (BSDE). Furthermore, we devise a comprehensive algorithm alongside a verification procedure to ascertain the optimal timing for network protection across various cyber attack scenarios. Our findings are illustrated through numerical approximations employing deep Galerkin methods for partial differential equations (PDEs). We visualize the optimal protection strategies in the context of two distinct attack scenarios: (1) a constant cyber attack, (2) an exogenous cyber attack strategy modeled with a Poisson process.

 

  • [Fintech][Contract] Alberto Gennaro, Thibaut Mastrolia.
    Delegated portfolio management with random default.
    We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager’s investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE’s driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

 

  • [Fintech] Thibaut Mastrolia, Tianrui Xu.
    Clearing time randomization and transaction fees for auction market design. 
    Flaws of a continuous limit order book mechanism raise the question of whether a continuous trading session and a periodic auction session would bring better efficiency. This paper wants to go further in designing a periodic auction when both a continuous market and a periodic auction market are available to traders. In a periodic auction, we discover that a strategic trader could take advantage of the accumulated information available along the auction duration by arriving at the latest moment before the auction closes, increasing the price impact on the market. Such price impact moves the clearing price away from the efficient price and may disturb the efficiency of a periodic auction market. We thus propose and quantify the effect of two remedies to mitigate these flaws: randomizing the auction’s closing time and optimally designing a transaction fees policy for both the strategic traders and other market participants. Our results show that these policies encourage a strategic trader to send their orders earlier to enhance the efficiency of the auction market, illustrated by data extracted from Alphabet and Apple stocks.

 

Published and Accepted Articles

 

 

 

 

 

 

 

 

  • [Fintech] Omar El Euch, Thibaut Mastrolia, Mathieu Rosenbaum, Nizar Touzi,
    Optimal make-take fees for market making regulation.
    Mathematical Finance, 31, pp109-148 (2021) and its corrigendum (pdf).

 

 

 

  • [Contract] Nicolás Hernández-Santibáñez, Thibaut Mastrolia.
    Contract Theory in a VUCA World.
    SIAM J. Control Optim. Vol. 57, No. 4, pp 3072-3100 (2019).

 

 

 

  • [Contract] Thibaut Mastrolia., Dylan Possamaï.
    Moral Hazard Under Ambiguity.
    Journal of Optimization Theory and Applications. Vol. 179, No. 2, pp 452-500 (2018).

 

 

  • [BSDE] Thibaut Mastrolia, Dylan Possamaï, Anthony Réveillac.
    On the Malliavin differentiability of BSDEs.
    Annales de l’Institut Henri Poincaré série Probabilités et Statistique. Vol. 53, No. 1, pp 464-492 (2017).

 

  • [BSDE] Thibaut Mastrolia, Dylan Possamaï, Anthony Réveillac.
    Density analysis of BSDEs.
    Annals of Probability. Vol. 44, No. 4, pp 2817-2857 (2016).