Mathematicians crack optimal decision-making under uncertainty with jumps
Researchers have solved a decades-old mathematical puzzle about making optimal decisions when outcomes involve sudden, unpredictable shifts—a problem that affects everything from financial derivatives pricing to real options valuation in M&A. The breakthrough offers new tools for modeling scenarios where barriers constrain outcomes, directly applicable to portfolio optimization and risk management in volatile markets.
Originaltitel: Optimal stopping of BSDEs with constrained jumps and related double obstacle PDEs
<p>We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman–Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope, arising from an optimal stopping problem—where the underlying process is the first component in the solution to a stopped backward stochastic differential equation (BSDE) with jumps and a constraint on the jumps process—to a viscosity solution for the PDE. Leveraging this Feynman–Kac representation, we subsequently prove existence and uniqueness of viscosity solutions in the non-local setting by employing a contraction argument. This approach also introduces a novel form of non-linear Snell envelope and expands the probabilistic representation theory for PDEs. </p>