Here are some interesting things that have come up in my research. suppose there is an SDE of the form [$]dX(t)=\mu(t) X^{\beta}(t) dt + \sigma(t) X^{\gamma}(t) dz(t)[$] that we want to simulate with this new Ito-Taylor density simulation method. We noticed that a lot of SDEs of this form could easily be simulated with this method after transformation into Bessel coordinates. While some non-linear SDEs like stochastic volatility SDEs could not be simulated in a satisfactory manner. Here I try to explain this anomaly and I will like to state a few observations.

When we transformed SDEs of the type given above in SDE equation into Bessel coordinates, the numerically calculated first derivative of transformed density with respect to normal variable[$]\frac{\partial Y}{\partial Z}[$] when calculated from density mass equality, remains almost constant and does not take a significant second derivative. Here Y represents Bessel transformed variable from SDE variable X. For all those SDEs that become linear after converting to Bessel coordinates, we can use several methods including the method in my original program. Here the criteria for linearity is that [$]\frac{\partial Y}{\partial Z}[$] is almost constant and [$]\frac{{\partial}^{2} Y}{\partial {Z}^2}[$] is almost negligible. In the equations that are non-linear like stochastic volatility equation that takes a non-negligible second derivative with normal variable, we need to use this numerically calculated second or higher derivatives with possibly hermite polynomials to appropriately calculate existing variance and innovations(I earlier called it simulation variance) variance and their ratios. We cannot easily use the original coordinates since in the original coordinates, the SDE variable continues to take many higher derivatives with normal. I am sure with a slightly better understanding of the method, we could also use original coordinates for density simulation as well. again the reason for my being able to successfully simulate the densities of many simpler linear processes was that when I converted them into transformed coordinates, all higher derivatives with respect to normal variable disappeared and only the first derivative remained. In general, after the appropriate transformations, the number of significant higher derivatives with respect to normal variable sharply decreases. I believe that these numerically calculated higher derivatives of transformed densities with normal variable have to be used with variance of hermite polynomials to satisfactorily calculate the existing and innovations variance.