The natural logarithm of the short rate in the given equation is normally distributed. This is because the equation represents a time-dependent version of the Vasicek model, which assumes that the short rate is normally distributed. The Vasicek model is a mathematical model that describes the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives, and it is named after Oldrich Vasicek, who introduced it in 1977. The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In finance and economics, the normal distribution is often used to describe, at least approximately, any variable that tends to cluster around the mean.

Choice A is incorrect. Although the model is referred to as a lognormal model, it does not imply that the distribution of natural logarithm short rates follows a lognormal distribution. The term "lognormal" in this context refers to the fact that the model uses the natural logarithm of short rates, not their distribution.

Choice C is incorrect. The standard normal distribution would only be applicable if there were no time-dependent factors in our equation and if our data was standardized (i.e., mean of 0 and standard deviation of 1). However, this is not the case here as we have time-dependent factors and no information about standardization.

Choice D is incorrect. The Bernoulli distribution applies to binary outcomes (success/failure or 1/0) which does not fit with our continuous variable – natural logarithm short rates.