| Preface |
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xiii | |
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1 | (107) |
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1 | (6) |
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Trajectories of stochastic processes |
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2 | (1) |
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Jumps of stochastic processes |
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3 | (3) |
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When are stochastic processes equal? |
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6 | (1) |
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Measurability of Stochastic Processes |
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7 | (22) |
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Filtration, adapted, and progressively measurable processes |
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8 | (5) |
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13 | (6) |
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Stopped variables, σ-algebras, and truncated processes |
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19 | (4) |
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23 | (6) |
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29 | (63) |
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30 | (5) |
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35 | (2) |
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The quadratic variation of discrete time martingales |
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37 | (5) |
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The downcrossings inequality |
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42 | (4) |
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Regularization of martingales |
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46 | (3) |
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The Optional Sampling Theorem |
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49 | (9) |
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Application: elementary properties of Levy processes |
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58 | (22) |
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Application: the first passage times of the Wiener processes |
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80 | (11) |
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Some remarks on the usual assumptions |
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91 | (1) |
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92 | (16) |
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Stability under truncation |
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93 | (1) |
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94 | (10) |
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Convergence of local martingales: uniform convergence on compacts in probability |
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104 | (2) |
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Locally bounded processes |
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106 | (2) |
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Stochastic Integration with Locally Square-Integrable Martingales |
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108 | (71) |
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The Ito--Stieltjes Integrals |
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109 | (29) |
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Ito--Stieltjes integrals when the integrators have finite variation |
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111 | (6) |
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Ito--Stieltjes integrals when the integrators are locally square-integrable martingales |
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117 | (7) |
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Ito--Stieltjes integrals when the integrators are semimartingales |
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124 | (2) |
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Properties of the Ito--Stieltjes integral |
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126 | (1) |
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126 | (2) |
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Integration by parts and the existence of the quadratic variation |
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128 | (6) |
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The Kunita--Watanabe inequality |
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134 | (4) |
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The Quadratic Variation of Continuous Local Martingales |
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138 | (8) |
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Integration when Integrators are Continuous Semimartingales |
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146 | (21) |
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The space of square-integrable continuous local martingales |
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147 | (4) |
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Integration with respect to continuous local martingales |
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151 | (11) |
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Integration with respect to semimartingales |
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162 | (1) |
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The Dominated Convergence Theorem for stochastic integrals |
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162 | (2) |
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Stochastic integration and the Ito--Stieltjes integral |
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164 | (3) |
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Integration when Integrators are Locally Square-Integrable Martingales |
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167 | (12) |
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The quadratic variation of locally square-integrable martingales |
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167 | (4) |
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Integration when the integrators are locally square-integrable martingales |
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171 | (5) |
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Stochastic integration when the integrators are semimartingales |
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176 | (3) |
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The Structure of Local Martingales |
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179 | (46) |
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182 | (25) |
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Predictable stopping times |
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182 | (6) |
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Decomposition of thin sets |
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188 | (2) |
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The extended conditional expectation |
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190 | (2) |
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Definition of the predictable projection |
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192 | (2) |
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The uniqueness of the predictable projection, the predictable section theorem |
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194 | (7) |
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Properties of the predictable projection |
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201 | (3) |
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Predictable projection of local martingales |
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204 | (2) |
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Existence of the predictable projection |
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206 | (1) |
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207 | (12) |
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Predictable Radon--Nikodym Theorem |
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207 | (6) |
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Predictable Compensator of locally integrable processes |
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213 | (4) |
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Properties of the Predictable Compensator |
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217 | (2) |
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The Fundamental Theorem of Local Martingales |
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219 | (3) |
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222 | (3) |
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General Theory of Stochastic Integration |
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225 | (67) |
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Purely Discontinuous Local Martingales |
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225 | (10) |
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Orthogonality of local martingales |
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227 | (5) |
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Decomposition of local martingales |
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232 | (2) |
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Decomposition of semimartingales |
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234 | (1) |
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Purely Discontinuous Local Martingales and Compensated Jumps |
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235 | (11) |
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Construction of purely discontinuous local martingales |
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240 | (4) |
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Quadratic variation of purely discontinuous local martingales |
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244 | (2) |
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Stochastic Integration With Respect To Local Martingales |
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246 | (8) |
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Definition of stochastic integration |
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248 | (2) |
