Greestings! ]]>

Although the Caratheodory Measurable set condition doesn't work anymore, from $(1)$ it seems $m$ can be constructed iteratively from $\mathcal{E}$ to $\lambda{E}$.

]]>Could you provide a reference for construction of a Brownian bridge via random walks? I assume it would be by conditioning a random walk to end up near zero.

]]>Hi pascalcule,

This is just a simple consequence of normality. Hölder, Cauchy-Schwarz, etc, are not needed.

As *B_t – B_s* is normal with zero mean, it is equal to σX for a standard normal X. So, its p’th moment is σ^pE[X^p]. As was noted, σ^2 is bounded by a multiple of t – s, and E[X^p] is a constant (depending only on p), giving the inequality you quote.

I am a bit perplex on the way you derive the Hölder continuity of the Brownian bridge. Precisely this sentence : “For any times {0\le s\le t\le1} the covariance structure of a Brownian bridge shows that {B_t-B_s} has variance bounded by {t-s} and, hence,

“.

I know how to obtain this property thanks to Cameron Martin theorem but how do you get this inequality from the previous one on the order 2 moment? It is probably an inequality like Cauchy Schwarz or Hölder but I don’t see the right one.

Thanks again for your answer.

]]>Not in full generality. E.g., reflecting Brownian motion has discontinuous local time (at level 0), but is a Brownian motion with (singular) drift.

However, in many cases, Girsanov transforms show that continuity will hold if the drift is of the form , for locally square-integrable .