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FINM345A09Lecture6corrected

# FINM345A09Lecture6corrected - 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 6 (Corrected Post-Lecture) More Compound Jump-Diffusion Calculus 6:30-9:30 pm, 02 November 2009, Kent 120 in Chicago 7:30-10:30 pm, 02 November 2009 at UBS Stamford 7:30-10:30 am, 03 November 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 Lecture6–page1 Floyd B. Hanson

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Normal-Uniform Hybrid Marks: { Continuing unfinished part of Lecture 5. } The very, very thin tails of the normal is consequence of the insistence on infinite domain for exact integrals and for using the large number of statistical tests and methods available. Just truncating the normal to finite range does not fatten the tails noticeably. However, an alternate idea is the combine the truncated normal and the uniform distribution, i.e., φ (nuq) Q ( q )= p u b - a + p n φ n ( q ; μ n , σ 2 n ) Φ n ( a, b ; μ n , σ 2 n ) U ( q ; ( a, b )) , (6.1) where a< 0 <b , Φ n ( a, b ; μ n , σ 2 n ) is the distribution on ( a, b ) , while p u and p n are the respective uniform and normal probabilities such that p u + p n =1 . FINM 345/Stat 390 Stochastic Calculus Lecture6–page2 Floyd B. Hanson
MATLAB Mark Simulations: Uniform on (a, < b) : Q (uq) = a+(b-a) * rand = unifrnd(a,b) . Normal for (mu,sigma) : Q (nq) = mu+sigma * randn = normrnd(mu,sigma) . Double-Uniform for (a < 0 < b;p1) : Q (duq) = binornd(1,p1) * unifrnd(a,0) +(1-binornd(1,p1)) * unifrnd(0,b) . Double-Exponential for (mu1 < 0 < mu2;p1) : Q (deq) = binornd(1,p1) * exprnd(-mu1) +(1-binornd(1,p1)) * exprnd(mu2) . Normal-Uniform for (a < 0 < b;mu,sigma,pu) : Q (nuq) = binornd(1,pu) * unifrnd(a,b)+(1-binornd(1,pu)) * AcceptedOnly { normrnd(mu,sigma) (a,b) } . For fixed probability values, the binornd(1,p * ) can be replaced by just p * , where p * = p1 or pu . FINM 345/Stat 390 Stochastic Calculus Lecture6–page3 Floyd B. Hanson

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FinM 345 Stochastic Calculus: 6. More Compound Jump-Diffusion Calculus: 6.1. State-Dependent Compound Jump-Diffusions: (Beginning the section corresponding to the nonlinear mark-jump-diffusions and linear simulations cancelled in Lecture 5.) 6.1.1 State-Dependent Generalizations for Compound Poisson: The space-time Poisson process is generalized to include state-dependence with X ( t ) in both the jump-amplitude and the Poisson measure, such that the jump-amplitude counter is d Π( t ; X ( t ) , t )= Q h ( X ( t ) , t, q ) P ( dt , dq; X ( t ) , t ) (6.2) on the Poisson mark space Q with Poisson random measure P ( dt , dq; X ( t ) , t ) , which helps to describe the space-time Poisson mechanism and related calculus. FINM 345/Stat 390 Stochastic Calculus Lecture6–page4 Floyd B. Hanson
The space-time state-dependent Poisson mark, Q = q , is again the underlying random variable for the state-dependent and mark-dependent jump-amplitude coefficient h ( x, t, q ) . The double time t arguments of d Π , dP and P are not considered redundant for modeling applications, since the first time t or right-continuous time set dt = [ t, t + dt ) is the usual Poisson jump process implicit time dependence, similarly dq = [ q, q +

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