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Steele M.J. — Stochastic Calculus and Financial Applications
Steele M.J. — Stochastic Calculus and Financial Applications

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Название: Stochastic Calculus and Financial Applications

Автор: Steele M.J.

Аннотация:

The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It(tm) integral and aims to provide a development that is honest and complete without being pedantic. With the It(tm) integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Corrected second printing

Год издания: 2001

Количество страниц: 301

Добавлена в каталог: 13.03.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L^{p}$ space      18
$SF^{+}$      249
$\k{C}$inlar, E.      vi
$\mathcal{A}$      252
$\mu$ problem      159
$\pi-\lambda$ theorem      208
$\sigma$-field      43
Adapted      50
Admissibility      252
Admissible strategies      252
Admissible strategies, uniqueness      254
Alternative fields      106
American option      244 290
Analysis and synthesis      111
Approximation, finite time set      209
Approximation, in $\mathcal{H}^{2}$      90
Approximation, operator      90
Approximation, theorem      90
Arbitrage      153
Arbitrage, risk-free bond      249
Artificial measure method      44
Asian options      258
Augmented filtration      50
Axler, S.      288
Bachelier, L.      29
Bass, R.      287 290
Baxter      290
Benedito, J.J.      286
Bessel's inequality      281
Binomial arbitrage      155
Binomial arbitrage, reexamination      233
Black - Scholes formula      158
Black - Scholes formula, via martingales      241
Black - Scholes model      156
Black - Scholes PDE, and Feynman - Kac representation      274
Black - Scholes PDE, and Feynman-Kac      271
Black - Scholes PDE, CAPM argument      162
Black - Scholes PDE, general drift      274
Black - Scholes PDE, hedged portfolio argument      160
Black - Scholes PDE, how to solve      182
Black - Scholes PDE, simplification      186
Black - Scholes PDE, uniqueness of solution      187
Borel - Cantelli Lemma      27 279
Borel field      60
Box algebra      124
Box calculus      124
Box calculus, and chain rule      123
Brown, Robert      120
Brownian bridge      41
Brownian bridge, as It$\hat{o}$ integral      141
Brownian bridge, construction      41
Brownian bridge, SDE      140
Brownian motion, covariance function      34
Brownian motion, definition      29
Brownian motion, density of maximum      68
Brownian motion, geometric      137 138
Brownian motion, H$\ddot{o}$lder continuity      63
Brownian motion, hitting time      56
Brownian motion, killed      264
Brownian motion, L$\acute{e}$vy's characterization      204
Brownian motion, not differentiable      63
Brownian motion, planar      120
Brownian motion, recurrence in $\mathbb{R}^{2}$      122
Brownian motion, ruin probability      55
Brownian motion, scaling and inversion laws      40
Brownian motion, time inversion      59
Brownian motion, wavelet representation      36
Brownian motion, with drift      118
Brownian motion, writes your name      229
Brownian paths, functions of      216
Burchfield, J.D.      289
Calculations, organization of      186
Campbell, J.Y.      288
CAPM      162
Capturing the past      191
Carslaw, H.S.      289
casinos      7 285
Cauchy sequence      280
Central limit theorem      279
Central limit theorem, via embedding      78
Characteristic function      30
Chebyshev's inequality      279
Chung, K.L.      vi 287
Churchill, W.      66
Coefficient matching      137 157
Coffin state      264
Coin tossing, unfair      5
Complete metric space      280
Complete orthonormal sequence      33 283
Completeness      252
Completeness, of $L^{2}$      280
Completeness, of model      252
Conditional expectation      278
Conditional expectation, as contraction      48
Conditional expectation, continuity of      201
Conditional expectation, definition      45
Conditional expectation, existence      46
Conditional expectation, uniform integrability      48
Conservation law      170
Constitutive law      170
Contingent claim      252
Covariance and independence      41
Covariance function      32
Covariance function, calculation      38
Cox, J.      290
Credit constraint      248
Credit constraint supermartingale      248
Dambis, D.E.      290
DeMoivre - Laplace approximation      72
Density argument      200 210
Derivative security      155
Dewynne, J.      289
Difference operator      4
Diffusion equation      169
Diffusion equation, derivation      171
Diffusion equation, nonuniqueness example      178
Diffusion equation, solution methods      172
Diffusion equation, uniqueness question      178
Diffusion Equation, Uniqueness Theorem      181
Diffusion equation, with constant coefficients      183
Discounting      235
Dominated Convergence Theorem      278
Dominated convergence theorem, conditional      279
Donsker's invariance principle      71
Doob's decomposition      28
Doob's inequalities      19
Doob's stopping time theorem, local martingales      105
Drift removal      222
Drift swapping      224
Dubins, L.      287 290
Dudley's representation theorem      194
Dudley's representation theorem, nonuniqueness      196
Dudley, R.M.      287 290
Duffie, D.      vi 290
Durrett, R.      287 290
Edgar, G.A.      290
Einstein, A.      29
Embedding theorem      76
Epstein, R.A.      285
Equivalent martingale measure      236
