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FINM345A09Lecture2Corr - FinM 345/Stat 390 Stochastic...

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FinM 345/Stat 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: [email protected] Master of Science in Financial Mathematics Program University of Chicago Lecture 2 (Corrected October 6, 2009) 6:30-9:30 pm, 05 October 2009, Kent 120 in Chicago 7:30-10:30 pm, 05 October 2009 at UBS Stamford 8:30-11:30 am, 06 October 2009 at Spring in Singapore FINM 345/Stat 390 Stochastic Calculus Lecture2–page1 Floyd B. Hanson
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Continuation of Lecture 1: 1.4. Time-Dependent (NonHomogeneous) Poisson Process: Financial markets are very time-dependent, so modelers need to think critically about constant coefficient models, understanding that in some cases time-dependence of coefficients may be difficult to estimate, but perhaps not much more difficult to analyze. Thus, consider λ = λ ( t ) so the Poisson process P(t) will be nonstationary . Thus, the Poisson parameter differential is d Λ( t ) λ ( t ) dt , while the integral parameter, assuming Λ(0) = 0 as in the constant jump rate case, is Λ( t ) = t 0 λ ( s ) ds. FINM 345/Stat 390 Stochastic Calculus Lecture2–page2 Floyd B. Hanson
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Then, the Poisson parameter increment is defined by ΔΛ( t ) Λ( t + Δ t ) - Λ( t ) = t t t λ ( s ) ds. Thus, ΔΛ( t ) λ ( t t only when Δ t 1 , i.e., is small, but if not the integral must be used. The temporal Poisson distributions Prob[ dP ( t ) = k ] = p k [1:3] ( t )) for the three cases Δ P [1:3] ( t ) = [ dP ( t ) , Δ P ( t ) , P ( t )] and parameters ΔΛ [1:3] ( t ) = [ d Λ( t ) , ΔΛ( t ) , Λ( t )] , are the same Φ Δ P i ( t ) ( k ; ΔΛ i ( t )) = e - ΔΛ i ( t ) (ΔΛ i ( t )) k k ! , for i = 1:3 and k = 0, 1, 2, . . . jumps, t 0 and Δ t 0 . ( Hint: In MATLAB, 1: n =[ j ] 1 × n is a row-vector. ) FINM 345/Stat 390 Stochastic Calculus Lecture2–page3 Floyd B. Hanson
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Note that all three Poisson processes are increment processes, even Δ P 3 ( t ) = P ( t ) = P ( t ) - P (0) , where P (0) 0 . Also, Λ( t ) is continuous as integrals with λ ( t ) > 0 for t > 0 . While the basic statistics for the set of Poisson increment processes are similar to the simple constant rate case, i.e., E[Δ P i ( t )] = ΔΛ i ( t ) = Var[Δ P i ( t )] . However, the exponential distribution of the interjump times are much more complicated, but see Hanson’s (2007), pp. 22-23, and cited background references. Some theory becomes more complicated in the nonstationary case, but changing the clock from t to Λ( t ) , by changing the rate to the constant one, will remove most of the difficulties (see text, p. 22). FINM 345/Stat 390 Stochastic Calculus Lecture2–page4 Floyd B. Hanson
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1.5. Martingale Properties of Markov Processes — Expectations Conditioned on the Past: Simple Definition 1.7: A martingale M(t) is a stochastic process that principally satisfies E[ M ( t ) | M ( s ) , 0 s < t ] = M ( s ) , with some technical side conditions in probability space that M(t) is absolutely integrable , i.e., E[ | M ( t ) | ] < on [0,T] for some finite horizon time T < . (Comment: The term Martingale comes from horse racing and abstractly symbolizes a fair game since E [ M ( t ) - M ( s ) | M ( s )] = 0 , 0 s < t, i.e., there being no net gain on the average conditioned on past data. Alternately, E M ( t ) | M ( t )] = 0 , t 0 .) FINM 345/Stat 390 Stochastic Calculus Lecture2–page5 Floyd B. Hanson
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