# Proof of time translation invariance of Brownian Motion. Missing assumption?

Proposition: Let us consider a Brownian motion $$W(t)$$, $$t\geq0$$. For fixed $$t_0\geq0$$, the stochastic process $$\widetilde{W}(t)=W(t+t_0)-W(t_0)$$ is also a Brownian Motion.

Proof: Let us take properties $$1.$$, $$3.$$ and $$4.$$ in here for granted as to stochastic process $$\widetilde{W}(t)$$ and focus on the proof of property $$2.$$ for $$\widetilde{W}(t)$$. First, consider that for any $$s: $$\widetilde{W}(t)-\widetilde{W}(s)=W(t+t_0)-W(s+t_0)\tag{1}$$ To check property $$2.$$, we may assume that $$t_0>0$$.
Then, for any $$0\leq t_1, we have $$0. By property $$2.$$ for $$W(t)$$, $$W(t_k+t_0)-W(t_{k-1}+t_0)$$, $$k=1,2,\ldots,n$$ are independent random variables. Thus, by $$(1)$$, the random variables $$\widetilde{W}(t_k)-\widetilde{W}(t_{k-1})$$, $$k=1,2,\ldots,n$$ are independent and so $$\widetilde{W}(t)$$ satisfies property $$2.$$.

Starting from $$0, I would say that property $$2.$$ does not necessary apply for $$W_t$$ in correspondence of $$t=1$$, since, given the above assumptions, I cannot be sure that, considering $$W(t_1+t_0)-W(t_0+t_0)$$, $$(t_1+t_0)>(t_0+t_0)$$, that is, in other words, that $$W(t_1+t_0)-W(t_0+t_0)$$ can be actually considered as an increment in time.

So, I would say that there is a missing assumption so as that $$(1)$$ holds true for all $$k\geq1$$ and not just for $$k>1$$, that is $$t_1>t_0\tag{2}$$ Would you agree with me as to the fact that assumption $$(2)$$ is necessary so as that $$(1)$$ holds true for $$k=1$$? If not, why am I mistaken?

• We know, by assumption, that $t_0>0$. Indeed, suppose (for instance) $t_1=0$ and $t_0=1.2$. With such values, you would have that $(t_1+t_0)<(t_0+t_0)$, so $W(t_1+t_0)-W(t_0+t_0)$ would not be an increment in time of the Brownian motion $W$ @TSF Sep 25, 2020 at 13:12
• Sorry, I cannot get it. Could you please explicit why, with $k=1$, could I consider $W(t_1+t_0)-W(t_0+t_0)$ as an increment in time? @TSF Sep 25, 2020 at 14:14
• I just realized there is probably an error in your definition after all. Where did you find this "proof"? Sep 25, 2020 at 16:26
• It is from "Introduction to Stochastic Integration" by Kuo (2006). Are you referring to the same error I am pointing at? @TSF Sep 25, 2020 at 18:36
• It's more than that. Look at how increments are defined on the wikipedia page. Sep 26, 2020 at 19:02

I think you should just rename $$t_0$$ to be $$\tau$$ and see if it doesn't fix things. You seem to be mixing the translating in time with the increments in time themselves.
Proof: Let us take properties $$1.$$, $$3.$$ and $$4.$$ in here for granted as to stochastic process $$\widetilde{W}(t)$$ and focus on the proof of property $$2.$$ for $$\widetilde{W}(t)$$. First, consider that for any $$s: $$\widetilde{W}(t)-\widetilde{W}(s)=W(t+\tau)-W(s+\tau)\tag{1}$$ To check property $$2.$$, we may assume that $$\tau>0$$.
Then, for any $$0\leq t_0, we have $$0<\tau\leq t_0+\tau. By property $$2.$$ for $$W(t)$$, $$W(t_k+\tau)-W(t_{k-1}+\tau)$$, $$k=1,2,\ldots,n$$ are independent random variables. Thus, by $$(1)$$, the random variables $$\widetilde{W}(t_k)-\widetilde{W}(t_{k-1})$$, $$k=1,2,\ldots,n$$ are independent and so $$\widetilde{W}(t)$$ satisfies property $$2.$$.
• Hence, in my reference, according to your restating, there is "just" the omission of the red part below? $$0<t_0\color{red}{\leq t_0+t_0}\leq t_1+t_0<\ldots<t_n+t_0$$ @TSF Sep 27, 2020 at 14:31
• No, it's not the same as that, I already answered you in the other comment. That condition doesn't make sense. The issue is calling the translation time, $\tau$, by the name $t_0$. This makes it seem like the first increment is $t_0$ when it's really $t_1$. $t_0$ is not an increment, it is the starting place. Do you see $\tau+\tau$ anywhere in my answer? Sep 27, 2020 at 14:33
• So, if you are just replacing $t_0$ with $\tau$, I would just expect $0<\tau<t_1+\tau<\ldots<t_n+\tau$ and not $0<\tau\color{red}{\leq t_0+\tau}\leq t_1+\tau\leq\ldots<t_n+\tau$ @TSF Sep 27, 2020 at 14:38
• I included a $t_0$ since it's how it was used in the definition. If you want to start from $t_1$ that's fine too but then you need to consider only $k=2,3,\ldots,n$. Do you understand what the mistake was now? Labelling both the translation and the increments as $t$ with a subscript confuses what conditions have to hold. Sep 27, 2020 at 14:40
• Maybe I am on the way. Sorry for my confusion. So, If I refer the the original reference, could we say that $\color{red}{\text{red}}$ part and $\color{blue}{\text{blue}}$ part below are two different "objects"? $$\widetilde{W}(t)-\widetilde{W}(s)=W(t+\color{red}{t_0})-W(s+\color{red}{t_0})\tag{1}$$ $$0<\color{green}{t_0}\leq t_1+\color{blue}{t_0}<\ldots<t_n+\color{blue}{t_0}\tag{2}$$ Sep 27, 2020 at 14:59