Equivalent martingale measure, existence and uniqueness      241
Equivalent martingale measure, uniqueness      240
Equivalent measures      220 236
Euler, L.      289
Existence and uniqueness theorem for SDEs      142
Exponential local martingales      224
Exponential martingales      225
Fatou's lemma      278
Feller, W.      285
Feynman - Kac formula      263
Feynman - Kac representation and Black - Scholes PDE      271
Feynman - Kac representation for diffusions      270
Feynman - Kac representation theorem      265
Filtration      50
Filtration, standard Brownian      50
Financial frictions      153
First step analysis      1
Forward contracts      154
Fourier transform method      172
Frazier, M.W.      286
Freedman, D.      vi
Friedman, B.      287
Fristedt, B.      291
Gaussian miracle      30
Gaussian process      32
Gaussian tail bounds      42
General Case, U.S. Army      291
Generating functions      7
Girsanov theorem, for standard processes      223
Girsanov theorem, simplest      219
Girsanov theory      213
Girsanov, I.V.      213
Gray, L.      291
Gronwall's lemma      150
H$\ddot{o}$lder continuity      62
H$\ddot{o}$lder inequality      21
H$\ddot{o}$lder inequality, applied to Novikov      231
H$\ddot{o}$lder inequality, or Roger inequality      287
H$\ddot{o}$lder inequality, proof via Jensen      44
Harmonic functions      120
Harrison, J.M.      vi 290 291
Heat kernel      174
Heat operator      174
Hilbert space      210 280
Hitting time, biased random walk      6
Hitting time, Brownian motion      56
Hitting time, density      69
Hitting time, simple random walk      4
Hitting time, sloping line      219
Howison, S.      289
Ikeda, N.      287
Importance sampling      213 290
Incompleteness example      260
Independence and covariance      41
Induction      229 289
Informative increments proposition      193
Integral sign      85
Interest rate (two-way)      154
Invariance principle      71
It$\hat{o}$ formula, location and time      116
It$\hat{o}$ formula, simplest case      111
It$\hat{o}$ formula, vector version      121
It$\hat{o}$ integral, $\mathcal{L}^{2}_{LOC}$      95
It$\hat{o}$ integral, as Gaussian process      101
It$\hat{o}$ integral, as martingale      83
It$\hat{o}$ integral, as process      82
It$\hat{o}$ integral, definition      79
It$\hat{o}$ integral, pathwise interpretation      87
It$\hat{o}$ isometry      80 85
It$\hat{o}$ isometry, conditional      82 131
It$\hat{o}$ isometry, counterexample in $\mathcal{L}^{2}_{LOC}$      211
It$\hat{o}$ isometry, on $\mathcal{H}^{2}[0,T]$      82
It$\hat{o}$, K.      286 288
Jaeger, J.C.      289
Jensen's inequality      18 287
John, F.      289
K$\ddot{o}$rner, T.W.      285 289
Karatzas, I.      vi 286-288 290
Kloeden, P.E.      288
Kreps, D.      290
L$\acute{e}$vy - Bachelier formula      219 230
L$\acute{e}$vy's Arcsin Law      267
L$\acute{e}$vy's modulus of continuity theorem      65
L$\acute{e}$vy, P.      287
Landau, E.      286
Lebesgue integral      277
Leibniz      85
Leveraging an abstraction      44
Li, W.V.      vi 288
Lin, C.C.      285 289
Lipster, R.S.      290
Lo, A.W.      vi 288
localization      96 114 225
Localization, discrete time example      23
Look-back options      258
Looking back      2 57 118 119 138 142 186 201 229
MacKinlay, A.C.      288
Market price of risk      239
Markov's inequality      279
Martingale representation theorem      197
Martingale representation theorem, in $L^{1}$      211
Martingale transform      13
Martingale, $L^{1}$-bounded      24
Martingale, $L^{p}$      27
Martingale, continuous      50
Martingale, convergence theorem      22
Martingale, creation of      221
Martingale, local      103
Martingale, PDE condition      116 121
Martingale, with respect to a sequence      11
Matching coefficients in product process      139
Maximal inequality, Doob's      19
Maximal inequality, Doob's in continuous-time      52
Maximal sequence      19
Maximum principle      179
Maximum principle, harmonic functions      189
Maximum principle, parabolic      179
Maximum principle, via integral representation      189
McKean, H.P.      286 287
Mean shifting identity      214
mesh      128
Meyer, Y.      286 287
Mice, diffusion of      169
Modes of convergence      58
Monotone class theorem      209
Monotone Convergence Theorem      278
Monte Carlo      66
Monte Carlo, improved      214
Multivariate Gaussian      30 41
Musiela      290
Neveu, J.      286
Newton's Binomial Theorem      8
No early exercise condition      245
Normal distribution, in $\mathbb{R}^{d}$      30
Novikov condition      225 290
Novikov condition, lazy man's      231
Novikov condition, sharpness      230
Numerical calculation and intuition      7
Occupation time of an interval      274
Omnia      85
Ornstein - Uhlenbeck process      138
Ornstein - Uhlenbeck process, SDE      138
Ornstein - Uhlenbeck process, solving the SDE      140
Orthonormal sequence      281
P$\acute{o}$lya's question      77
P$\acute{o}$lya, G.      2 77 246 286 289 291
Parallelogram law      283
Parseval's identity      34 282
Persistence of identity      89 98
Phelan, M.      vi
Pitman, J.      vi
Platen, E.      288
Pliska, S.R.      290
Polarization trick      134
Portfolio properties      243
Portfolio weights      237 243
Portfolio weights, abstract to concrete      261
Power series      7 225 226
Pozdnyakov's example      260
Pozdnyakov, V.      vi 260
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