arXiv subjects
Publications and source records attributed to Zhuoshu Wu.
In this article we provide a valuation formula for a defaultable perpetual Russian option in the Black-Scholes market where the default time is modelled as the last passage time of the running maximum of the stock price. In this setting, default occurs when the stock price fails to exceed its historical maximum, leading to a non-stopping time that depends on the path of the underlying asset.
We study the valuation of an American put option with a random time horizon given by the last exit time of the underlying asset from a fixed level. Since this random time is not a stopping time, the problem falls outside the classical optimal stopping framework. Using enlargement of filtrations and the associated Azéma supermartingale, we transform the problem into an equivalent optimal stopping problem with a semi-continuous, time-dependent gain function whose partial derivatives exhibit singular behaviour. The resulting formulation introduces significant analytical challenges, including the loss of smoothness of the optimal stopping boundary. We develop new arguments to characterise the continuation and stopping regions, establishing monotonicity of the free boundary under suitable conditions, and analyse the regularity of the value function. In particular, we derive nonlinear integral equations that uniquely characterise both the free-boundary and the value function. Our results extend the classical theory of American options to a class of problems with random horizons and provide a framework for incorporating default-type features modelled by last exit times.
We propose a mean field game (MFG) framework to model the evolution of renewable energy production in competitive electricity markets. Producers interact through the spot price while optimising their profits under production, installation, and capacity adjustment costs, as well as the generation uncertainty. We first formulate the market as an $N$-player stochastic differential game and analyse its mean field game limit as $N\to\infty$. We characterise the representative producer's optimal control via forward-backward stochastic differential equations (FBSDEs) derived from the stochastic maximum principle and determine the corresponding equilibrium spot price. We establish existence and uniqueness of solutions to the FBSDEs and prove that the MFG admits a unique equilibrium. We then extend the model to a Stackelberg mean field game to incorporate the role of a social planner. The planner's optimisation problem leads to an extended Hamilton-Jacobi-Bellman (HJB) system, for which we prove existence and uniqueness of viscosity solutions. Finally, we implement a deep learning-based numerical scheme to approximate the equilibrium and investigate the impact of policy interventions on capacity dynamics. Our results highlight how optimal subsidy design depends on prevailing market conditions and can mitigate both capacity shortages and overproduction.