# SCF-2006-L10 - Stochastic Calculus in Finance Stochastic...

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Stochastic Calculus in Finance Stochastic Integration

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Simple Process i. all for ) E(Z and ) ,..., 1 , ( the to adapted is } {Z sequence The t if Z T t if C such that ) ,..., 1 , (Z variables random of sequence a and ... 0 : partition a exist There : properties following the satisfied it if simple be to said is ]) , 0 [ , (C C process stochastic The ) 0 , ( to adapted is 0) t , (X X if Motion Brownian to adapted be to said is X process A process. Motion Brownian the for filtration natural the be 0 t ), , ( Let 2 i t i 1 - i i t i 1 1 0 n t t t t 1 - i < = < = = = = < < < < = = = = - n i t t Z n i T t t t t T t t t s B i n n n s τ σ
Examples of Simple Process process. simple a is ... t 0 partition a for ,..., 1 , t if , Z T t if , C process The process. simple a is hence function, step a is ,..., 1 , n 1 - i if , n 1 - i T t if , 1 ) ( f function tic determinis The 1 1 0 1 - i i t n 1 1 T t t t n i t t B B Z n i n i t n n t n n i t t n i n = < < < < = = < = = = = = < = - = - - -

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Ito integral for Simple Processes = = = - = - - T 0 1 1 s ) ( C by given is T] [0, on C process simple a of integral stochastic Ito The 1 1 n i n i i i t t t s B Z B B C dB i i i

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Ito Stochastic Integral = - = = - + = = - 0 1 i i 1 1 0 ] , 0 [ t 0 s 1 - k 0 Z where ) ( ) ( C by given is t t], [0, on C process simple a of integral stochastic Ito The 1 B B B Z B Z dB s I C dB t t i k i t t k i i T s t s s k k
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## This note was uploaded on 05/07/2009 for the course SCF scf taught by Professor Scf during the Spring '09 term at Indian Institute Of Management, Ahmedabad.

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SCF-2006-L10 - Stochastic Calculus in Finance Stochastic...

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