alan bain - stochastic calculus

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www.t r adi n g- sof t war e- col l ect i on .comi i i .gon ch@gmai l .com, Sk ype: i i i .gon ch FFOORR SSAALLEE && EEXXCCHHAANNGGEE wwwwww..ttrraaddiinngg--ssooffttwwaarree--ccoolllleeccttiioonn..ccoomm S Su ub bs sc cr ri ib be e f fo or r F FR RE EE E d do ow wn nl lo oa ad d m mo or re e s st tu uf ff f. . M Mi ir rr ro or rs s: : w ww ww w. .f fo or re ex x- -w wa ar re ez z. .c co om m w ww ww w. .t tr ra ad de er rs s- -s so of ft tw wa ar re e. .c co om m C Co on nt ta ac ct ts s a an nd dr re ey yb bb br rv v@ @g gm ma ai il l. .c co om m a an nd dr re ey yb bb br rv v@ @h ho ot tm ma ai il l. .c co om m a an nd dr re ey yb bb br rv v@ @y ya an nd de ex x. .r ru u S Sk ky yp pe e: : a an nd dr re ey yb bb br rv v I IC CQ Q: : 7 70 09 96 66 64 43 33 3 www.t r adi n g- sof t war e- col l ect i on .comi i i .gon ch@gmai l .com, Sk ype: i i i .gon ch1. IntroductionThe following notes aim to provide a very informal introduction to Stochastic Calculus, andespecially to the Ito integral. They owe a great deal to Dan Crisans Stochastic Calculusand Applications lectures of 1998; and also much to various books especially those ofL. C. G. Rogers and D. Williams, and Dellacherie and Meyer. They have also benettedfrom lecture notes of T. Kurtz. They arent supposed to be a set of lecture notes; andits important to remember this if you are reading this as a part III student with a viewto being examined in this course. Many important parts of the course have been omittedcompletely.The present notes grew out of a set of typed notes which I produced when revisingfor the Part III course; combining the printed notes and my own handwritten notes intoa consistent text. Ive subsequently expanded them inserting some extra proofs from agreat variety of sources. The notes principally concentrate on the parts of the course whichI found hard; thus there is often little or no comment on more standard matters; as asecondary goal they aim to present the results in a form which can be readily extendedDue to their evolution, they have taken a very informal style; in some ways I hope thismay make them easier to read.The addition of coverage of discontinuous processes was motivated by my interest inthe subject, and much insight gained from reading the excellent book of J. Jacod andA. N. Shiryaev.The goal of the notes in their current form is to present a fairly clear approach to the Itointegral with respect to continuous semimartingales. The various alternative approaches tothis subject which can be found in books tend to divide into those presenting the integraldirected entirely at Brownian Motion, and those who wish to prove results in completegenerality for a semimartingale. Here at all points clarity has hopefully been the main goalhere, rather than completeness; although secretly the approach aims to be readily extendedto the discontinuous theory. I make no appology for proofs which spell out every minutedetail, since on a rst look at the subject the purpose of some of the steps in a proofoften seems elusive. Id especially like to convince the reader that the Ito integral isnt thatmuch harder in concept than the Lebesgue Integral with which we are all familiar. Themotivating principle is to try and explain every detail, no matter how trivial it may seemonce the subject has been understood!Passages enclosed in boxes are intended to be viewed as digressions from the maintext; usually describing an alternative approach, or giving an informal description of whatis going on feel free to skip these sections if you nd them unhelpful.These notes contain errors with probability one. I always welcome people telling meabout the errors because then I can x them! I can be readily contacted by email asafrb2@statslab.cam.ac.uk. Also suggestions for improvements or other additions arewelcome.Alan Bain[i]www.t r adi n g- sof t war e- col l ect i on .comi i i .gon ch@gmai l .com, Sk ype: i i i .gon ch2. Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . i2. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii3. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 13.1. Probability Space . . . . . . . . . . . . . . . . . . . . . . . 13.2. Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 14. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.1. Stopping Times . . . . . . . . . . . . . . . . . . . . . . . 45. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.1. Local Martingales . . . . . . . . . . . . . . . . . . . . . . 86. Total Variation and the Stieltjes Integral . . . . . . . . . . . . . 116.1. Why we need a Stochastic Integral . . . . . . . . . . . . . . . 116.2. Previsibility . . . . . . . . . . . . . . . . . . . . . . . . . 126.3. Lebesgue-Stieltjes Integral . . . . . . . . . . . . . . . . . . . 137. The Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.1. Elementary Processes . . . . . . . . . . . . . . . . . . . . . 157.2. Strictly Simple and Simple Processes . . . . . . . . . . . . . . 158. The Stochastic Integral . . . . . . . . . . . . . . . . . . . . . 178.1. Integral for H L and M /2 . . . . . . . . . . . . . . . . 178.2. Quadratic Variation . . . . . . . . . . . . . . . . . . . . . 198.3. Covariation . . . . . . . . . . . . . . . . . . . . . . . . . 228.4. Extension of the Integral to L2(M) . . . . . . . . . . . . . . . 238.5. Localisation . . . . . . . . . . . . . . . . . . . . . . . . . 268.6. Some Important Results . . . . . . . . . . . . . . . . . . . . 279. Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . 2910. Relations to Sums . . . . . . . . . . . . . . . . . . . . . . . 3110.1. The UCP topology . . . . . . . . . . . . . . . . . . . . . 3110.2. Approximation via Riemann Sums . . . . . . . . . . . . . . . 3211. Itos Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.1. Applications of Itos Formula . . . . . . . . . . . . . . . . . 4011.2. Exponential Martingales . . . . . . . . . . . . . . . . . . . 4112. Levy Characterisation of Brownian Motion . . . . . . . . . . . 4613. Time Change of Brownian Motion . . . . . . . . . . . . . . . 4813.1. Gaussian Martingales . . . . . . . . . . . . . . . . . . . . 4914. Girsanovs Theorem . . . . . . . . . . . . . . . . . . . . . . 5114.1. Change of measure . . . . . . . . . . . . . . . . . . . . . 5115. Stochastic Dierential Equations . . . . . . . . . . . . . . . . 5316. Relations to Second Order PDEs . . . . . . . . . . . . . . . . 5816.1. Innitesimal Generator . . . . . . . . . . . . . . . . . . . . 5816.2. The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . 5916.3. The Cauchy Problem . . . . . . . . . . . . . . . . . . . . 6116.4. Feynman-Kac Representation . . . . . . . . . . . . . . . . . 63[ii]www.t r adi n g- sof t war e- col l ect i on .comi i i .gon ch@gmai l .com, Sk ype: i i i .gon chContents iii17. Stochastic Filtering . . . . . . . . . . . . . . . . . . . . . . . 6517.1. Signal Process . . . . . . . . . . . . . . . . . . . . . . . 6517.2. Observation Process . . . . . . . . . . . . . . . . . . . . . 6517.3. The Filtering Problem . . . . . . . . . . . . . . . . . . . . 6517.4. Change of Measure . . . . . . . . . . . . . . . . . . . . . 6617.5. The Unnormalised Conditional Distribution . . . . . . . . . . . 7217.6. The Zakai Equation . . . . . . . . . . . . . . . . . . . . . 7317.7. Kushner-Stratonowich Equation . . . . . . . . . . . . . . . . 7918. Discontinuous Stochastic Calculus . . . . . . . . .

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