Comparing Probability Estimates and Standard Errors for Diffusion Maxima

# Compare errors in estimating a probability


# Compare errors in estimating a probability












Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion.

Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ the maximum of $X_t$ over $[t_{j-1}, t_{j}]$ and $M_{i,j}$ the maximum of $X_t$ over $[t_{i}, t_{j}]$and $H$ a constant. Let $\epsilon_j$ the error made in estimating $P[M_j<H]$ and define two quantities $S_1 = P[M_{1,N}<H] = \prod_{j=1}^{N}P[M_j<H]$ and $S_2 = P[M_{U,N}<H] = \prod_{j=U}^{N}P[M_j<H]$. The error in estimating $S_1$ is $\prod_{j=1}^{N}\epsilon_j$ and the error in estimating $S_2$ is $\prod_{j=U}^{N}\epsilon_j$.

Could we compare the estimation error of $S_1$ and $S_2$ ?

## Answer by Dave Harris (score 0, accepted)

https://quant.stackexchange.com/a/67838

You can’t compare them meaningfully because specific errors don’t have a meaning. For example, imagine two sets index 1 to U and 1 to N, U<N, of coin flips. If U=10 and the first ten coin tosses are all heads and N=1000000, and 500,000 are heads, there are different errors, but they are due to chance alone. The differences in errors do not provide information about the data generating process.

However, you are stipulating the process. The population parameter you are looking for is the standard error. Except for scaling effects caused by the impact of time on the diffusion process, the standard error will be identical in both cases. The sample estimate of the standard error will differ, but it is the population parameter that you care about.

Because the first set is a subset of the second, there is no information in the first that is not in the second, and the second contains more information, so the first estimation should not be performed. You would be throwing away information.

Your problem involves extreme value theory. The diffusion process does matter because the measurement is sensitive to the nature of the overall equation. A more detailed answer isn’t possible without greater specification about the nature of the equation.

You can use sample statistics to test or estimate population parameters; the individual sample statistics do not have any comparative meaning since they are inherently wrong according to your specification. Since your model is in the real numbers, the probability that your model with estimates in lieu of true values will be correct is inherently a measure zero event. As such, unless you actually know the parameters, the value of one sample versus another is uninteresting.