endobj 272 0 obj This book is intended as a beginning text in stochastic processes for stu-dents familiar with elementary probability calculus. A change of measure of a stochastic process is a method of shifting the probability distribution into another probability distribution. << /S /GoTo /D (Outline0.18.2.144) >> endobj endobj << /S /GoTo /D (Outline0.4.2.30) >> 134 0 obj Stochastic Calculus for Finance Brief Lecture Notes Gautam Iyer Gautam Iyer, 2017. c 2017 by Gautam Iyer. endobj /Length 15 endobj 141 0 obj Proposition 2.4. A Brief Introduction to Stochastic Calculus 3 2 Stochastic Integrals We now discuss the concept of a stochastic integral, ignoring the various technical conditions that are required to make our de nitions rigorous. 210 0 obj endstream 246 0 obj 138 0 obj 78 0 obj p©WÝB¹àA™}k. 258 0 obj << /S /GoTo /D (Outline0.6.2.45) >> (A Mathematical Formulation of the Option Pricing Problem) (-Fields) 245 0 obj 118 0 obj endobj endobj endobj 22 0 obj (The General Case) 150 0 obj /Trans << /S /R >> endobj << What does given a s- eld mean? 170 0 obj Whether it’s to pass that big test, qualify for that big promotion or even master that cooking technique; people who rely on dummies, rely on it to learn the critical skills and relevant information necessary for success. << /S /GoTo /D (Outline0.16) >> << /S /GoTo /D (Outline0.9.1.77) >> << /S /GoTo /D (Outline0.8.2.70) >> u�G�\X%9D�%���ٷ�F��1+j�F�����˜h�Vޑ����V�.�DС��|nB��T������T���G�d������O��p�VD���u^})�GC�!���_0��^����t7h�W�س���E�?�y�n/��ߎ9A&=9T�+!�U9њ�^��5� $%�m�n0h��ۧ������L(�ǎ� ���f'q�u�|��ou��,g��3���Q.�D�����g�&���c��1b����Tv����R�� Taking limits of random variables, exchanging limits. 267 0 obj << /S /GoTo /D (Outline0.3.1.18) >> endobj (Martingales) 98 0 obj 101 0 obj /A << /S /GoTo /D (Navigation169) >> endobj (The General It\364 Stochastic Integral) (Basic Concepts from Probability Theory) (Linear It\364 SDE with Multiplicative Noise) �F)��r�Ӕ,&. 253 0 obj endobj endobj endobj 222 0 obj 162 0 obj %���� 149 0 obj 221 0 obj endobj stream Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equi­ librium," and in later papers he used the machinery of stochastic calculus to begin investigation of this issue. derivation of the Black-Scholes equation for option values Wenyu Zhang (Cornell) Ito’s Lemma May 6, 2015 3 / 21 endobj 61 0 obj endobj endobj endobj endobj << /S /GoTo /D (Outline0.5) >> << /S /GoTo /D (Outline0.4.1.25) >> endobj (The It\364 Lemma: Stochastic Analogue of the Chain Rule) endobj << /S /GoTo /D (Outline0.7.1.51) >> (Notations) Stochastic Integrals The stochastic integral has the solution ∫ T 0 W(t,ω)dW(t,ω) = 1 2 W2(T,ω) − 1 2 T (15) This is in contrast to our intuition from standard calculus. (A Short Excursion into Finance) (Simple Processes) endobj 77 0 obj /MediaBox [0 0 362.835 272.126] endobj (The Milstein Approximation) endobj endobj (Why does the Riemann-Stieltjes Approach fail?) /Resources 270 0 R endobj üÄ%òÓ_1š6œô\®l¨C!