# Early estimate the sign of the drift in a generalized Wiener process

I posting here my problem, perhaps somebody can point me how to proceed further :)

[The challenge]

I have an electronic system that can be modeled as a Wiener process with a drift $$\mu$$:

$$X_t = \mu t + \sigma W_t$$

$$E[X_t] = \mu t$$ , $$Var[X_t] = \sigma \sqrt t$$

where the value of the drift $$\mu$$ is unknown and can be either positive or negative. The index $$t$$ can be seen as the time evolving form the initial time $$t_0$$, where $$X_0 = 0$$.

In my physical system I can only observe the $$X_t$$ evolution over time $$t$$.

My goal is to minimize the amount of time $$t_{detection}$$, required to estimate the sign of the drift $$\mu$$, with a given confidence interval $$CI$$.

[The current (naive) approach]

The system signal-to-noise-ratio is: $$SNR(t) = \frac{\mu \sqrt t}{3 \sigma}$$ for a CI of $$3 \sigma$$ = 99.7%

Intuitively, if I wait long enough, I could reach any arbitrary SNR level as the signal power is proportional to $$t$$, whereas the noise power only to the $$\sqrt t$$.

However, in my scenario my $$t_{detection}$$ is constrained, and with the current values I am off by a factor of x100 outside specification with this approach.

[The question]

Is there a way to reduce the $$t_{detection}$$ time for the same $$CI$$ confidence interval?

I am not familiar with Kalman filter, but I suspect it may help here? Or perhaps by looking at the derivate of $$X_t$$?

[Remarks]

You are talking to an analog designer. Please bear with me... and thank you for any help!

• Is $\mu$ constant? And just to see if I got it right, you are after the CI of the estimate? – A_A Jun 24 '19 at 6:34
• look at en.m.wikipedia.org/wiki/Sequential_probability_ratio_test. do you know the standard deviation? – user28715 Jun 24 '19 at 15:39
• yes! $\mu$ is constant. yes, I'm looking for the estimate and its CI. Also the standard deviation is known. – groviere Jun 26 '19 at 20:28