Dynamic Markov Bridges and Market Microstructure by Umut Çetin & Albina Danilova

Author:Umut Çetin & Albina Danilova
Language: eng
Format: epub
ISBN: 9781493988358
Publisher: Springer New York

3.its quadratic covariation with β is 0.

Observe that this construction is a generalisation of the above static bridge with random terminal condition. In particular, the last condition is analogous to V  being independent of B.

A naive approach would be to consider the following SDE:

which has a solution given by

It is easy to see that the first term above converges to β 1 while the second tends to 0. Also by construction the quadratic covariation of X with β is 0. However, X is not a Brownian motion in its own filtration. One way to see this is to use Theorem 3.​4 in conjunction with Corollary 3.​1. Another way is to observe that a process satisfying all of the conditions above cannot exist.

Indeed, if X is a process with properties 1)–3) as above, then

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