2023/12/01 by Andrea Bovo, Bovo, Andrea, Tiziano De Angelis +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #49J40 #60G40 #91A05 #91A15 #93E20 #FOS: Economics and business #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Finance (q-fin.MF) #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2312.00613
openalex publication_date 2023/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter γ>0. For each γ>0 the approximating game is non-degenerate and admits a value uγ and an optimal strategy τγ_* for the stopper. Letting γ→ 0 we prove convergence of uγ to a function v, which identifies the value of the original game. We also construct explicitly optimal stopping times θγ_* for uγ, related but not equal to τγ_*, which converge almost surely to an optimal stopping time θ_* for the game with degenerate dynamics.