Furthermore it can be shown that given  we have :

, which when depicted using diagrams looks like that shown in Figure 6.6. In simple word it implies that it is the present value which only dictates the future and the values from the past have no bearing on the future. Thus it is only the present which effects the future.

Figure 6.6: A hypothetical example of present affecting the future

Thus given what ever has happened in the past, only the present, i.e.,  will effect the future, i.e., , where . Hence the distribution of the future, given the past and present is equal to the distribution of the future given the present only and this is what we mean by the no-memory process.

Few important facts about Brownian motion

1. Sample paths every where are continuous, but no where are they differentiable.
2. Any level may be attained with probability 1.
3. The expected time taken to reach any level is .