Complements on Brownian Motion
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In the first part of this chapter, we present the definition of local time and the associated Tanaka formulae, first for Brownian motion, then for more general continuous semi-martingales. In the second part, we give definitions and basic properties of Brownian bridges and Brownian meander. This is motivated by the fact that, in order to study complex derivative instruments, such as passport options or Parisian options, some knowledge of local times, bridges and excursions with respect to BM in particular and more generally for diffusions, is useful. We give some applications to exotic options, in particular to Parisian options.
KeywordsBrownian Motion Local Time Implied Volatility Stochastic Volatility Model Local Martingale
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