The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent $\delta\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $\delta\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.Comment: Keywords: local martingales, local stochastic solutions, degenerate Cauchy problems, Feynman-Kac formula, necessary and sufficient condition for uniqueness, comparison principl

Information Elicitation from Decentralized Crowd Without Verification

Information Elicitation Without Verification (IEWV) refers to the problem of eliciting high-accuracy solutions from crowd members when the ground truth is unverifiable. A high-accuracy team solution (aggregated from members' solutions) requires members' effort exertion, which should be incentivized properly. Previous research on IEWV mainly focused on scenarios where a central entity (e.g., the crowdsourcing platform) provides incentives to motivate crowd members. Still, the proposed designs do not apply to practical situations where no central entity exists. This paper studies the overlooked decentralized IEWV scenario, where crowd members act as both incentive contributors and task solvers. We model the interactions among members with heterogeneous team solution accuracy valuations as a two-stage game, where each member decides her incentive contribution strategy in Stage 1 and her effort exertion strategy in Stage 2. We analyze members' equilibrium behaviors under three incentive allocation mechanisms: Equal Allocation (EA), Output Agreement (OA), and Shapley Value (SV). We show that at an equilibrium under any allocation mechanism, a low-valuation member exerts no more effort than a high-valuation member. Counter-intuitively, a low-valuation member provides incentives to the collaboration while a high-valuation member does not at an equilibrium under SV. This is because a high-valuation member who values the aggregated team solution more needs fewer incentives to exert effort. In addition, when members' valuations are sufficiently heterogeneous, SV leads to team solution accuracy and social welfare no smaller than EA and OA