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Unformatted text preview: J. Wissel Financial Engineering with Stochastic Calculus I Fall 2010 Assignment Sheet 5 1. Let W ( t ), t ≥ 0 be a Brownian motion for a filtration ( F t ) t ≥ and set M ( t ) = max ≤ u ≤ t W ( u ) . a) Show that the two-dimensional process X ( t ) = ( W ( t ) ,M ( t ) ) is a Markov process for the filtration ( F t ) t ≥ , but the process M ( t ) is not. b) Let Y ( t ) = R t W ( u ) du . Is Y ( t ) is a Markov process for the filtration ( F t ) t ≥ ? Justify your answer. Remark. You may use that the process Z ( t ) = W ( t + s )- W ( s ), t ≥ 0, is independent of F s . 2. Consider a continuous-time market with a bond and a stock. Prices at time t ≥ 0 are given by stochastic processes B ( t ) and S ( t ) adapted to a filtration ( F t ) t ≥ . We assume that B ( t ) > 0. Trading takes place only at times 0 = t < t 1 < ... < t m = T , and we assume that we hold η ( t j ) units of bond and Δ( t j ) units of stock during [ t j ,t j +1 ) for all j ≥ 0. We trade in a self-financing way, which means0....
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- Brownian Motion, Stochastic process, Markov process, J. Wissel, continuous-time trading strategies