So, in the last video, we spoke about a hypothetic scenario in which the exponential as a growth that we obtained in the geometric Brownian motion would hold only approximately for sufficiently small times, while for longer times, it would approach a constant rather than infinity due to some saturation effects. Now, I would like to show you such a model. This model is known in physics and biology as the Verhulst model and it's shown in equation eight on the slide. It's a model for a variable Xt that can describe, for example, the size of the population as a function of time. This equation is what is called an ordinary differential equation. Let's assume that we want to use this equation to describe stock prices instead. So, that now Xt is the stock price and we rewrite this equation as shown in the second form in equation eight. Now, we can say that the whole expression in parentheses can be interpreted as a state dependent return. This return shown here in equation nine has a state dependence that it depends linearly on Xt. It has a negative coefficient in front of it. This describes the effect of saturation. In an original Verhulst whose model, saturation in the growth of population occurs because the population competes for the same limited resource such as food, water, territory, and so on. We could imagine a world in which a similar thing could happen to an equity value as an effect of saturation in the market. We can think of some effects of saturation in the market failure of a stock. So, that the market will eventually stabilize around a certain regime determined by this saturation rules. We can think of market saturation in conditions when capital supply to the market keeps a constant phase that can be, in particular, zero or non-zero, and no new information is available to the market anymore. in this case, equation nine suggests that returns in such model will be diminishing. They would approach zero in finite time. We will see how exactly it all plays out a bit later, but before doing that, let me pause for a moment and discuss a special case of equation eight. Let's take another look at the Verhulst model that I repeat on the slide. Later on, we will extend this model to negative values of parameter Kappa, but for now we assume that Kappa is positive here. Now, let's assume that values of Xt that we are interested in are much smaller than the ratio Theta over Kappa, and in this case, we can neglect Xt relatively to the first term in the parentheses in the right hand side of the Verhulst equation. This means that for such regime of small fields, our dynamics is approximately linear. It's shown here in equation 10. Now, this looks exactly the same as our previous equation in the GBM model when we turn the noise off. This is an equation of a linear growth where now parameter Theta plays the role over some of a risk free rate RF and the weighted sum of signal Zt as we had before. If we now integrate this equation, we again obtain an exponential growth, as shown on the right of this equation. So, this looks the same as in the classical financial equilibrium models, but the key difference is that in a Verhulst model, such exponential growth only applies at an initial stage of the dynamics only. Let's assume that we just started looking at the given population and parameters, Theta and Kappa are fixed. In addition, the initial value Xnot is much less than the ratio, Theta over Kappa. Then, the dynamics for short enough times would appear linear to us. This linear approximation should be good as long as running value Xt obtained as a solution to this linear approximation is still much smaller than the ratio, Theta over Kappa. So, for short times and until this condition is violated as a result of the growth, the dynamics on this model will look exactly like an exponential growth. We can also look at the opposite limit, when Xt is much larger than the ratio, Theta over Kappa. In this case, in this region, we can neglect this ratio in parenthesis and an approximation to the dynamics will be as shown in equation 12. This equation can be easily solved and the solution is shown in the right hand side of the same equation. It shows that the solution approaches a constant that we call C here as time goes to infinity. By looking back at the equation, we can figure out that this constant is actually equal to the fraction, Theta over Kappa. We can also make another interesting observation here. We can try to ensure the regime when X is much larger than Theta over Kappa by redefining the variable X. So, we can set X to be equal to the ratio, Theta/Kappa plus a new variable Yt, which we can assume to be large. Now, we can substitute this back to the equation of the model and get a corresponding equation for a variable Yt, and this equation is shown here in equation 14. So, what we can see here is it's almost the same as the original equation, but the sign of Theta is now reversed. This suggests that some interesting symmetries are present in the model, and we will talk more about symmetries of this model and similar models in our follow-up videos.
