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Consider the Cramér–Lundberg model $$\hspace{8em}R(t)=u+c\,t-\sum_{j=1}^{N(t)}V_{j}\,,\hspace{8em}(1)$$ where $c$ and $u$ are positive constants, $N(t)$ is a Poisson process with a rate $\lambda$ (in other words, $N(t_{2})-N(t_{1})$ is Poisson-distributed with a mean $(t_{2}-t_{1})\lambda$), and the $V_{j}$'s are independent identically distributed random variables (i.i.d. RVs) whose common probability distribution is Pareto (of Type I), with a shape parameter $\boldsymbol{\alpha\in (0,1)}$. Namely, the cumulative distribution function is $$\hspace{6em}F(x)= \begin{cases} 1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}\,. \end{cases}\hspace{6em}(2) $$ Recall that if $\boldsymbol{\alpha<1}$, then the distribution has an infinite mean.

In actual fact, I am particularly interested in the case $\alpha=1/2$. Also, we may set $x_\mathrm{m}=1$.

The ruin time $\boldsymbol{\tau}$ is the 'earliest' time the reserve $R$ turns negative. More precisely, $$ \hspace{8em}\tau=\inf\{t>0\,:\,R(t)<0\}\,.\hspace{8em}(3) $$

Usually, the first order of business when analyzing a Cramér–Lundberg model is to compute the infinite-time ruin probability $\boldsymbol{\psi}$, $$\hspace{11em}\psi=\mathbb{P}(\tau<\infty)\,.\hspace{11em}(4)$$ But when Pareto-distributed claims have an infinite mean, that probability is known to be 1, no matter what values $c$ and $u$ have. So $R(t)$ is certain to turn negative if one waits long enough. See Kortschak, Loisel, and Ribereau, 'Ruin Problems with Worsening Risks or with Infinite Mean Claims', Stoch. Model. 31, 119–152 (2015), here.

In fact, Kortschak, Loisel, and Ribereau show that in order to make $\psi$ less than 1, one would have to modify the model by making $c$ time-dependent. In particular, $c$ would have to be increasing with time fast enough, namely as $c\sim t^\beta$ with $\beta>1/\alpha$. This is consistent with the estimate of the upper $q$-quantiles (one of which is the median, $q=0.5$) of $S_{n}=\sum_{j=1}^{n}V_{j}$ given in the Appendix, below.

But for my purposes, $c$ should remain time-independent, and the infinite-time ruin probability is always 1.

My questions

  1. Does the value of $R(t)/t$, when conditioned on $\boldsymbol{\tau>t}$, converge to any finite value as $t\to\infty$? If it doesn't for general values of $c$ and $u$, does it perhaps converge for some special values of these parameters? If so, what does it converge to?
  2. Is there anything known about the distribution of ruin times $\tau$, such as their probability distribution, mean value (if it exists), or median (or other quantiles)?
  3. Is there anything known about the behavior of $R(t)$ prior to ruin? In particular, do we know anything about the distribution of the quantity $R^{*}={\displaystyle \lim_{t\to \tau^{-}} R(t)}$, the value of the reserve just prior to ruin—its probability distribution, mean value (if it exists), or median (or other quantiles)?
  4. Finally, if nothing is in fact yet known about these things, how one might go about computing some of them? What might be some promising strategies?

Prior research

I looked at the following references (among many others!) from ruin theory literature, but apart from Kortschak, Loisel, and Ribereau, all of them assume that the $V_{j}$'s have at least a finite mean.

Asmussen and Steffensen, Risk and Insurance.
Asmussen, 'Rare Events in the Presence of Heavy Tails', Ch 10 in Stochastic Networks, Sigman and Yao, eds.
Kortschak, Loisel, and Ribereau, 'Ruin Problems with Worsening Risks or with Infinite Mean Claims', Stoch. Model. 31, 119–152 (2015).
Embrechts and Veraverbeke, 'Estimates for the probability of ruin with special emphasis on the possibility of large claims', Insur. Math. Econ. 1, 55–72 (1982).
von Bahr, 'Asymptotic ruin probabilities when exponential moments do not exist', Scand. Actuar. J. 1975, 6–10 (1975).

From non-risk theory literature, I found the following interesting, as it deals with Pareto distributions with infinite means:

Zaliapin, Kagan, and Schoenberg, 'Approximating the Distribution of Pareto Sums', Pure Appl. Geophys. 162, 1187–1228 (2005).

Motivation

The genealogy of this question consists of prior questions of mine on various StackExchanges. In order from the most recent to the oldest: here, here, here, here, and here.

Appendix: estimates of the median and other upper q-quintiles of the sum $\boldsymbol{S_{n}=\sum_{j=1}^{n}V_{j}}$.

I'll be following the discussion in Zaliapin, Kagan, and Schoenberg, 'Approximating the Distribution of Pareto Sums', Pure Appl. Geophys. 162, 1187–1228 (2005), here.

Our starting point is that $S_{n}=\sum_{j=1}^{n}V_{j}$ satisfies the Generalized Central Limit Theorem $$ \lim_{n\to\infty}\mathbb{P}\left(\frac{S_{n}-b_{n}}{n^{1/\alpha}C_{\alpha}}<x\right)=F_{\alpha}(x)\,, $$ where $F_{\alpha}(x)$ is a stable cumulative distribution function with index $\alpha$ (in Mathematica, it is ${\scriptstyle\texttt{StableDistribution[1, $\alpha$, 1, 0, 1]}}$, see here). If $0<\alpha<1$, we further have $b_{n}=0$ and $C_{\alpha}=\left[\Gamma(1-\alpha)\,\cos(\pi\alpha/2)\right]^{1/\alpha}$, where $\Gamma(x)$ is the gamma function. Let $x_{q}$ be the solution of the equation $F_{\alpha}(x_{q})=q$. Then we have the following estimate for $z_{q}$, the upper $q$-quintile of $S_{n}$: $z_{q}\approx x_{q}\,C_{\alpha}\,n^{1/\alpha}+b_{n}$ (for example, setting $q=0.5$ gives an estimate for the median). In the special case $\alpha=1/2$, the following approximation of $F_{\alpha}$ is accurate to better than 0.1%, as long as $x\geqslant 1$: $F_{1/2}(x)\approx 1-\frac{\sqrt{{2}/{\pi }}}{\sqrt{x}}+\frac{1}{3 \sqrt{2 \pi } x^{3/2}}-\frac{1}{20 \sqrt{2 \pi } x^{5/2}}+\frac{1}{168 \sqrt{2 \pi } x^{7/2}}$. For $\alpha=1/2$, we have that $z_{n}\approx \frac{\pi}{2}x_{q}n^{2}$. The median corresponds to $q=0.5$, for which $x_{0.5}=2.19811$; for $q=0.98$, we have $x_{0.98}=1591.22$.

The accuracy of these estimates (compared to numerical simulations) grows with increasing $n$, and, for a fixed $n$, it grows with increasing $q$. For $\alpha=1/2$, the estimates are accurate to better than 2% when $n\geqslant 10$ and $q\geqslant 0.5$.

Zaliapin, Kagan, and Schoenberg give several other estimates of increasing precision, but the scaling $\sim n^{1/\alpha}$ remains in all of them.

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