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If you find it helpful, you may assume that $v_{\text{D}}$ ('D' for 'deterministic' or 'degenerate') and $v_{\text{P}}$ ('P for Pareto) have some particular values. For example, you might even assume that $v_{\text{D}}=v_{\text{P}}=1$.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulties are that (a) the mean is infinite and (b) ruin happens always, with probability one.¹ Given this, it is not clear how to interpret $c$, $\mu$, and $\eta$. In particular, it is not clear what principle should govern the choice $\mu_{V}$. My conjecture below is, in part, an attempt to answer that question.

^{In the context of insurance, provided the ruin probability isn't 1, $\eta=0$ leads to premium rates ($c$) that are break-even for the insurer. A nonzero $\eta$ leads to net profit.}

1. Is my conjecture a good way to make precise the 'vague hypothesis'? Is there a better way?
2. Is the conjecture true? How would one prove it? It seems like the paper by Kortschak, Loisel, and Ribereau I referenced above would be a good start, but they are only considering the ruin probabilities (which they make finite by making $c$ grow with time).
3. If the conjecture is true, what is the value of $\mu_{V}$? Is there an analytic expression? If not, can we at least say how it compares to $v_{\text{D}}$ and $v_{\text{P}}$?
4. How can one characterize the magnitude of ruin time $n_{\text{R}}$? For example, is there such a thing as the mean value of it? Note that this question is independent of whether my conjecture is true.

If you find it helpful, you may assume that $v_{\text{D}}$ ('D' for 'deterministic' or 'degenerate') and $v_{\text{P}}$ ('P for Pareto) have some particular values. For example, you might even assume that $v_{\text{D}}=v_{\text{P}}=1$.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulties are that (a) the mean is infinite and (b) ruin happens always, with probability one.¹ Given this, it is not clear how to interpret $c$, $\mu$, and $\eta$. In particular, it is not clear what principle should govern the choice of $\mu_{V}$. My conjecture below is, in part, an attempt to answer that question.

^{In the context of insurance, provided that the $V_{j}$'s have finite means $\mathbb{E}(V)$ and that the ruin probability isn't 1, the setting of $\mu_{V}=\mathbb{E}(V)$ and $\eta=0$ leads to premium rates $c$ that are break-even for the insurer. If $\eta$ is raised from zero, it leads to an expected net profit.}

1. Is my conjecture a good way to make precise the 'vague hypothesis'? Is there a better way?
2. Is the conjecture true? How would one prove it? It seems like the paper by Kortschak, Loisel, and Ribereau I referenced above would be a good start, but they are only considering the ruin probabilities (which they make finite by making $c$ grow with time).
3. If the conjecture is true, what is the value of $\mu_{V}$? Is there an analytic expression? If not, can we at least say how it compares to $v_{\text{D}}$ and $v_{\text{P}}$?
4. Is it actually true that the infinite-time ruin probability is 1? How would one prove it?
5. How can one characterize the magnitude of ruin time $n_{\text{R}}$? For example, is there such a thing as the mean value of it? Note that this question is independent of whether my conjecture is true.

In particular, the probability density function is given by $$\hspace{4em} f(v)=(1-q)\,\delta(v-v_{\text{D}})+q\,\theta(v-v_{\text{P}})\,\,\frac{({\sqrt{v_{\text{P}}}}/{2})}{v^{3/2}}\,,\hspace{4em}(2) $$ where $\delta$ is the Dirac delta function, $\theta$ is the Heaviside step function, $v_{\text{D}},\,v_{\text{P}}>0$, and $0<q\ll 1$.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulties are that (a) the mean is infinite and (b) ruin happens always, with probability 1 (I am almost sure that this is so—see below). Given this, it is not clear how to interpret $c$, $\mu$, and $\eta$. In particular, it is not clear what principle should govern the choice $\mu_{V}$. My conjecture below is, in part, an attempt to answer that question.

The Pareto distribution

In Eq. (2), I picked the Pareto shape parameter $\alpha$ to be $1/2$ (the pdf of the Pareto distribution is usually written as $\alpha\,x_{m}^{\alpha}/x^{\alpha+1}$ for $x\geqslant x_{m}$, and $0$ otherwise). But any $\alpha\in(0,1)$ would do, as it would also result in an infinite mean. (Of course, the mean is also infinite when $\alpha=1$. But the divergence is then logarithmic, which might make that case qualitatively different from the rest in some way.)

In particular, the probability density function, defined for $v\in [0,\,\infty)$, is given by $$\hspace{4em} f(v)=(1-q)\,\delta(v-v_{\text{D}})+q\,\theta(v-v_{\text{P}})\,\,\frac{({\sqrt{v_{\text{P}}}}/{2})}{v^{3/2}}\,,\hspace{4em}(2) $$ where $\delta$ is the Dirac delta function, $\theta$ is the Heaviside step function, $q$ is a real parameter such that $0<q\ll 1$, and the parameters $v_{\text{D}}$ and $v_{\text{P}}$ are positive real numbers.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulties are that (a) the mean is infinite and (b) ruin happens always, with probability one.¹ Given this, it is not clear how to interpret $c$, $\mu$, and $\eta$. In particular, it is not clear what principle should govern the choice $\mu_{V}$. My conjecture below is, in part, an attempt to answer that question.

