FINM345A09Lecture4CorrSims - 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 4 (from Singapore) Jump & Jump-Diffusion Stochastic Calculus 6:30-9:30 pm, 19 October 2009, Kent 120 in Chicago 7:30-10:30 pm, 19 October 2009 at UBS Stamford 7:30-10:30 am, 20 October 2009 at Spring in Singapore Copyright c ± 2009 by the Society for Industrial and Applied Mathematics, and Floyd B. Hanson. FINM 345/Stat 390 Stochastic Calculus Lecture4–page1 Floyd B. Hanson
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FinM 345 Stochastic Calculus: 4. Jump & Jump-Diffusion Stochastic Calculus: 4.1. Poisson Jump Calculus Basic Chain Rules: The Poisson process is quite different from the continuous diffusion process, primarily because of the discontinuity property of the Poisson process and the property that multiple jumps are highly unlikely during small increments in time Δ t . 4.1.1. Jump Calculus Rule for h ( dP ( t )) : Thus, the most basic rule is the zero-one law (ZOL) for jumps for dP ( t ) in precision- dt compact differential form, ( dP ) m ( t ) dt = zol dP ( t ) , (4.1) provided the integer m 1 , the case m =0 being trivial. FINM 345/Stat 390 Stochastic Calculus Lecture4–page2 Floyd B. Hanson
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An immediate generalization of this law is the following corollary. Corollary 4.1 Zero-One Jump Law for h ( dP ( t )) : h ( dP ( t )) dt = zol h (1) dP ( t ) + h (0)(1 - dP ( t )) , (4.2) with probability one, provided the function h ( p ) is right-continuous, such that values h (0) and h (1) exist and are bounded. Proof : This follows by simple substitution of the zero-one jump law , h ( dP ( t )) dt = zol h (1) , dP ( t )=1 h (0) , dP ( t )=0 dt = zol h (1) dP ( t )+ h (0)(1 - dP ( t )) , dP ( t ) = 0 or dP ( t ) = 1 with probability one to precision- dt . ± FINM 345/Stat 390 Stochastic Calculus Lecture4–page3 Floyd B. Hanson
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Formally, the differential dP ( t ) can be treated as a condition to test whether there has been a jump . This form (4.2) of the zero-one law suggests another extension of the jump function definitions (B.178-B.179). For example, recall in (B.185) for a jump in a transformation F of the state process X at t 1 , [ F ]( X ( t 1 ) ,t 1 ) = F ( X ( t + 1 ) + 1 ) - F ( X ( t - 1 ) - 1 ) . Definition 4.1. Jump Function [ h ]( dP ( t )) : [ h ]( dP ( t )) dt = zol h ( dP ( t )) - h (0) (4.3) to precision- dt , provided h ( p ) is right-continuous, such that values h (0) and h ( dP ( t )) exist and are bounded. FINM 345/Stat 390 Stochastic Calculus Lecture4–page4 Floyd B. Hanson
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With this definition, version (4.2) of the zero-one law can immediately be written. Corollary 4.2 Zero-One Jump Law for h ( dP ( t )) with Jump Function : h ( dP ( t )) dt = zol h (0) + [ h ]( dP ( t )) (4.4) in terms of the jump function [ h ]( dP ( t )) . Alternatively, the jump function is written as [ h ]( dP ( t )) dt = zol ( h (1) - h (0)) dP ( t ) . (4.5) FINM 345/Stat 390 Stochastic Calculus Lecture4–page5 Floyd B. Hanson
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4.1.2. Jump Calculus Rule for H ( P ( t ) ,t ) : Equations (4.4 – 4.5) are a primitive differential chain rule for functions of only the Poisson differential dP ( t ) .
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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.

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FINM345A09Lecture4CorrSims - FinM 345/Stat 390 Stochastic...

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