Outrunning the Omega Clock: A Singular Control Problem for Dividend Optimisation with Ruin and Time-in-Distress Default

TL;DR

The paper extends the classical dividend problem by introducing an occupation-time–based default (the omega clock) that can trigger default before ruin at zero, modelling distress through $\omega_t^y = q \int_0^t I_{\{X_s^D<y\}} ds$ and a random horizon. Treating this as a singular stochastic control with state-dependent discounting $r+q I_{\{X_t^D<y\}}$, the authors derive closed-form solutions via a bespoke guess-and-verify approach. They uncover three regimes: a supercritical regime where the policy reduces to the fully penalised case, a subcritical regime with a single boundary, and a critical regime with two disjoint payout regions, revealing a rich and novel structure including disconnected action/inaction sets. The results demonstrate how occupation-time risk can delay dividends under increased distress and provide explicit multi-barrier strategies and transition behavior across regimes, contributing to dividend theory and risk management under persistence-based default.

Abstract

This paper extends the classical dividend problem by incorporating a novel, path-dependent mechanism of firm default. In the traditional framework, ruin occurs when the surplus process first reaches zero. In contrast, default in our model may also arise when the surplus spends an excessive amount of time below a distress threshold, even without ever hitting zero. This occupation-time-based default criterion captures financial distress more realistically, as prolonged periods of low liquidity or capitalisation may trigger regulatory intervention or operational failure. The resulting optimisation problem is formulated as a new singular stochastic control problem with discontinuous state-dependent discounting and killing. We provide a complete analytical solution via a bespoke sequential guess-and-verify method and identify three distinct classes of optimal dividend strategies corresponding to different parameter regimes of the dual-ruin structure. Notably, for certain distress thresholds, the optimal policy features disconnected action and inaction regions. We further show that, unlike in the classical dividend problem, higher effective discounting induced by occupation time below a distress level can lead to delayed, rather than earlier, dividend payments.

Figures (1)

• Figure 1: Value functions $V$ (top) and first derivatives $V'$ (bottom) for the three different regimes and the parameters $\mu = 0.1, \sigma = 0.1, r = 0.02, q = 0.1$. The three regimes result from different values of $y$ and are characterised by the pair $(y_l, y_u) \approx (1.7461, 4.4292)$. On the left, we have $y = y_l\cdot0.9\leq y_l$ and the optimal dividend policy is to pay out dividends above the barrier $b^*(y)\approx 1.8717$. In the middle, we have $y = 0.9 \cdot y_l + 0.1 \cdot y_u \in (y_l,y_u)$ and the optimal dividend policy is to pay out dividends in the disjoint set $[b^*_{r+q},\underline{b}^*(y)]\cup[\overline{b}^*(y),\infty)\approx[0.2626,1.1394]\cup[ 2.3147,\infty)$. On the right we have $y = y_u \cdot 1.001\geq y_u$ and the optimal dividend policy is to pay out dividends above $b^*_{r+q} \approx0.2626$, as in the classical case with ruin at zero and constant discount rate $r+q$.

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 3.1
• proof
• Lemma 3.2