This book highlights recent developments in mathematical control theory and its applications to finance. It presents a collection of original contributions by distinguished scholars, addressing a large spectrum of problems and techniques. Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, ranging from "pure" areas of mathematics up to applied sciences like finance. Stochastic optimal control is a well established and important tool of mathematical finance. Other branches of control theory have found comparatively less applications to financial problems, but the exchange of ideas and methods has intensified in recent years. This volume should contribute to establish bridges between these separate fields. The diversity of topics covered as well as the large array of techniques and ideas brought in to obtain the results make this volume a valuable resource for advanced students and researchers.
Author(s): Andrey Sarychev, Albert Shiryaev, Manuel Guerra, Maria do Rosario Grossinho
Edition: 1
Publisher: Springer
Year: 2008
Language: English
Pages: 434
MATHEMATICAL CONTROL THEORY AND FINANCE......Page 1
Springerlink......Page 0
Half-title......Page 2
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 6
Contents......Page 8
List of Contributors......Page 12
1 Introduction......Page 15
2 Curvature......Page 19
3 Results......Page 21
4 Sketch of proofs......Page 22
References......Page 26
1 Introduction......Page 29
2 Analysis......Page 30
2.1 0 < D < r......Page 33
3 The integral equations......Page 36
4 Discussion......Page 38
References......Page 40
1 Introduction......Page 43
2.1 Turbulence......Page 44
2.2 Stylized features of finance and turbulence......Page 45
3.1 Volterra type processes......Page 46
3.2 Volatility modulated Volterra processes......Page 48
3.3 Time change and VMVP......Page 49
4 Time change in stationary processes......Page 50
5 Universality in turbulence......Page 51
5.2 Theoretical considerations......Page 52
6.1 Finance......Page 53
6.2 Turbulence......Page 54
7 Increment processes......Page 55
8 Time change in finance and turbulence......Page 56
8.2 Turbulence......Page 57
9.2 Refined universality model......Page 58
10 Concluding remarks......Page 59
A The normal inverse Gaussian law......Page 60
References......Page 62
1 Introduction......Page 69
2 Calculus on time scales......Page 71
3 External dynamical equivalence......Page 73
4 Function universes......Page 76
5 Conditions of equivalence......Page 77
6.1 Conclusions......Page 81
References......Page 82
1 Introduction......Page 85
2 Lie group symmetries......Page 88
3 The special case λ(S) = ωS^k......Page 92
4 The special case λ = ωS......Page 94
4.1 Exact invariant solutions in case of a fixed relation between variables S and t......Page 98
5 Properties of invariant solutions......Page 104
Acknowledgments......Page 107
References......Page 108
1 Introduction......Page 109
2 Reduction to standard form......Page 112
3 The free-boundary problem......Page 114
4 The result and proof......Page 116
References......Page 126
1 Introduction......Page 127
2 Stochastic demand model......Page 130
2.1 Parameterisations......Page 131
3 Optimization problem......Page 133
3.1 Deterministic case......Page 134
4 Stochastic market average premium......Page 135
4.1 Infinite market......Page 136
4.2 Finite market......Page 137
5 Stochastic exposure......Page 142
5.1 Linear utility function......Page 143
5.2 Exponential utility function......Page 145
6 Conclusions......Page 148
References......Page 149
1 Introduction......Page 151
2 Discrete-time MDPs with average rewards per unit time......Page 153
3 Continuous-time MDPs......Page 155
4 Nonatomic MDPs......Page 159
Acknowledgement......Page 161
References......Page 162
1 Introduction......Page 163
2 Basic definitions and results on time scales......Page 164
3 Main results......Page 167
References......Page 173
1 Introduction......Page 175
2 Preliminaries......Page 176
2.1 Classical Fourier descriptors for contours (the circle group)......Page 177
2.2 The Fourier Transform on locally compact unimodular groups......Page 178
2.3 General definition of the generalized Fourier descriptors, from those over the circle group......Page 182
3 The generalized Fourier descriptors for the motion group M 2......Page 184
4.1 Chu and Tannaka categories, Chu and Tannaka dualities......Page 185
4.2 Generalized Fourier descriptors over compact groups......Page 187
5.2 Representations, Fourier transform and generalized Fourier descriptors over M 2,N......Page 189
5.3 The cyclic-lift from L²(R²) to L²(M 2,N )......Page 190
5.4 Fourier transform, generalized Fourier descriptors of cyclic lifts over M 2,2n+1......Page 191
5.5 Completeness of the discrete generalized Fourier descriptors......Page 194
6 Conclusion......Page 198
References......Page 199
1 Introduction, examples, position of the problems......Page 201
1.1 A few academic examples of kinematic systems subject to motion planning......Page 202
1.2 The subriemannian cost, the metric complexity and the interpolation-entropy......Page 203
