I just came across this stochastic process (link):

$dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is bounded between zero and one.

Is there any discrete time analog of this process? I tried:

$Y_{t+1} = (a-bY_t) + c \sqrt{Y_t(1-Y_t)}w_{t+1}$, where $w_t$ is a normal random variable with mean zero and variance 1, but I guess this process might jump of bounds...


2 Answers 2


I believe that the process you postulate has a Beta conditional distribution. If my memory serves me well, I have encountered it in the book by Liptser and Shiryayev "Statistics of Random Processes" as the evolution of the conditional probability in a HMM. This was 10+ years ago, therefore I might be well off.

In that case you should be sampling from Beta to discretize


My mistake, the stationary distribution is Beta, not the conditional one. Therefore you will not be able to evolve from Beta exactly. The diffusion you postulate is called 'Jacobi diffusion', see Forman and Sørensen, case 6, at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1150110

I suspect that you might be able to use the stationary PDF to produce an approximate scheme to discretize.

Update 2:

Actually, let me change the notation slightly and write $$d Y_t = \theta (\mu-Y_t)\ dt + \sqrt{2\alpha\theta Y_t(1-Y_t)}\ dB_t$$ which we know has a stationary distribution $Y_\infty\sim B\left(\frac{\mu}{\alpha}, \frac{1-\mu}{\alpha}\right)$.

Now, use the change of variable $X_t = f(Y_t) = 2 \arcsin\sqrt{Y_t}$, which leads to the diffusion $$dX_t = \frac{\theta (\mu -\alpha/2) - (\alpha-1) \theta \sin^2 (X_t/2)}{|\sin X_t|}\ dt + \sqrt{2\alpha\theta}\ dB_t$$

You can simulate this as $$x_{t+\delta} = x_t + \frac{\theta (\mu -\alpha/2) - (\alpha-1) \theta \sin^2 (x_t/2)}{|\sin x_t|}\ \delta + \sqrt{2\alpha\theta\delta}\ \epsilon_t,\quad \epsilon_t\sim N(0,1)$$ and then transform back to produce the paths $$y_t = \sin^2 (x_t/2)$$


You are not going to get a process which stays within bounds if your increments are normal random variables, which have an unbounded distribution.

You probably want to look at some kind of random walk, where the increment is a discrete distribution. In other words you have a finite list of values which you add or subtract, each with a positive probability. The distribution could change based on $t$ and the previous value of $Y_t$.

One approach which I think is very promising is to add some fraction of the difference between $Y_t$ and either 0 or 1, depending on the direction.

Eg $Y_{t+1} = (a' - b'Y_t) + c'(1 - Y_t)$ with probability $p(t, Y_t)$, and $Y_{t+1} = (a' - b'Y_t) - c'Y_t$ with probability $1 - p(t, Y_t)$. It should be easy to find a function $p(t, Y_t)$ such that the variance of $Y_{t+1} - Y_t$ is exactly what you want, and then to adjust $a'$ and $b'$ to make the mean of $Y_{t+1} - Y_t$ the same or similar to the continuous case.


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