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Unformatted text preview: 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) 6:309:30 pm, 05 October 2009, Kent 120 in Chicago 7:3010:30 pm, 05 October 2009 at UBS Stamford 8:3011:30 am, 06 October 2009 at Spring in Singapore FINM 345/Stat 390 Stochastic Calculus — Lecture2–page1 — Floyd B. Hanson Continuation of Lecture 1: 1.4. TimeDependent (NonHomogeneous) Poisson Process: • Financial markets are very timedependent, so modelers need to think critically about constant coefficient models, understanding that in some cases timedependence 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 ) = Z t λ ( s ) ds. FINM 345/Stat 390 Stochastic Calculus — Lecture2–page2 — Floyd B. Hanson • Then, the Poisson parameter increment is defined by ΔΛ( t ) ≡ Λ( t + Δ t ) Λ( t ) = Z t +Δ t t λ ( s ) ds. Thus, ΔΛ( t ) ∼ λ ( t )Δ 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 (Λ [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 ≥ and Δ t ≥ . ( Hint: In MATLAB, 1: n =[ j ] 1 × n is a rowvector. ) FINM 345/Stat 390 Stochastic Calculus — Lecture2–page3 — Floyd B. Hanson • Note that all three Poisson processes are increment processes, even Δ P 3 ( t ) = P ( t ) = P ( t ) P (0) , where P (0) ≡ . Also, Λ( t ) is continuous as integrals with λ ( t ) > for t > . • 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. 2223, 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 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 ) , ≤ s < t ] = M ( s ) , with some technical side conditions in probability space...
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This note was uploaded on 04/19/2010 for the course FIN 390 taught by Professor Hansen during the Fall '09 term at University of Illinois, Urbana Champaign.
 Fall '09
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