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L15 - Lecture 15 Girsanov Theorem and Risk Neutral...

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Lecture 15 Girsanov Theorem and Risk Neutral Valuation Assume the basic setting in a BS market: a riskless asset B following (14.1), a risky asset S following (14.2) with the Brownian filtration {F t } , defined in a probability space (Ω , F , P ). 15.1 Value processes, self-financing strategies, arbitrage An adapted process h = { ( h 0 t , h 1 t ) : 0 t T } is called a dynamic portfolio in which h 0 t B t represents the balance of bank account B at time t and h 1 t is the number of shares of stock S at time t . In general, we should define predictability in continuous-time and require h to be a predictable process. However, it is not necessary here due to the continuous sample paths of Brownian motion W , i.e. if h is adapted to the Brownian filtration, then it is predictable. V = { V t } is called the value process of a portfolio h where V t = h 0 t B t + h 1 t S t , t [0 , T ] . (15.1) It is intuitive to call h a self-financing strategy if dV t = h 0 t dB t + h 1 t dS t (15.2) at every t . But a formal definition is required. Definition 15.1 Assume R T 0 E | h 0 t | dt + R T 0 Eh 2 1 t dt < . h is called a self-financing strategy if V t = V 0 + Z t 0 h 0 u dB u + Z t 0 h 1 u dS u (15.3) for 0 < t T with probability one.
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