Finiteness of the forward price at infinity

Determine whether the boundary value v*(τ,∞) := lim_{y→∞} v*(τ,y) of the forward price function v*(τ,y) = E^y[Y_τ] is finite for all τ > 0 when the underlying asset price Y is modeled as a non-negative strict local martingale diffusion solving dY_t = σ(Y_t) dβ_t with absorbing boundary at 0 under a risk-neutral measure, as formulated in equation (2.1).

Background

In markets where the discounted asset price is a strict local martingale (a bubble), the forward price v*(τ,y) = E^y[Y_τ] is concave in y and differs from the classical solution v(τ,y) = y. The paper develops boundary conditions at infinity based on an integral representation of v*(τ,∞) and provides sufficient conditions ensuring v*(τ,∞) < ∞.

Ekström and Tysk (2009) gave volatility growth conditions to guarantee finiteness, and Proposition 3 in this paper weakens those conditions. Despite these advances, the general question of whether v*(τ,∞) is always finite across the admissible class of strict local martingale diffusions remains unresolved.