## An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and MedicineThis book is a systematic, rigorous, and self-consistent introduction to the theory of continuous-time stochastic processes. But it is neither a tract nor a recipe book as such; rather, it is an account of fundamental concepts as they appear in relevant modern applications and literature. We make no pretense of it being complete. Indeed, we have omitted many results, which we feel are notdirectly relatedtothemain themeorthatare availablein easilyaccessible sources. (Thosereaderswhoareinterestedinthehistoricaldevelopmentofthe subject cannot ignore the volume edited by Wax (1954). ) Proofs are often omitted as technicalities might distract the reader from a conceptual approach. They are produced whenever they may serve as a guide to the introduction of new concepts and methods towards the app- cations; otherwise, explicit references to standard literature are provided. A mathematically oriented student may ?nd it interesting to consider proofs as exercises. The scope of the book is profoundly educational, related to modeling re- world problems with stochastic methods. The reader becomes critically aware oftheconceptsinvolvedincurrentappliedliterature,andismoreoverprovided with a ?rm foundation of the mathematical techniques. Intuition is always supported by mathematical rigor. Our book addresses three main groups: ?rst, mathematicians working in a di?erent ?eld; second, other scientists and professionals from a business or academic background; third, graduate or advanced undergraduate students of a quantitative subject related to stochastic theory and/or applications. |

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### Contents

3 | |

8 | |

13 Expectations | 15 |

14 Independence | 19 |

15 Conditional Expectations | 26 |

16 Conditional and Joint Distributions | 35 |

17 Convergence of Random Variables | 41 |

18 Exercises and Additions | 44 |

44 Kolmogorov Equations | 185 |

45 Multidimensional Stochastic Differential Equations | 194 |

46 Stability of Stochastic Differential Equations | 196 |

47 Exercises and Additions | 203 |

The Applications of Stochastic Processes | 209 |

Applications to Finance and Insurance | 210 |

51 ArbitrageFree Markets | 212 |

52 The Standard BlackScholes Model | 216 |

Stochastic Processes | 51 |

22 Stopping Times | 58 |

23 Canonical Form of a Process | 59 |

24 Gaussian Processes | 60 |

25 Processes with Independent Increments | 61 |

26 Martingales | 63 |

27 Markov Processes | 72 |

28 Brownian Motion and the Wiener Process | 90 |

29 Counting Poisson and Lévy Processes | 102 |

210 Marked Point Processes | 111 |

211 Exercises and Additions | 118 |

The Itô Integral | 127 |

32 Stochastic Integrals as Martingales | 139 |

33 Itô Integrals of Multidimensional Wiener Processes | 143 |

34 The Stochastic Differential | 146 |

35 Itôs Formula | 149 |

36 Martingale Representation Theorem | 150 |

37 Multidimensional Stochastic Differentials | 152 |

38 Exercises and Additions | 155 |

Stochastic Differential Equations | 161 |

42 The Markov Property of Solutions | 176 |

43 Girsanov Theorem | 182 |

53 Models of Interest Rates | 222 |

54 Contingent Claims under Alternative Stochastic Processes | 227 |

55 Insurance Risk | 230 |

56 Exercises and Additions | 236 |

Applications to Biology and Medicine | 239 |

Continuous Approximation of Jump Models | 250 |

IndividualBased Models | 253 |

64 Neurosciences | 270 |

65 Exercises and Additions | 275 |

Appendices | 280 |

Measure and Integration A1 Rings and 𝛔Algebras | 281 |

A2 Measurable Functions and Measure | 284 |

A3 Lebesgue Integration | 288 |

A4 LebesgueStieltjes Measure and Distributions | 292 |

A5 Stochastic Stieltjes Integration | 296 |

Convergence of Probability Measures on Metric Spaces | 297 |

B2 Prohorovs Theorem | 304 |

Maximum Principles of Elliptic and Parabolic Operators | 313 |

C2 Maximum Principles of Parabolic Equations | 315 |

Stability of Ordinary Differential Equations | 321 |

324 | |

### Other editions - View all

An Introduction to Continuous-Time Stochastic Processes: Theory, Models, and ... Vincenzo Capasso,David Bakstein No preview available - 2012 |

### Common terms and phrases

absolutely continuous assumptions BC(R Black–Scholes Brownian motion called characteristic function compact consider constant convergence countable counting process defined Definition denoted density distribution function du(t equivalent exists a unique filtration finite finite-dimensional distributions following theorem function f Gaussian given Hence identically distributed independent increments inequality initial condition Itô's formula Lebesgue measure Lemma Let f Let Q Let X)er Lévy process mapping Markov process Markov property martingale measurable space metric space ML ML ML Moreover nonnegative O-algebra obtain P-integrable particle Poisson process probability law probability measure probability space process X)er Proof Proposition random vector RCLL Remark right-continuous satisfies sequence Show solution space Q stochastic differential equation stochastic integral stochastic process subset surely with respect t e R+ t∈R+ Theorem 4.4 trajectory transition probability Wiener process XN(t Xt)t∈R+