www.TechnicalBooksPDF.com MEASURE, PROBABILITY, AND MATHEMATICAL FINANCE www.TechnicalBooksPDF.com www.TechnicalBooksPDF.com MEASURE, PROBABILITY, AND MATHEMATICAL FINANCE A Problem-Oriented Approach Guojun Gan Manulife Financial Toronto, ON, Canada Chaoqun Ma School of Business Administration Hunan University Changsha, Hunan, P.R China HongXie Manulife Financial Toronto, ON, Canada WILEY www.TechnicalBooksPDF.com Copyright© 2014 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section I 07 or I 08 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright 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www.TechnicalBooksPDF.com CONTENTS Preface XVll Financial Glossary xxii PART I Sets and Sequences 1.1 1.2 1.3 1.4 1.5 MEASURE THEORY Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes 8 13 Measures 15 2.1 Basic Concepts and Facts Problems 15 2.2 2.3 Hints 20 2.4 Solutions 21 18 vii www.TechnicalBooksPDF.com viii CONTENTS 2.5 Bibliographic Notes 28 Extension of Measures 29 3.1 3.2 3.3 3.4 3.5 29 30 32 32 36 Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes Lebesgue-Stieltjes Measures 37 4.1 4.2 4.3 4.4 4.5 37 39 41 41 45 Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes Measurable Functions 47 5.1 5.2 5.3 5.4 5.5 47 48 50 51 56 Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes Lebesgue Integration 57 6.1 6.2 6.3 6.4 6.5 57 59 62 Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes 64 76 The Radon-Nikodym Theorem 77 7.1 7.2 7.3 7.4 7.5 77 Basic Concepts and Facts Problems Hints Solutions Bibliographic Notes 79 80 80 83 85 LP Spaces www.TechnicalBooksPDF.com REFERENCES Stroock, D and Varadhan, S (1979) Springer, New York, NY 701 Multidimensional Diffusion Processes Svoboda, S (2004) Interest Rate Modelling Palgrave Macmillan, Gordonsville, VA Takacs, L (1960) Stochastic Processes: Problems and Solutions Wiley, New York Taylor, S (1973) Introduction to Measure and Integration Cambridge University Press, Cambridge, UK Tijms, H (2003) A First Course in Stochastic Models Wiley, Hoboken, NJ Topper, J (2005) Financial Engineering with Finite Elements Wiley, Hoboken, NJ Vasicek, (1977) An equilibrium characterization of the term structure Journal of Financial Economics, 5(2): 177-188 Vaught, R (1995) Set Theory: An Introduction Birkhauser, Boston, 2nd edition Vestrup, E (2003) The Theory of Measures and Integration Wiley, Hoboken, NJ Watsham, T and Parramore, K (1997) Quantitative Methods in Finance Thomson, London, UK Weir, A (1979) General integration and measure Cambridge University Press, Cambridge, UK Wheeden, R and Zygmund, A (1977) Measure and Integral: An Introduction to Real Analysis CRC Press, Boca Raton, FL Wiersema, U (2008) Brownian Motion Calculus Wiley, Hoboken, NJ Williams, D (1991) Probability with Martingales Cambridge University Press, Cambridge, UK Wilmott, P (2001) Paul Wilmott Introduces Quantitative Finance Wiley, Hoboken, NJ, 2nd edition Wilmott, P (2006) Paul Wilmott on Quantitative Finance Volume Set Wiley, Hoboken, NJ, 2nd edition Wilmott, P., Howison, S., and Dewynne, J (1995) The Mathematics of Financial Derivatives: A Student Introduction Cambridge University