Elementary stochastic calculus with finance in view:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2008
|
| Edition: | Reprinted |
| Series: | Advanced series on statistical science & applied probability
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 212 S. graph. Darst. |
| ISBN: | 9810235437 9789810235437 |
Staff View
MARC
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| 245 | 1 | 0 | |a Elementary stochastic calculus with finance in view |c Thomas Mikosch |
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| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2008 | |
| 300 | |a IX, 212 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804138478387068928 |
|---|---|
| adam_text | Contents
Reader Guidelines 1
1
Preliminaries
5
1.1 Basic
Concepts from Probability Theory
............. 6
1.1.1
Random Variables
..................... 6
1.1.2
Random Vectors
...................... 14
1.1.3
Independence and Dependence
.............. 19
1.2
Stochastic Processes
........................ 23
1.3
Brownian Motion
.......................... 33
1.3.1
Defining Properties
..................... 33
1.3.2
Processes Derived from Brownian Motion
........ 40
1.3.3
Simulation of Brownian Sample Paths
.......... 44
1.4
Conditional Expectation
...................... 56
1.4.1
Conditional Expectation under Discrete Condition
... 56
1.4.2
About
σ
-Pields
.......................
62
1.4.3
The General Conditional Expectation
.......... 67
1.4.4
Rules for the Calculation of Conditional Expectations
. 70
1.4.5
The Projection Property of Conditional Expectations
. 74
1.5
Martingales
............................. 77
1.5.1
Defining Properties
..................... 77
1.5.2
Examples
.......................... 81
1.5.3
The Interpretation of a Martingale as a Fair Game
... 84
2
The Stochastic Integral
87
2.1
The Riemann and Riemann-Stieltjes Integrals
.......... 88
2.1.1
The Ordinary Riemann Integral
.............. 88
2.1.2
The Riemann-Stieltjes Integral
.............. 92
2.2
The
Ito
Integral
........................... 96
2.2.1
A Motivating Example
................... 96
viii CONTENTS
2.2.2
The
Ito
Stochastic Integral for Simple Processes
.... 101
2.2.3
The General
Ito
Stochastic Integral
............ 107
2.3
The
Ito
Lemma
........................... 112
2.3.1
The Classical Chain Rule of Differentiation
....... 113
2.3.2
A Simple Version of the
Ito
Lemma
........... 114
2.3.3
Extended Versions of the
Ito
Lemma
........... 117
2.4
The Stratonovich and Other Integrals
.............. 123
3
Stochastic Differential Equations
131
3.1
Deterministic Differential Equations
............... 132
3.2
Ito
Stochastic Differential Equations
............... 134
3.2.1
What is a Stochastic Differential Equation?
....... 134
3.2.2
Solving
Ito
Stochastic Differential Equations by the
Ito
Lemma
........................... 138
3.2.3
Solving
Ito
Differential Equations via Stratonovich Cal¬
culus
............................. 145
3.3
The General Linear Differential Equation
............ 150
3.3.1
Linear Equations with Additive Noise
.......... 150
3.3.2
Homogeneous Equations with Multiplicative Noise
. . . 153
3.3.3
The General Case
..................... 155
3.3.4
The Expectation and Variance Functions of the Solution
156
3.4
Numerical Solution
......................... 157
3.4.1
The
Euler
Approximation
................. 158
3.4.2
The Milstein Approximation
............... 162
4
Applications of Stochastic Calculus in Finance
167
4.1
The Black-Scholes Option Pricing Formula
........... 168
4.1.1
A Short Excursion into Finance
.............. 168
4.1.2
What is an Option?
.................... 170
4.1.3
A Mathematical Formulation of the Option Pricing Pro¬
blem
............................. 172
4.1.4
The Black and Scholes Formula
.............. 174
4.2
A Useful Technique: Change of Measure
............. 176
4.2.1
What is a Change of the Underlying Measure?
..... 176
4.2.2
An Interpretation of the Black-Scholes Formula by Chan¬
ge of Measure
........................ 180
Appendix
185
Al
Modes of Convergence
....................... 185
A2 Inequalities
. . ........................... 187
CONTENTS ix
A3
Non-Differentiability and Unbounded Variation of Brownian Sam¬
ple Paths
..............................188
A4
Proof of the Existence of the General
Ito
Stochastic Integral
. . 190
A5 The Radon-Nikodym Theorem
..................193
A6 Proof of the Existence and Uniqueness of the Conditional Ex¬
pectation
..............................194
Bibliography
195
Index
199
List of Abbreviations and Symbols
209
|
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| author_facet | Mikosch, Thomas |
| author_role | aut |
| author_sort | Mikosch, Thomas |
| author_variant | t m tm |
| building | Verbundindex |
| bvnumber | BV035217071 |
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| classification_tum | MAT 606f WIR 160f WIR 522f |
| ctrlnum | (OCoLC)634365232 (DE-599)BVBBV035217071 |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | Reprinted |
| format | Book |
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| id | DE-604.BV035217071 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:28:50Z |
| institution | BVB |
| isbn | 9810235437 9789810235437 |
| language | English |
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| physical | IX, 212 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | World Scientific |
| record_format | marc |
| series | Advanced series on statistical science & applied probability |
| series2 | Advanced series on statistical science & applied probability |
| spelling | Mikosch, Thomas Verfasser aut Elementary stochastic calculus with finance in view Thomas Mikosch Reprinted Singapore [u.a.] World Scientific 2008 IX, 212 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced series on statistical science & applied probability 6 Finanzwirtschaft (DE-588)4017214-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 gnd rswk-swf Stochastische Analysis (DE-588)4132272-1 s Finanzwirtschaft (DE-588)4017214-4 s DE-604 Finanzmathematik (DE-588)4017195-4 s 1\p DE-604 Advanced series on statistical science & applied probability 6 (DE-604)BV011932321 6 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017023208&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Mikosch, Thomas Elementary stochastic calculus with finance in view Advanced series on statistical science & applied probability Finanzwirtschaft (DE-588)4017214-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| subject_GND | (DE-588)4017214-4 (DE-588)4017195-4 (DE-588)4132272-1 |
| title | Elementary stochastic calculus with finance in view |
| title_auth | Elementary stochastic calculus with finance in view |
| title_exact_search | Elementary stochastic calculus with finance in view |
| title_full | Elementary stochastic calculus with finance in view Thomas Mikosch |
| title_fullStr | Elementary stochastic calculus with finance in view Thomas Mikosch |
| title_full_unstemmed | Elementary stochastic calculus with finance in view Thomas Mikosch |
| title_short | Elementary stochastic calculus with finance in view |
| title_sort | elementary stochastic calculus with finance in view |
| topic | Finanzwirtschaft (DE-588)4017214-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Stochastische Analysis (DE-588)4132272-1 gnd |
| topic_facet | Finanzwirtschaft Finanzmathematik Stochastische Analysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017023208&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011932321 |
| work_keys_str_mv | AT mikoschthomas elementarystochasticcalculuswithfinanceinview |