Introduction: The Dilemma of Large-Scale Liquidation
In modern financial markets, institutional investors face a critical, highly practical optimization problem regarding how a large portfolio position should be liquidated before a strict deadline. Executing a large sell order all at once is costly because it pushes market prices in an unfavourable direction, a phenomenon known as price impact that intensifies with trading speed. Conversely, trading too slowly introduces unacceptable exposure to market risk. Finding the optimal balance between trading costs and risk while navigating fluctuating market liquidity is a quintessential problem at the intersection of stochastic control, applied probability, and financial mathematics. At the University of Wuppertal, M.Sc. Petr Petrov, working under the supervision of Prof. Thomas Kruse, is tackling this challenge in his PhD project by developing advanced mathematical frameworks for multi-asset optimal trade execution.
Mathematical Formulation and Terminal Constraints
The project models the investor’s problem as a linear-quadratic stochastic control problem with random coefficients, where the investor’s current inventory position acts as the state, her trading speed serves as the control, and market coefficients governing risk, costs, and price impact vary randomly over time. The objective is to minimize the expected cost, combining quadratic trading costs and running risk penalties with a terminal cost. A defining mathematical feature of Petr’s work is the incorporation of a nonstandard terminal constraint, meaning that at the final time deadline, the portfolio must lie within a prescribed linear subspace which may itself be random and revealed only at the deadline, with full liquidation appearing as a special case of this constraint.
Why the Multidimensional Setting Matters
Institutional investors rarely manage assets in isolation, instead overseeing interconnected portfolios where trading one stock impacts the prices of related assets through cross-asset price impact. This impact is captured via a full matrix rather than isolated scalar values, allowing the model to reflect shifting market liquidity dynamically. Beyond simple full liquidation, the investor may aim to achieve a specific target balance among assets at the deadline. While single-asset execution is well understood, the multidimensional, constrained version has historically been studied mostly under deterministic cross-impact. Petr’s project extends this framework to include random cross-impact, general linear terminal targets, and complex interaction terms, introducing fresh technical depth to the analysis.
The Singular BSDE Formulation and Dual Approach
To solve this class of problems, the project relies on advanced backward stochastic differential equations. The minimal cost is represented as a quadratic function of the current position, weighted by a random, matrix-valued process that solves a Riccati backward stochastic differential equation running backwards in time with a martingale component. Once solved, the optimal trading strategy is obtained in feedback form. Because ending outside the target subspace must carry infinite cost, the terminal condition is singular, and Petr constructs a solution via penalization by replacing the strict constraint with a penalty parameter and studying the convergence of the resulting standard equations as the penalty tends to infinity. An alternative strategy employs a Lagrange multiplier to enforce the linear terminal constraint, yielding a dual stochastic linear-quadratic control problem that lacks uniform convexity and leads to a backward stochastic differential equation with a singular generator, offering an independent mathematical challenge currently under active investigation.
Conclusion and Industrial Relevance
By bridging abstract stochastic analysis with real-world financial engineering, this PhD project exemplifies the core mission of the European Consortium for Mathematics in Industry by deploying rigorous mathematical modeling to solve complex economic challenges. Understanding how multi-asset portfolios can be efficiently liquidated under uncertainty provides valuable insights for algorithmic trading, asset management, and risk governance in modern industries.