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Properties of stochastic integration |
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250 | (4) |
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Stochastic Integration With Respect To Semimartingales |
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254 | (23) |
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Integration with respect to special semimartingales |
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257 | (4) |
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Linearity of the stochastic integral |
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261 | (1) |
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262 | (2) |
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264 | (13) |
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The Proof of Davis' Inequality |
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277 | (15) |
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Discrete-time Davis' inequality |
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279 | (8) |
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287 | (5) |
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292 | (59) |
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The Doob--Meyer Decomposition |
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292 | (16) |
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292 | (7) |
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Dellacherie's formulas and the natural processes |
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299 | (4) |
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The sub- super- and the quasi-martingales are semimartingales |
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303 | (5) |
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Semimartingales as Good Integrators |
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308 | (6) |
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Integration of Adapted Product Measurable Processes |
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314 | (5) |
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Theorem of Fubini for Stochastic Integrals |
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319 | (9) |
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Martingale Representation |
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328 | (23) |
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351 | (109) |
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Ito's Formula for Continuous Semimartingales |
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353 | (6) |
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Some Applications of the Formula |
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359 | (18) |
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Zeros of Wiener processes |
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359 | (7) |
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Continuous Levy processes |
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366 | (2) |
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Levy's characterization of Wiener processes |
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368 | (5) |
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Integral representation theorems for Wiener processes |
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373 | (2) |
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375 | (2) |
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Change of Measure for Continuous Semimartingales |
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377 | (17) |
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Locally absolutely continuous change of measure |
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377 | (1) |
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Semimartingales and change of measure |
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378 | (2) |
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Change of measure for continuous semimartingales |
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380 | (2) |
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Girsanov's formula for Wiener processes |
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382 | (4) |
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Kazamaki--Novikov criteria |
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386 | (8) |
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Ito's Formula for Non-Continuous Semimartingales |
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394 | (23) |
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Ito's formula for processes with finite variation |
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398 | (3) |
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The proof of Ito's formula |
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401 | (10) |
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Exponential semimartingales |
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411 | (6) |
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Ito's Formula For Convex Functions |
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417 | (43) |
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Derivative of convex functions |
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418 | (4) |
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Definition of local times |
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422 | (7) |
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429 | (9) |
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Local times of continuous semimartingales |
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438 | (7) |
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Local time of Wiener processes |
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445 | (5) |
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450 | (7) |
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Theorem of Dvoretzky Erdos and Kakutani |
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457 | (3) |
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Processes with Independent Increments |
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460 | (87) |
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460 | (36) |
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461 | (3) |
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Compound Poisson processes generated by the jumps |
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464 | (8) |
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Spectral measure of Levy processes |
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472 | (8) |
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Decomposition of Levy processes |
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480 | (6) |
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Levy--Khintchine formula for Levy processes |
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486 | (3) |
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Construction of Levy processes |
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489 | (2) |
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Uniqueness of the representation |
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491 | (5) |
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Predictable Compensators of Random Measures |
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496 | (12) |
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Measurable random measures |
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497 | (4) |
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Existence of predictable compensator |
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501 | (7) |
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Characteristics of Semimartingales |
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508 | (5) |
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Levy--Khintchine Formula for Semimartingales with Independent Increments |
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513 | (25) |
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Examples: probability of jumps of processes with independent increments |
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513 | (5) |
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518 | (5) |
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Semimartingales with independent increments |
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523 | (7) |
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Characteristics of semimartingales with independent increments |
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530 | (4) |
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534 | (4) |
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Decomposition of Processes with Independent Increments |
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538 | (9) |
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547 | (47) |
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Results from Measure Theory |
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547 | (12) |
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The Monotone Class Theorem |
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547 | (3) |
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Projection and the Measurable Selection Theorems |
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550 | (1) |
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551 | (4) |
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Interpretation of Stopped σ-algebras |
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555 | (4) |
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559 | (20) |
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559 | (8) |
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Existence of Wiener Processes |
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567 | (4) |
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Quadratic Variation of Wiener Processes |
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571 | (8) |
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579 | (15) |
| Notes and Comments |
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594 | (3) |
| References |
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597 | (6) |
| Index |
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603 | |
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