On the properties of $r$-excessive mappings for a class of diffusions

Abstract

We consider the convexity and comparative static properties of a class of r-harmonic mappings for a given linear, time-homogeneous and regular diffusion process. We present a set of weak conditions under which the minimal r-excessive mappings for the considered diffusion are convex and under which an arbitrary nontrivial r-excessive mapping is convex on the regions where it is r-harmonic. Consequently, we are able to present a set of usually satisfied conditions under which increased volatility increases the value of r-harmonic mappings. We apply our results to a class of optimal stopping problems arising frequently in studies considering the pricing of perpetual American contingent claims and state a set of conditions under which the value function is convex on the continuation region and, consequently, under which increased volatility unambiguously increases the value function and expands the continuation region, thus postponing the rational exercise of the claim.

On Threshold Strategies and the Smooth-Fit Principle for Optimal Stopping Problems

“…We recall that the convexity of the value function and its connection with volatility misspecification has recently been studied in depth; see [9], [18], [11], [7], and [1]. A general result comes from these papers: the value function is convex if either • δ is constant, that is, (e −(r−δ)t X t ) t≥0 is a local martingale [9], [18], [11], [7], or…”

Leveraged Investments and Agency Conflicts When Prices are Mean Reverting

“…Theorem 1 in Alvarez (2003) states that, provided infinity is a natural boundary for the process X (a standing assumption in the present paper) then the fundamental solutions φ and ψ are convex if and only if the auxiliary function θ(z) := rz+λσ(z)−α(z) is non-decreasing, i.e. when r+λσ…”

Taxation and rotation age under stochastic forest stand value

“…This observation is of interest since it demonstrates that the sign of the relationship between forest value volatility and the optimal rotation policy is a process-specific, not a payoff-specific, property (cf. Alvarez 2003). The effect of risk aversion on the optimal harvesting threshold and thereby on the expected length of the rotation period is now summarized in the following.…”