Continuous financial processes revisited
Photo credit: Reynaldo #brigworkz Brigantty
The discovery of prior work
I first became interested in this topic in 2016 and derived all of the results in the loans and FIRE/investing sections between 2016 and 2019. At that time, it seemed to me odd that these results were not more commonly known. Odder still, they seemed not to have even been previously discovered despite not requiring math beyond a second or third year engineering curriculum. Back then, I searched and I did not find anything resembling prior work. In 2026, I discovered most of the results here had indeed been previously discovered and published. A 1968 paper by Richard Beckwith wonderfully titled “Continuous financial processes” [1] is exactly what I might have written were I a better author and writing for an academic journal instead of a blog. It captures all the key ideas beautifully almost exactly 50 years before I put any effort into the matter. This paper is criminally under appreciated having only 4 citations, none of which are directly related to or building on his work.
It is also worth noting that some extremely similar results were shown in the context of annuities pricing as early as 1869, 1870 [2,3]. These results were never (to my knowledge) explicitly applied to loans but they are similar enough to count as prior work. Beckwith [1] makes no mention of them as references, likely because he was unaware of their existence. More recently, I have found a couple books [4,5,6] that contain the many of the key results in the loans and investing sections. Once again, these were published before I started working on such problems but I was unaware of it until a decade later.
In a burst of exhuberance, when I discovered the Beckwith paper, I rewrote the loans and investing sections of this blog into a more academic format for the Journal of Financial and Quantitative Analysis (the original journal which published the Beckwith paper). It was only after I finished that process that I was reminded of the cost prohibitive nature of academic publishing. Seeing as I do not have thousands of dollars around for article processing charges, I have elected to share the resulting PDF here for anyone who would prefer it in that abbreviated format.
Basic mathematics
The following formula is called a linear first order ordinary differential equation (ODE):
$$ \frac{dy}{dt} = a(t) y(t) + b(t) $$In the special case where \(b(t) = 0\), we call it a homogeneous ODE. In the special case where \(a(t) = \text{constant}\) and \(b(t) = \text{constant}\), we call it an autonomous ODE. Because autonomous and homogeneous ODEs are separable, they are relatively easy to solve. By separable, we mean we can rearrange the ODE so all the \(y\) is on one side and all the \(t\) is on the other.
$$\begin{align} \frac{dy}{dt} &= a y + b \\ \frac{dy}{a y + b } &= dt\\ \int \frac{dy}{a y + b } &= \int dt\\ \frac{1}{a} \log(a y + b) &= t + c \\ y(t) &= c e^{at} - \frac{b}{a} \end{align}$$Everything in the loans and FIRE sections of this blog are exhaustive explorations of this equation using different boundary conditions or numerical approximations. If it seems repetitive, that is because it is.
Normally when using mathematics to model the world, the numerical solution is an approximation of a continuous process. However, finance is one of the few domains where the numerical solution is the “true” answer and the continuous solution is the approximation. I dedicated one entire article to showing the error of this approximation is negligible in all relevant conditions.
References
[1] “Continuous Financial Processes,” Journal of Financial and Quantitative Analysis, vol. 3, pp. 113-133, 1968.
[2] “On an Improved Theory of Annuities and Assurances,” Journal of the Institute of Actuaries and Assurance Magazine, vol. 15, pp. 95-125, 1869.
[3] “A Table for determining the Amounts, &c., of Continuous Annuities Certain,” Journal of the Institute of Actuaries and Assurance Magazine, vol. 15, pp. 432—446, 1870.
[4] Introduction to Financial Mathematics, 1 Ed., 2015.
[5] “Chapter 1.6,” in Introduction to Financial Mathematics, 2 Ed., pp. 90-93, 2025.
[6] Financial Mathematics: A Practical Guide for Actuaries and Other Business Professionals, 2 Ed., 2005.