ÃFu‚ÂzYBĀ´Æ(ìWá&Tm§¦¡ð¦ÉÚoƒŠr¤%ƒ•Ÿq¸g¬ÝçfÇòcSƒ%´5 V2L¥L+1#»snÿjµlŒCN@ UT=¬Wä << /S /GoTo /D (Outline0.14.2.117) >> endobj 62 0 obj If the current closing price is 108, the stochastic is 80 -- that is, 100 times the result of 8 divided by 10. << /S /GoTo /D (Outline0.7.2.56) >> << /S /GoTo /D (Outline0.19.1.158) >> Jan.29: Stochastic processes in continuous time (martingales, Markov property). << /S /GoTo /D (Outline0.15.1.121) >> I learned the Ito’s lemma, but I can only use that to derive things, I don’t know how to integrate things with that; when others do it, especially when professors do it, it looks so easy and everything is a blur but when I need to integrate something by myself, I can’t. It has been called the fundamental theorem of stochastic calculus. /Font << /F25 275 0 R /F27 276 0 R >> 46 0 obj endobj In 1969, Robert Merton introduced stochastic calculus into the study of finance. 13 0 obj /D [267 0 R /XYZ 10.909 272.126 null] 178 0 obj endobj << /S /GoTo /D (Outline0.11) >> << /S /GoTo /D (Outline0.8) >> SDEs Consider the SDE X˙ (t) = FX(t)+BZ(t) This is a Langevin equation A problem is that we want to think of Z(t) as being the derivative of a Wiener process, but the Wiener process is Allow me to give my take on this question. endobj endobj stream (The Stratonovich and Other Integrals) endobj /Filter /FlateDecode endobj 130 0 obj 186 0 obj endobj endobj endobj endobj 161 0 obj (The It\364 Integral) 25 0 obj It is used to model systems that behave randomly. << STOCHASTIC CALCULUS 5 As H k2 n is F k2 n-measurable, it follows that H n t is previsible. (Conditional Expectation) << /S /GoTo /D (Outline0.3) >> x��UMs� ��W�њi,B��I�'�����N�,'�آ��!���V�I*ۇ�����.��Px;�Ad62Y�O�(. (The General Linear Differential Equation) Holding H(t) shares at each time tleads to a pro t of Z T 0 (1) H(t)S0(t)dt if Sis di erentiable, but in many cases it is not. << Chapters 1 to 4 4.1 Show that if Aand B belongs to the ˙-algebra Fthen also BnA 2F(for de nition of ˙-algebra, see De nition 1.3). Because X(t j) X(t j 1) is Normally distributed with mean zero and variance t=n, i.e. << /S /GoTo /D (Outline0.18.1.140) >> Used in Ito’s calculus, which extends the methods of calculus to stochastic processes Applications in mathematical nance e.g. Thus we begin with a discussion on Conditional Expectation. STOCHASTIC CALCULUS: BASIC TOPICS. 126 0 obj Since t n "tas n!1, it follows that H t n!H t as n!1by left-continuity. Recall that a stochastic process is a probability distribution over a set of paths. Because Brownian motion is nowhere differentiable, any stochastic process that is driven by Brownian motion is nowhere differentiable. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions. (Simulation via the Functional Central Limit Theorem) 49 0 obj By Lillian Pierson . /Rect [99.247 2.007 201.906 8.519] >> endobj << /S /GoTo /D (Outline0.19.3.163) >> 54 0 obj 190 0 obj 225 0 obj endobj endobj /Filter /FlateDecode (Numerical Solutions) endobj Geometric Brownian motion can be thought of as the stochastic analog of the exponential growth function. 241 0 obj 10 0 obj << /S /GoTo /D (Outline0.10.2.88) >> endobj endobj (The Forward Risk Adjusted Measure and Bond Option Pricing) (Homogeneous Equations with Multiplicative Noise) 266 0 obj 86 0 obj I enjoyed Peter’s answer and my answer will mostly be akin to his (minus all the equations). 280 0 obj (The It\364 Stochastic Integrals) << /S /GoTo /D (Outline0.6.1.42) >> /Length 506 endobj 109 0 obj /Filter /FlateDecode endobj 165 0 obj (Processes Related to Brownian Motion) Has been called the fundamental theorem of stochastic calculus deals with integration of a stochastic with... 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