Continuous Time - Brownian Motion

Continuous Time - Brownian Motion - 1 Temple University...

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Unformatted text preview: 1 Temple University Continuous Time Finance Brownian Motion 2 Continuous Time Finance Brownian Motion 3 Brownian Motion In 1827, while examining pollen grains suspended in water under a microscope, Brown observed minute particles executing a continuous jittery motion. Robert Brown 17731858 Jan Ingenhousz 1730-1799 One only has to place a drop of alcohol in the focal point of a microscope and introduce a little finely ground charcoal therein, and one will see these corpuscules in a confused, continuous and violent motion, as if they were animalcules which move rapidly around. 4 Brownian Motion Although atoms and molecules were still open to objection in 1905, Einstein predicted that the random motions of molecules in a liquid impacting on larger suspended particles would result in irregular, random motions of the particles, which could be directly observed under a microscope. Perrin did the experimental work to test Einstein's predictions that matter is made of atoms and molecules. Jean Baptiste Perrin 1870 1942 5 Brownian Motion 3.2.1-3.2.4 Symmetric Random Walks 6 Symmetric Random Walk 1,2,... k , if 1 if 1 .... 1 j j 3 2 1 = = =- = = = = k j j k j X M T H X 7 MATLAB Random Walk function paths = RandomWalk (p,NRandom) % rand('twister',0); paths = zeros(1,NRandom); for k =1:NRandom-1 if rand < p step = 1; else step = -1; end paths(1,k+1) = paths(1,k) + step; end plot(1:NRandom,paths(1,:)); 10 20 30 40 50 60 70 80 90 100-12-10-8-6-4-2 2 4 6 8 Increments 10 20 30 40 50 60 70 80 90 100-12-10-8-6-4-2 2 4 6 s) coin tosse g overlappin- non on (depend t independen are ) and ) increments The k integers choose l m k l M (M M (M m l-- < < 9 Bernoulli Random Variable ) 1 ( 4 ) 1 2 ( 1 ) ( 1 ) 1 ( ) 1 ( * 1 1 2 ) 1 )( 1 ( * 1 p- 1 1 p 1 2 2 2 2 2 2 p p p EX EX VarX p p EX p p p EX X- =-- =- = =-- + =- =-- + = - = 1 ) ( 1 ) 1 ( * 1 ) 1 ( * 1 1 1 2 2 2 1 2 2 1 2 2 2 1 2 1 2 1 2 1 =- = =- + = =- + = - = EX EX VarX EX EX X 10 Increments k l VarX M ar(M EX M (M X M (M j k l j k l j k l- = =- = =- =- < + = + = + = l 1 k j l 1 k j l 1 k j ) V ) E ) for l, k integers choose 11 Martingale Property K K k l K k k l k K k k l k K k l k l M M M M E M F M M E F M E F M M E F M M M E F M E = +- = +- = +- = +- = )] [( ] | ) [( ] | [ ] | ) [( ] | ) [( ] | [ 12 Quadratic Variation k M M M M k M M M M j k j j k j k j j k =- = =- =- =- = 2 1 1 2 1 1 ) ( ] , [ )) ( ) ( ( ) ( ] , [ 13 Brownian Motion 3.2.5-3.2.6 Scaled Random Walk 14 Scaled Random Walk 1,2,... k , if 1 if 1 .......
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Continuous Time - Brownian Motion - 1 Temple University...

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