# Best estimate for Stochastic difference equation

On the subject of Stochastic differential equations. If we consider the difference equation $$\Delta x(t_n) = x(t_n) \Delta t + f(t_n) \Delta t$$ where we consider $f(t_n) \Delta t$, the driving term as a Wiener process (random process). Given the value of $x$ at time $t_n$, the value at time $t_{n+1}$ is then $$x(t_n + \Delta t) = x(t_n) + \Delta x(t_n) \\= x(t_n) + x(t_n) \Delta t + f(t_n) \Delta t$$ In a text it states that "upon obtaining $\Delta x (t_n)$, the best esitmate of $x$ is $\frac{\Delta x(t_n)}{\Delta t}$". Can anyone see what is meant by this?

Thanks.

• Do you see your question as more largely based in the probability / statistics aspect (stochastic DE), or in the math? I'm wondering if this would be a better fit on the Mathematics SE site. Aug 2 '17 at 14:05

You have $x(t_n)+\Delta x(t_n)=x(t_n)+x(t_n)\Delta t+f(t_n)\Delta t$, so that:
$$\frac{\Delta x(t_n)}{\Delta t}=x(t_n)+f(t_n).$$
Usually Weiner processes are defined to have mean 0, so that the left hand side is indeed the best estimate of $x(t_n)$.
• Thanks for your response, so you are interpretting 'best estimate' to mean the quantity which has the same mean as the value $x(t_n)$? Aug 4 '17 at 11:00