# How would you show that Brownian motion is Markov only using the fact that the Brownian bridge is Markov?

Try to forget everything you know about Brownian motion.

You are told there is a process called Brownian bridge on $$[0,1]$$. It is Gaussian with mean $$0$$ and covariance $$s(1-t)$$ for $$s and can be written as $$B_t=W_t - tW_1$$ where $$(W_t)$$ is a continuous process with $$W_0=0$$, called Brownian motion. Of course, from what you were told about the Brownian bridge, you know that $$(W_t)$$ is Gaussian with mean $$0$$ (this is not true, see remark of John Dawkins and EDIT below).

You are told $$(B_t)$$ is Markov with regard to its own filtration. How would you show that $$(W_t)$$ is Markov with regard to its own filtration ?

EDIT: In light of the remark of John Dawkins, we should indeed say that $$(W_t)$$ is itself Gaussian with mean $$0$$, otherwise, other processes than the real Brownian motion could be solution of $$B_t = W_t - t W_1$$.

• The law of Brownian bridge is the regular conditional distribution of Brownian motion. Maybe u can use this to calculate the probability and then show the Markov property. May 31, 2021 at 3:06
• You cannot use that as you know nothing about Brownian motion besides the information above. May 31, 2021 at 13:27
• N.B. The covariance of the Brownian bridge is $s(1-t)$ for $0\le s\le t\le 1$. May 31, 2021 at 15:42
• Thank you, I corrected the typo May 31, 2021 at 16:03

1. You don't need to be told that $$(B_t)$$ is Markov; this follows from the form of the covariance function.
2. Why does it follow from $$B_t=W_t-tW_1$$, $$0\le t\le 1$$, that $$(W_t)$$ is Gaussian, or even that $$E[W_t]=0$$?
3. If $$(W_t)$$ is one solution of $$B_t=W_t-tW_1$$, $$0\le t\le 1$$, then so is $$\tilde W_t:=W_t+tZ$$, where $$Z$$ is any random variable on the probability space where $$B$$ and $$W$$ are defined. If $$Z$$ were to depend on the "future" of $$W$$ in some way, this might disturb the Markov property of $$\tilde W$$.
• You are right, I should say that $(W_t)$ is Gaussian, otherwise we can indeed create counter-examples, thank you! May 31, 2021 at 17:58