Stochastic Control

Signature approach for pricing and hedging path-dependent options with frictions ArXiv ID: 2511.23295 “View on arXiv” Authors: Eduardo Abi Jaber, Donatien Hainaut, Edouard Motte Abstract We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case. ...

Competitive optimal portfolio selection under mean-variance criterion ArXiv ID: 2511.05270 “View on arXiv” Authors: Guojiang Shao, Zuo Quan Xu, Qi Zhang Abstract We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent’s utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the optimization framework. To address this game-theoretic problem, we first reformulate the mean-variance criterion as a constrained, non-homogeneous stochastic linear-quadratic control problem and derive the corresponding optimal feedback strategies. The existence of Nash equilibria is shown to depend on the well-posedness of a complex, coupled system of equations. Employing decoupling techniques, we reduce the well-posedness analysis to the solvability of a novel class of multi-dimensional linear backward stochastic differential equations (BSDEs). We solve a new type of nonlinear BSDEs (including the above linear one as a special case) using fixed-point theory. Depending on the interplay between market and competition parameters, three distinct scenarios arise: (i) the existence of a unique Nash equilibrium, (ii) the absence of any Nash equilibrium, and (iii) the existence of infinitely many Nash equilibria. These scenarios are rigorously characterized and discussed in detail. ...

On Bellman equation in the limit order optimization problem for high-frequency trading ArXiv ID: 2510.15988 “View on arXiv” Authors: M. I. Balakaeva, A. Yu. Veretennikov Abstract An approximation method for construction of optimal strategies in the bid & ask limit order book in the high-frequency trading (HFT) is studied. The basis is the article by M. Avellaneda & S. Stoikov 2008, in which certain seemingly serious gaps have been found; in the present paper they are carefully corrected. However, a bit surprisingly, our corrections do not change the main answer in the cited paper, so that, in fact, the gaps turn out to be unimportant. An explanation of this effect is offered. ...

FinFlowRL: An Imitation-Reinforcement Learning Framework for Adaptive Stochastic Control in Finance ArXiv ID: 2509.17964 “View on arXiv” Authors: Yang Li, Zhi Chen, Steve Y. Yang, Ruixun Zhang Abstract Traditional stochastic control methods in finance rely on simplifying assumptions that often fail in real world markets. While these methods work well in specific, well defined scenarios, they underperform when market conditions change. We introduce FinFlowRL, a novel framework for financial stochastic control that combines imitation learning with reinforcement learning. The framework first pretrains an adaptive meta policy by learning from multiple expert strategies, then finetunes it through reinforcement learning in the noise space to optimize the generation process. By employing action chunking, that is generating sequences of actions rather than single decisions, it addresses the non Markovian nature of financial markets. FinFlowRL consistently outperforms individually optimized experts across diverse market conditions. ...

Optimal Exit Time for Liquidity Providers in Automated Market Makers ArXiv ID: 2509.06510 “View on arXiv” Authors: Philippe Bergault, Sébastien Bieber, Leandro Sánchez-Betancourt Abstract We study the problem of optimal liquidity withdrawal for a representative liquidity provider (LP) in an automated market maker (AMM). LPs earn fees from trading activity but are exposed to impermanent loss (IL) due to price fluctuations. While existing work has focused on static provision and exogenous exit strategies, we characterise the optimal exit time as the solution to a stochastic control problem with an endogenous stopping time. Mathematically, the LP’s value function is shown to satisfy a Hamilton-Jacobi-Bellman quasi-variational inequality, for which we establish uniqueness in the viscosity sense. To solve the problem numerically, we develop two complementary approaches: a Euler scheme based on operator splitting and a Longstaff-Schwartz regression method. Calibrated simulations highlight how the LP’s optimal exit strategy depends on the oracle price volatility, fee levels, and the behaviour of arbitrageurs and noise traders. Our results show that while arbitrage generates both fees and IL, the LP’s optimal decision balances these opposing effects based on the pool state variables and price misalignments. Lastly, we find the optimal fee level for the representative LP when they play the exit strategy we derived. This work contributes to a deeper understanding of dynamic liquidity provision in AMMs and provides insights into the sustainability of passive LP strategies under different market regimes. ...

