Vigirdas Mackevicius,
V Mackevicius
John Wiley & Sons
e druk, 2011
9781848213111
Introduction to Stochastic Analysis – Integrals and Differential Equations
Integrals and Differential Equations
Specificaties
Gebonden, 276 blz.
|
Engels
John Wiley & Sons |
e druk, 2011
ISBN13: 9781848213111
Rubricering
Onderdeel van serie
ISTE
Verwachte levertijd ongeveer 16 werkdagen
Samenvatting
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes.
The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.
Specificaties
ISBN13:9781848213111
Taal:Engels
Bindwijze:gebonden
Aantal pagina's:276
Uitgever:John Wiley & Sons
Serie:ISTE
Inhoudsopgave
<p>Preface 9</p>
<p>Notation 13</p>
<p>Chapter 1. Introduction: Basic Notions of Probability Theory 17</p>
<p>1.1. Probability space 17</p>
<p>1.2. Random variables 21</p>
<p>1.3. Characteristics of a random variable 21</p>
<p>1.4. Types of random variables 23</p>
<p>1.5. Conditional probabilities and distributions 26</p>
<p>1.6. Conditional expectations as random variables 27</p>
<p>1.7. Independent events and random variables 29</p>
<p>1.8. Convergence of random variables 29</p>
<p>1.9. Cauchy criterion 31</p>
<p>1.10. Series of random variables 31</p>
<p>1.11. Lebesgue theorem 32</p>
<p>1.12. Fubini theorem 32</p>
<p>1.13. Random processes 33</p>
<p>1.14. Kolmogorov theorem 34</p>
<p>Chapter 2. Brownian Motion 35</p>
<p>2.1. Definition and properties 35</p>
<p>2.2. White noise and Brownian motion 45</p>
<p>2.3. Exercises 49</p>
<p>Chapter 3. Stochastic Models with Brownian Motion and White Noise 51</p>
<p>3.1. Discrete time 51</p>
<p>3.2. Continuous time 55</p>
<p>Chapter 4. Stochastic Integral with Respect to Brownian Motion 59</p>
<p>4.1. Preliminaries. Stochastic integral with respect to a step process 59</p>
<p>4.2. Definition and properties 69</p>
<p>4.3. Extensions 81</p>
<p>4.4. Exercises 85</p>
<p>Chapter 5. Itô s Formula 87</p>
<p>5.1. Exercises 94</p>
<p>Chapter 6. Stochastic Differential Equations 97</p>
<p>6.1. Exercises 105</p>
<p>Chapter 7. Itô Processes 107</p>
<p>7.1. Exercises 121</p>
<p>Chapter 8. Stratonovich Integral and Equations 125</p>
<p>8.1. Exercises 136</p>
<p>Chapter 9. Linear Stochastic Differential Equations 137</p>
<p>9.1. Explicit solution of a linear SDE 137</p>
<p>9.2. Expectation and variance of a solution of an LSDE 141</p>
<p>9.3. Other explicitly solvable equations 145</p>
<p>9.4. Stochastic exponential equation 147</p>
<p>9.5. Exercises 153</p>
<p>Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155</p>
<p>10.1. Introduction 155</p>
<p>10.2. Backward and forward Kolmogorov equations 161</p>
<p>10.3. Stationary density of a diffusion process 172</p>
<p>10.4. Exercises 176</p>
<p>Chapter 11. Examples 179</p>
<p>11.1. Additive noise: Langevin equation 180</p>
<p>11.2. Additive noise: general case 180</p>
<p>11.3. Multiplicative noise: general remarks 184</p>
<p>11.4. Multiplicative noise: Verhulst equation 186</p>
<p>11.5. Multiplicative noise: genetic model 189</p>
<p>Chapter 12. Example in Finance: Black Scholes Model 195</p>
<p>12.1. Introduction: what is an option? 195</p>
<p>12.2. Self–financing strategies 197</p>
<p>12.3. Option pricing problem: the Black Scholes model 204</p>
<p>12.4. Black Scholes formula 206</p>
<p>12.5. Risk–neutral probabilities: alternative derivation of Black Scholes formula 210</p>
<p>12.6. Exercises 214</p>
<p>Chapter 13. Numerical Solution of Stochastic Differential Equations 217</p>
<p>13.1. Memories of approximations of ordinary differential equations 218</p>
<p>13.2. Euler approximation 221</p>
<p>13.3. Higher–order strong approximations 224</p>
<p>13.4. First–order weak approximations 231</p>
<p>13.5. Higher–order weak approximations 238</p>
<p>13.6. Example: Milstein–type approximations 241</p>
<p>13.7. Example: Runge Kutta approximations 244</p>
<p>13.8. Exercises 249</p>
<p>Chapter 14. Elements of Multidimensional Stochastic Analysis 251</p>
<p>14.1. Multidimensional Brownian motion 251</p>
<p>14.2. Itô s formula for a multidimensional Brownian motion 252</p>
<p>14.3. Stochastic differential equations 253</p>
<p>14.4. Itô processes 254</p>