^{1Disclaimer: I actually don't know for a fact that (b) is true, but I'm am almost certain that it is—see below.}

The Pareto distribution

In Eq. (2), I picked the Pareto shape parameter $\alpha$ to be $1/2$ (the pdf of the Pareto distribution is usually written as $\alpha\,x_{m}^{\alpha}/x^{\alpha+1}$ for $x\geqslant x_{m}$, and $0$ otherwise). But any $\alpha\in(0,1)$ would do, as it would also result in an infinite mean.²

^{2Of course, the mean is also infinite when $\alpha=1$. But the divergence is then logarithmic, which might make that case qualitatively different from the rest in some way.}

2. Note that the $V_{j}$'s are 'arriving' at regularly spaced intervals. This is different (and simpler) than in the standard Cramér–Lundberg model, where their arrival times instead come from a Poisson process (see e.g. here).
3. $u\geqslant 0$ is a fixed (i.e. independent of $n$) real number. I suspect that its value won't matter much for what interests me, and that it can be set to zero. I'm including it because people familiar with ruin theory will expect it.
4. $c=(\eta+1)\mu_{V}$.
5. $\mu_{V}>0$ is a fixed real number.
6. $\eta>0$ is a fixed real number.

The meaning of $\eta$ will become clear when I state my conjecture that is the actual subject of this post, below. Basically, it's a 'tolerance level' of sorts.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulty is that the mean is infinite, so it is not clear to what value $\mu_{V}$ should be set.

In the context of ruin theory,

We now introduce the notion of ruin time $\boldsymbol{n}_{\textbf{R}}$ (which in our case is a positive integer): for any given realization of the process $R_{n}$, $n_{\text{R}}$ is the lowest $n>0$ such that $R_{n}<0$.

In ruin theory, the basic question is what is the probability that ruin will occur in infinite time for a given $c$ and $u$. I don't have a proof, but I am fairly certain that this probability is 1 for the process in Eq. (1). I'm basing this on the fact that Kortschak, Loisel, and Ribereau ('Ruin Problems with Worsening Risks or with Infinite Mean Claims', Stoch. Model. 31, 119–152 (2015), here, see p. 120) say that in an actual risk process (meaning, a process in which the timing of claims comes from a Poisson process) in which the claim sizes are distributed according to a pure Pareto distribution with infinite mean, the infinite-time ruin probability is always 1.

One reason why I'm invoking ruin theory is that it helps to characterize what is meant by a 'catastrophic jump': it is a jump that results in ruin.

2. Note that the $V_{j}$'s are 'arriving' at regularly spaced intervals. This is different (and simpler) than in the standard Cramér–Lundberg model, where their arrival times instead come from a Poisson process (see e.g. here).
3. $u\geqslant 0$ is a fixed (i.e. independent of $n$) real number. I suspect that its value won't matter much for what interests me, and that it can be set to zero. I'm including it because people familiar with ruin theory will expect it.
4. $c=(\eta+1)\mu_{V}$.
5. $\mu_{V}>0$ is a fixed real number.
6. $\eta>0$ is a fixed real number.
7. If $R_{n}$ turns negative, we say that ruin has occured. (Intuitively, think of $R_{n}$ as the financial reserve of an insurance company at time $n$.)
8. For any given realization of the process $R_{n}$, the ruin time $\boldsymbol{n}_{\textbf{R}}$ (which in our case is a positive integer) is the lowest $n>0$ such that $R_{n}<0$.

The meaning of $\boldsymbol{\mu_{V}}$ and $\boldsymbol{\eta}$

The meaning of $\eta$ will become clear when I state my conjecture that is the actual subject of this post, below. Basically, it's a 'tolerance level' of sorts.

If the $V_{j}$'s had a finite mean, i.e. if the integral $\int_{0}^{\infty}x\,f(x)\,dx$ existed, then $\mu_{V}$ would be set to that mean. Our main difficulties are that (a) the mean is infinite and (b) ruin happens always, with probability 1 (I am almost sure that this is so—see below). Given this, it is not clear how to interpret $c$, $\mu$, and $\eta$. In particular, it is not clear what principle should govern the choice $\mu_{V}$. My conjecture below is, in part, an attempt to answer that question.

The Pareto distribution

In Eq. (2), I picked the Pareto shape parameter $\alpha$ to be $1/2$ (the pdf of the Pareto distribution is usually written as $\alpha\,x_{m}^{\alpha}/x^{\alpha+1}$ for $x\geqslant x_{m}$, and $0$ otherwise). But any $\alpha\in(0,1)$ would do, as it would also result in an infinite mean. (Of course, the mean is also infinite when $\alpha=1$. But the divergence is then logarithmic, which might make that case qualitatively different from the rest in some way.)

A bit more on the connection to ruin theory

In the context of ruin theory,

In ruin theory, the basic question is what is the probability that ruin will occur in infinite time for given values of $c$ and $u$; in other words, what is the probability that $n_{\text{R}}$ is finite. This probability is usually considered as a function of $u$ and denoted $\psi(u)$. I don't have a proof, but I am fairly certain that this probability is $\psi(u)=1$ for the process in Eq. (1), no matter what values $u$ and $c$ have. I'm basing this on the fact that Kortschak, Loisel, and Ribereau ('Ruin Problems with Worsening Risks or with Infinite Mean Claims', Stoch. Model. 31, 119–152 (2015), here, see p. 120) say that in an actual risk process (meaning, a process in which the timing of claims comes from a Poisson process) in which the claim sizes are distributed according to a pure Pareto distribution with infinite mean, the infinite-time ruin probability is always 1.

One reason why I'm formulating my question in terms of ruin theory is that it helps to characterize what is meant by a 'catastrophic jump': it is a jump that results in ruin.