Press, Cambridge, UK Wimmer, G and Altmann, G (1999) Thesaurus of Univariate Discrete Probability Distributions STAMM Verlag, Essen, Germany Wise, G and Hall, E (1993) Counterexamples in Probability and Real Analysis Oxford University Press, Oxford, UK 702 REFERENCES Yeh, J (2006) Real Analysis: Theory of Measure And Integration World Scientific Publishing, Singapore, 2nd edition Zakai, M ( 1967) Some moment inequalities for stochastic integrals and for solutions of stochastic differential equations Israel Journal ofMathematics, 5(3): 170176 List of Symbols A_K, 563,564 A AI, 563,564 A,563 szl, 611 do, 611 A=B,4 B ~A,3 B B, 16 \, 17 8, 38 A~B,4 AnB,4 A-B,4 A\B,4 AUB,4 Ac,4 B(Rd), 38 a B(E), 16 Be(p), 227 Beta(p, q), 240 f3(t), 630 B(n,p), 227 B(¢), 593 B(p, q), 241 Bo(:R), 38 Bo(Il.d), 38 B(S), 16 a.s., 128 c a.e., 17 a I b, 144 An-+ A, Ant A, An-J A,4 f b, 145 A(¢), 593 703 704 LIST OF SYMBOLS C [a, b], 478 C(a, h), 240 C(ry, ¢), 594 Corr(X, Y), 163 Cov(X, Y), 162 Cx(t), 218 g-T, 302 g-T• 302 g-t-• 274 g-t+• 274 D V, 17 aD, 547 D.i,b,aG(x1, X2, , Xd), 38 f(t, T, Y), 648 g-{, 275 D.[M] , 478 D.xoã 257 D(ry, Â), 594 d(I), 18 Dw, 402 apQ 548 g;, , HG(N, m, n), 228 668 d-system, 17 E E, 531 E , 272 t£: , 272 0, 3, 16 EQ 563,583 esssupX, 611 Ex, 532 E(X;F), 161 E(XIQ), 174 E(XI911i), 176 Exp(>.), 241 E(XIZ1, Z2, .), 174 E[YIH], 174 E[YIX], 174 F g-, 17, 127 g-00, 334 r,58 fn ~J,98 Fn F, 257 J+, 58 * G r(o:,B),241 r(x), 240 gcd(X, Y), 156 G(p), 228 r(o:), 241 H ~ dQ , 667 dQTI dQ F(t, T), 629 F(t; T, S), 647 F(t, T, Y), 648 I I, 17 11,4 fa I: J:fA J(t)dMt 465 f(t)dBt 422 f(t)dBt(w), 422 fdf.l, 58 fdf.l, 58 fs fdf.1, 58 I(x, A), 390 I K KJ, 391 L £ (S, ~ f.l), 58, 61 L [a, b], 421 £ (0, g-, P), 421 £ (0, g-, P), 162 L~red([a, b](M) X n, 464 IL([a, b] x D), 463 Cad(n, L [a, b]), 477 L~d([a, b] X n), 431 Cad(n, L 2[a, b]), 453 > -,78 ) « f.l, 78 LIST OF SYMBOLS >.+,78 1>-1, 78 lim inf, 4, liminf En, 5, 128 limn-+oo An, limn-+oo Xn, lim sup, 4, lim sup En, 5, 128 LN(f.l, a ), 240 U(n,g:-,P), 161 Lpred(Sl, L [a, b] (M)), 465 (LP(S, ~' f.l), dv), 87 LP(S, ~' f.l), 85 Lt, 533 L(t, T), 667 £x, 129 M M,48 M(A), 403 Mw,403 mg:-, 190 (m~)+, 57 m~, 47,48 (M)t, 466 [M]f, 478 [1,,218 flp, 218 f.lo(f), 58 f.L(f), 58 f.l(f; A), 58 {l-integrable, 58 f.LK, 391 {l-null, 17 f.ll j_ f.l2, 78 f.l"' >., 78 f.lx, 162 fly, 162 Mx(t), 217 N ~ = {O,l, ,oo},317 N, 17, 18 NB(r,p), 228 N(d), 582 N(f.l, a ), 239 N(x), 240 211 , 128 n, 17, 127 (n,g:-,P), 17,127 p P, 17, 127 P, 144 P,463 Pa(a, h), 240 P(BIQ), 174 P(BIX = x), 174 iJ.l(f.L,a,x),240 iJ.l(x), 240 ¢x(t), 218 ?-integrable, 161 1r(R), 136 1r-system, 17 1r(X), 131 P-null, 151 P(B), 228 P(t, T), 629 Pt,u• 390 Q (!,6,279,282,459 Q*, 672 QT, 667 Qx, 532 R R, 6,16 :R, 115 R+, 114 r(t), 630 R(t, T), 629 R(t; T, S), 648 s , 18 (S, d), 86 sF+, 57 ~Q, 15 705 706 LIST OF SYMBOLS I:, 15 L;l (>9 I:2, 113 a(C), 16 a(I), 30 I:-measurab1e, 47 L:ll, 30 I:J/I: -measurab1e, 48 "-', 86 (S, I:), 16 (S, L:, fJ), 17 ST(v), 240 SVT, 303 S 1\ T, 303 X"' Pa(a, h), 240 X"' P(B), 228 X "' ST(v), 240 Xt 580 Xt- 402 Xt+• 402 [X]~Pl, 464 532 Xf, 532 (x, y), 87 X l_ y, 87 (X, Y)t, 478 T y T, 142 ~ 0}, 333,334 v Var(X), 162