FINM345A09Lecture3_1 - FINM 345/Stat 390 Stochastic...

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FINM 345/Stat 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: fhanson@uchicago.edu Master of Science in Financial Mathematics Program University of Chicago Lecture 3 (from Singapore) Diffusion Stochastic Calculus 6:30-9:30 pm, 12 October 2009, Kent 120 in Chicago 7:30-10:30 pm, 12 October 2009 at UBS Stamford 7:30-10:30 am, 13 October 2009 at Spring in Singapore FINM 345/Stat 390 Stochastic Calculus Lecture3–page1 Floyd B. Hanson
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FINM 345 Stochastic Calculus: 3. Jump-Diffusion Basic Stochastic Chain Rules: 3.1. Diffusion Calculus Basic Chain Rules: Recall most basic dt –precision rule : ( dW ) 2 ( t ) dt = dt . Higher order zero examples from summary Table 2.1.1: ( dW ) 3 ( t ) dt = 0 , dtdW ( t ) dt = 0 , ( dt ) 2 dt = 0 , etc. Preliminary forms for increments of general function G ( t ) G ( W ( t ) ,t ) : Δ G ( W ( t ) ) G ( W ( t t ) t ) - G ( W ( t ) ) , so with Δ W ( t ) W ( t + Δ t ) - W ( t ) , Δ G ( W ( t ) )= G ( W ( t )+Δ W ( t ) t ) - G ( W ( t ) ) . Similarly for the differential, dG ( W ( t ) G ( W ( t )+ dW ( t ) + dt ) - G ( W ( t ) ) . FINM 345/Stat 390 Stochastic Calculus Lecture3–page2 Floyd B. Hanson
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3.1.1. Special Diffusion Examples with G = G ( W ( t )) : Itˆo Cubic Integral using binomial expansion: Δ[ W 3 ]( t ) =( W W ) 3 ( t ) - W 3 ( t ) bin = (3 W 2 Δ W +3 W W ) 2 +(Δ W ) 3 )( t ) , so in the dt –precision limit as Δ t dt 0 , d ± W 3 ² ( t ) dt = ( 3 W 2 ( t ) dW ( t )+3 W ( t ) dt ) , where the 3 wdt is the Itˆo correction to the deter- ministic differential d ( w 3 ) = 3 w 2 dw . Solving for W 2 ( t ) dW ( t ) and Itˆo integrating, Z t t 0 W 2 ( s ) dW ( s ) int = dt 1 3 ( W 3 ( t ) - W 3 ( t 0 ) ) - Z t t 0 W ( s ) ds, where ³ int = dt ´ means integrating using dt –precision limits, a formal version of IMS equality ³ ims = ´ . FINM 345/Stat 390 Stochastic Calculus Lecture3–page3 Floyd B. Hanson
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Note that the Itˆo integral has only been reduced to an explicit exact term plus a Riemann integral of W ( t ) . Itˆo General Power Integral using the full binomial ex- pansion theorem (Online Appendix B.150), Δ ± W m +1 ² ( t ) = ( W W ) m +1 ( t ) - W m +1 ( t ) bin = m X i =0 m + 1 i ! W i ( t ) ( Δ W ) m +1 - i ( t ) , where the passage to the limit as Δ t dt 0 + and the dt –precision limits leading to the Itˆo integral form, Z t 0 W m ( s ) dW ( s ) and its reduction to an exact integral plus Riemann in- tegral has been left as an Exercise. FINM 345/Stat 390 Stochastic Calculus Lecture3–page4 Floyd B. Hanson
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Itˆo Exponential Integral : Using the laws of exponen- tials (LOE) and the first few terms of the exponential expansion (B.53), going directly to the formal differen- tial form and skipping the more general increment form to expedite applied stochastic calculations, d ± e W ² ( t )= ( e W + dW - e W ) ( t ) loe = ( e W ( e dW - 1 )) ( t ) dt = ³ e W ³ dW + 1 2 ( dW ) 2 ´´ ( t ) , neglecting differential forms that are zero in dt precision limit, such as dW 3 ( t ) dt = 0 , dtdW ( t ) dt = 0 , ( dt ) 2 dt = 0 and higher powers. FINM 345/Stat 390 Stochastic Calculus Lecture3–page5 Floyd B. Hanson
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Using the basic mean square limit differential form, ( dW ) 2 ( t ) dt = dt , so d ± e W ² ( t ) dt = ³ e W ³ dW + 1 2 dt ´´ ( t ) .
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FINM345A09Lecture3_1 - FINM 345/Stat 390 Stochastic...

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