Here is the theory behind my joint probability approximation (JPA) tool

I do not have a formal background in statistics and maybe less so in calculus. Since my new tool, what I am calling the joint probability approximation for lifetime ruin (JPA)[1], matches the output of the Kolmogorov equation pretty well, involves the combination of two PDFs for longevity and portfolio ruin, looks and smells like a practical application of  probability theory and integral calc, and at the end of the day works pretty darn well, I was guessing that it had some good theory behind it. It looks like this below (e.g., equation 20) is the theory which makes more sense now that I've coded it (but made no sense at all the first time I read it a couple months ago). This post, by the way, is for those few that might actually be interested in this kind of thing -- which might not exactly include the family and friends and otherwise supportive others that surround me here in FL[2]:

[This is from section 2 (The Probability of Lifetime Ruin. pages 2-5) of Ruined Moments in Your Life: How Good Are the Approximations?  by H. Huang, M. Milevsky and J. Wang York University1, Toronto, Canada 1 October 2003]

I had read this piece a few months back when I was trying to understand ruin and the Kolmogorov equation. I have to say that then and now most of this goes over my head.  On the other hand I will also say that now, having done it in practice in real life, it seems like it is actually quite a bit easier to do than the formal notation might imply.  I mean it took about an hour or two and one (one!) page of code to do what theorem/proof three above seems to imply.  And now that I've done that project, the calculus is a little easier to read and the aside about g_w(∞) makes perfect sense -- where before I would have no idea what he was talking about -- because that is precisely one of the things I did to make it work.

Now...if anyone wants to license my code, I'll warm up my ABA number for you.

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[1] I'm calling it joint because I am multiplying two probabilities and summing the product over a range so it seems like the right thing to say.  If there is a better way to describe this formally let me know; reread my first sentence.

[2] feel free to email me if you are interested. I'm just winging down here in la la land. I need a little help here and there and now and then with the large body of subject matter where I often find myself flying blind.