Conditional Volatility of a Discounted Brownian Integral Perpetuity

Summary

The document poses a stochastic calculus question about the conditional expectation of a powered perpetuity, formed by integrating an exponentially discounted function of Brownian motion over an infinite horizon. It defines this conditional expectation as a process and observes that, under suitable integrability conditions, it is a martingale with a Brownian representation whose instantaneous relative volatility is denoted by sigma.

The central question is whether that volatility decreases over time, perhaps in expectation. The proposed intuition is that as time advances, less of the discounted integral remains unknown, so its uncertainty might shrink. No derivation, result, or empirical evidence is supplied, and the claim is left unresolved. The answer may depend on parameter restrictions, the exponent, and the precise meaning of decreasing volatility; these conditions and the existence assumptions are not developed in the note.

Key ideas

• The process is a conditional expectation of a powered, discounted Brownian integral over an infinite horizon.
• Under suitable conditions, conditional expectations form a martingale and can have a Brownian stochastic representation.
• The note asks whether the martingale’s relative volatility declines over time.
• The intuition is that the remaining discounted integral becomes less uncertain as time passes.
• No proof is offered, and parameter and integrability conditions remain unspecified.

Tags

# Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$


# Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$












Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as

$$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big].$$

where $\eta\in\mathbb{R}^\star$. Then, $Y$ is a martingale (this seems obvious). I assume all conditions are met for the above integral to exist (c.f. Yor, 2002 for example). Therefore there exists a process $\{\sigma_t\}_{t\geq 0}$ such that

$$dY_t=Y_t\sigma_tdz_t.$$

I want to show that $\sigma_t$ is decreasing (in expectation, perhaps?). The intuition is that the discounting $e^{-ks}$ becomes stronger and stronger so the unknown part of the integral becomes less and less volatile... Sorry for the bad phraseology, I can't think of a better way of explaining it. Thanks!

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