1.3 The results for examples 2 and 3......Page 205
1.4 Content of the paper......Page 206
2.1 Normal coordinates and normal form......Page 208
2.3 Two crucial lemmas......Page 212
2.2 Nilpotent approximations along Γ......Page 210
3 The codimension one case......Page 213
4.1 One step bracket generating case......Page 216
4.2 Remaining cases......Page 218
5 Codimension more than three......Page 219
6 Conclusion......Page 222
References......Page 223
1.1 Example......Page 225
1.2 Reasons for Trading Compound and Instalment Options......Page 227
2 Valuation in the Black-Scholes Model......Page 228
2.1 The Curnow and Dunnett Integral Reduction Technique......Page 230
2.2 A Closed-Form Solution for the Value of an Instalment Option......Page 231
2.4 Forward Volatility Smile......Page 233
3 Instalment Options with a Continuous Payment Plan......Page 234
4.1 Implementational Aspects......Page 236
4.2 Performance......Page 237
4.3 Convergence......Page 238
A.1 The Package instalment.m......Page 239
B.1 The R Functions......Page 240
References......Page 242
1 Introduction......Page 245
2 Existence of relaxed minimizers in unbounded nonconvex case......Page 248
3 Lipschitzian regularity of non relaxed minimizers for Lagrange variational problem: brief account......Page 251
4.1 Main result on Lipschitzian regularity......Page 253
4.2 Reduction of the relaxed optimal control problem to a time-optimal control problem......Page 254
4.3 Loeb’s compactification of the set of control parameters......Page 255
4.4 Parameterization of the compactification......Page 257
4.5 Solutions of the relaxed Lagrangian and of the compactified time-optimal problems......Page 259
4.6 Pontryagin maximum principle, second Erdmann condition, normality and Lipschitzian regularity......Page 260
References......Page 264
1 Introduction......Page 265
2 Pricing formulae......Page 266
3 Approximations to BCP......Page 270
4 Accuracy of the approximation......Page 271
5.1 Deterministic interest rate......Page 274
References......Page 276
1 Preliminaries......Page 279
2 Spline cubatures for expectations of diffusion processes......Page 284
3 Termination at fixed time: A numerical recipe for fixed-boundary problems......Page 289
4 Termination at arbitrary time: A numerical recipe for optimal stopping and free-boundary problems......Page 294
5 Concluding remarks......Page 302
References......Page 304
1 Introduction......Page 307
2 The market structure......Page 309
2.1 The investor’s wealth......Page 310
3 The optimal portfolio......Page 311
3.1 The value function......Page 312
4 A general approximate solution for an incomplete market......Page 314
4.1 The maximum error......Page 317
5.1 Stochastic volatility......Page 318
5.2 Stochastic market price of risk......Page 320
6 Conclusion......Page 322
References......Page 323
1 Introduction......Page 325
2 Preliminaries......Page 326
3 Carleman linearization......Page 328
4 Linear observability of polynomial dynamical system......Page 332
5 Carleman bilinearization for polynomial control system......Page 334
References......Page 336
1 Introduction......Page 339
2 Calculus on time scales......Page 340
3 Indistinguishability relation......Page 343
4 Observability......Page 347
6 Conclusions and future works......Page 348
References......Page 349
1 Introduction......Page 351
2.1 Notation and regularity......Page 352
2.2 The finite dimensional sub–problem......Page 355
3.1 The maximized flow......Page 356
3.2 The second variations......Page 358
3.3 The invertibility of the flow......Page 362
3.4 Reduction to a finite–dimensional problem......Page 366
4 Appendix: Invertibility of piecewise C¹ maps......Page 368
References......Page 370
1 Motivation......Page 373
2 Setup......Page 375
2.1 Changing measure......Page 376
2.2 Pricing of electricity futures......Page 377
3 Illustration......Page 382
4.1 The EM-algorithm......Page 383
4.2 Application to the proposed model......Page 385
4.3 Estimation of the model on EEX data......Page 387
Acknowledgement......Page 388
References......Page 389
1 Brownian motion model......Page 391
2 Linear-penalty case......Page 392
3 Nonlinear-penalty case......Page 393
5 Some extensions......Page 397
References......Page 400
1 Introduction......Page 401
3 Main results......Page 403
3.1 Gâteaux differentiability......Page 404
3.2 Necessary optimality condition......Page 406
References......Page 409
1.1 Introduction. Motivation of the work......Page 411
1.2 Description of the data and points of investigation......Page 412
1.3 Applying extreme value theory......Page 413
1.4 Models for loss occurrences and trends testing......Page 414
1.5 Internal and external data: introduction to the bayesian methodology......Page 418
2.1 Monte Carlo simulations for aggregating operational risk......Page 421
2.2 Fast Fourier transformation......Page 422
3.1 Overview of results......Page 424
3.2 A note on the insurance of operational losses......Page 427
A Error bounds estimation......Page 428
References......Page 430
Workshop on Mathematical Control Theory and Finance......Page 433