<p>14.5. Itô s formula for multidimensional Itô processes 256</p>
<p>14.6. Linear stochastic differential equations 256</p>
<p>14.7. Diffusion processes 257</p>
<p>14.8. Approximations of stochastic differential equations 259</p>
<p>Solutions, Hints, and Answers 261</p>
<p>Bibliography 271</p>
<p>Index 273</p>
<p>Notation 13</p>
<p>Chapter 1. Introduction: Basic Notions of Probability Theory 17</p>
<p>1.1. Probability space 17</p>
<p>1.2. Random variables 21</p>
<p>1.3. Characteristics of a random variable 21</p>
<p>1.4. Types of random variables 23</p>
<p>1.5. Conditional probabilities and distributions 26</p>
<p>1.6. Conditional expectations as random variables 27</p>
<p>1.7. Independent events and random variables 29</p>
<p>1.8. Convergence of random variables 29</p>
<p>1.9. Cauchy criterion 31</p>
<p>1.10. Series of random variables 31</p>
<p>1.11. Lebesgue theorem 32</p>
<p>1.12. Fubini theorem 32</p>
<p>1.13. Random processes 33</p>
<p>1.14. Kolmogorov theorem 34</p>
<p>Chapter 2. Brownian Motion 35</p>
<p>2.1. Definition and properties 35</p>
<p>2.2. White noise and Brownian motion 45</p>
<p>2.3. Exercises 49</p>
<p>Chapter 3. Stochastic Models with Brownian Motion and White Noise 51</p>
<p>3.1. Discrete time 51</p>
<p>3.2. Continuous time 55</p>
<p>Chapter 4. Stochastic Integral with Respect to Brownian Motion 59</p>
<p>4.1. Preliminaries. Stochastic integral with respect to a step process 59</p>
<p>4.2. Definition and properties 69</p>
<p>4.3. Extensions 81</p>
<p>4.4. Exercises 85</p>
<p>Chapter 5. Itô s Formula 87</p>
<p>5.1. Exercises 94</p>
<p>Chapter 6. Stochastic Differential Equations 97</p>
<p>6.1. Exercises 105</p>
<p>Chapter 7. Itô Processes 107</p>
<p>7.1. Exercises 121</p>
<p>Chapter 8. Stratonovich Integral and Equations 125</p>
<p>8.1. Exercises 136</p>
<p>Chapter 9. Linear Stochastic Differential Equations 137</p>
<p>9.1. Explicit solution of a linear SDE 137</p>
<p>9.2. Expectation and variance of a solution of an LSDE 141</p>
<p>9.3. Other explicitly solvable equations 145</p>
<p>9.4. Stochastic exponential equation 147</p>
<p>9.5. Exercises 153</p>
<p>Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155</p>
<p>10.1. Introduction 155</p>
<p>10.2. Backward and forward Kolmogorov equations 161</p>
<p>10.3. Stationary density of a diffusion process 172</p>
<p>10.4. Exercises 176</p>
<p>Chapter 11. Examples 179</p>
<p>11.1. Additive noise: Langevin equation 180</p>
<p>11.2. Additive noise: general case 180</p>
<p>11.3. Multiplicative noise: general remarks 184</p>
<p>11.4. Multiplicative noise: Verhulst equation 186</p>
<p>11.5. Multiplicative noise: genetic model 189</p>
<p>Chapter 12. Example in Finance: Black Scholes Model 195</p>
<p>12.1. Introduction: what is an option? 195</p>
<p>12.2. Self–financing strategies 197</p>
<p>12.3. Option pricing problem: the Black Scholes model 204</p>
<p>12.4. Black Scholes formula 206</p>
<p>12.5. Risk–neutral probabilities: alternative derivation of Black Scholes formula 210</p>
<p>12.6. Exercises 214</p>
<p>Chapter 13. Numerical Solution of Stochastic Differential Equations 217</p>
<p>13.1. Memories of approximations of ordinary differential equations 218</p>
<p>13.2. Euler approximation 221</p>
<p>13.3. Higher–order strong approximations 224</p>
<p>13.4. First–order weak approximations 231</p>
<p>13.5. Higher–order weak approximations 238</p>
<p>13.6. Example: Milstein–type approximations 241</p>
<p>13.7. Example: Runge Kutta approximations 244</p>
<p>13.8. Exercises 249</p>
<p>Chapter 14. Elements of Multidimensional Stochastic Analysis 251</p>
<p>14.1. Multidimensional Brownian motion 251</p>
<p>14.2. Itô s formula for a multidimensional Brownian motion 252</p>
<p>14.3. Stochastic differential equations 253</p>
<p>14.4. Itô processes 254</p>
<p>14.5. Itô s formula for multidimensional Itô processes 256</p>
<p>14.6. Linear stochastic differential equations 256</p>
<p>14.7. Diffusion processes 257</p>
<p>14.8. Approximations of stochastic differential equations 259</p>
<p>Solutions, Hints, and Answers 261</p>
<p>Bibliography 271</p>
<p>Index 273</p>