FinFlowRL: An Imitation-Reinforcement Learning Framework for Adaptive Stochastic Control in Finance ArXiv ID: 2510.15883 “View on arXiv” Authors: Yang Li, Zhi Chen Abstract Traditional stochastic control methods in finance struggle in real world markets due to their reliance on simplifying assumptions and stylized frameworks. Such methods typically perform well in specific, well defined environments but yield suboptimal results in changed, non stationary ones. We introduce FinFlowRL, a novel framework for financial optimal stochastic control. The framework pretrains an adaptive meta policy learning from multiple expert strategies, then finetunes through reinforcement learning in the noise space to optimize the generative process. By employing action chunking generating action sequences rather than single decisions, it addresses the non Markovian nature of markets. FinFlowRL consistently outperforms individually optimized experts across diverse market conditions. ...

Two Stochastic Control Methods for Mean-Variance Portfolio Selection of Jump Diffusions and Their Relationship ArXiv ID: 2508.01138 “View on arXiv” Authors: Qiyue Zhang, Jingtao Shi Abstract This paper is concerned with the maximum principle and dynamic programming principle for mean-variance portfolio selection of jump diffusions and their relationship. First, the optimal portfolio and efficient frontier of the problem are obtained using both methods. Furthermore, the relationship between these two methods is investigated. Specially, the connections between the adjoint processes and value function are given. ...

Optimal Execution under Liquidity Uncertainty ArXiv ID: 2506.11813 “View on arXiv” Authors: Etienne Chevalier, Yadh Hafsi, Vathana Ly Vath, Sergio Pulido Abstract We study an optimal execution strategy for purchasing a large block of shares over a fixed time horizon. The execution problem is subject to a general price impact that gradually dissipates due to market resilience. This resilience is modeled through a potentially arbitrary limit-order book shape. To account for liquidity dynamics, we introduce a stochastic volume effect governing the recovery of the deviation process, which represents the difference between the impacted and unaffected price. Additionally, we incorporate stochastic liquidity variations through a regime-switching Markov chain to capture abrupt shifts in market conditions. We study this singular control problem, where the trader optimally determines the timing and rate of purchases to minimize execution costs. The associated value function to this optimization problem is shown to satisfy a system of variational Hamilton-Jacobi-Bellman inequalities. Moreover, we establish that it is the unique viscosity solution to this HJB system and study the analytical properties of the free boundary separating the execution and continuation regions. To illustrate our results, we present numerical examples under different limit-order book configurations, highlighting the interplay between price impact, resilience dynamics, and stochastic liquidity regimes in shaping the optimal execution strategy. ...

Goal-based portfolio selection with mental accounting ArXiv ID: 2506.06654 “View on arXiv” Authors: Erhan Bayraktar, Bingyan Han Abstract We present a continuous-time portfolio selection framework that reflects goal-based investment principles and mental accounting behavior. In this framework, an investor with multiple investment goals constructs separate portfolios, each corresponding to a specific goal, with penalties imposed on fund transfers between these goals, referred to as mental costs. By applying the stochastic Perron’s method, we demonstrate that the value function is the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman equation system. Numerical analysis reveals several key features: the free boundaries exhibit complex shapes with bulges and notches; the optimal strategy for one portfolio depends on the wealth level of another; investors must diversify both among stocks and across portfolios; and they may postpone reallocating surplus from an important goal to a less important one until the former’s deadline approaches. ...

FlowOE: Imitation Learning with Flow Policy from Ensemble RL Experts for Optimal Execution under Heston Volatility and Concave Market Impacts ArXiv ID: 2506.05755 “View on arXiv” Authors: Yang Li, Zhi Chen Abstract Optimal execution in financial markets refers to the process of strategically transacting a large volume of assets over a period to achieve the best possible outcome by balancing the trade-off between market impact costs and timing or volatility risks. Traditional optimal execution strategies, such as static Almgren-Chriss models, often prove suboptimal in dynamic financial markets. This paper propose flowOE, a novel imitation learning framework based on flow matching models, to address these limitations. FlowOE learns from a diverse set of expert traditional strategies and adaptively selects the most suitable expert behavior for prevailing market conditions. A key innovation is the incorporation of a refining loss function during the imitation process, enabling flowOE not only to mimic but also to improve upon the learned expert actions. To the best of our knowledge, this work is the first to apply flow matching models in a stochastic optimal execution problem. Empirical evaluations across various market conditions demonstrate that flowOE significantly outperforms both the specifically calibrated expert models and other traditional benchmarks, achieving higher profits with reduced risk. These results underscore the practical applicability and potential of flowOE to enhance adaptive optimal execution. ...