Humboldt Universität zu Berlin
Institut für Mathematik
Bereich Stochastik

Hans Föllmer

Professor of Mathematics (Stochastics and Mathematical Finance)


(with Alexander Schied) Stochastic Finance. An Introduction in Discrete Time

Stochastic Finance: An Introduction in Discrete Time
New: Third revised and extended edition

xii + 544 pp.,
published January 2011

- now with more than 100 exercises

- new chapter on dynamic risk measures

- new sections on robust utility maximization and on efficient hedging with convex risk measures

Russian edition, 496 pp.,  MCCME, Moscow, ISBN 978-5-94057-346-3, published February 2008

Translated by Yuliya Mishura and Georgiy Shevshenko

Second revisedand extended edition, xi + 459 pp., published November 2004

• First edition, ix + 422 pp., published July 2002

de Gruyter Studies in Mathematics 27


•  The Axiomatic Approach to Risk Measures for Capital Determination (with Stefan Weber)
Annual Review of Financial Economics, Vol. 7, 301 - 337 (2015)
[pdf (556 KB)]

•  Spatial Risk Measures: Local Specification and Boundary Risk (with Clauda Klüppelberg)
In: Stochastic Analysis and Applications 2014 - In Honour of Terry Lyons, Eds. D. Crisan, B. Hambly and T. Zariphopoulou, Springer Proceedings in Mathematics & Statistics 100, 307-326 (2014)
[pdf (239 KB)]

•  Consistent Risk Measures and a non-linear Extension of Backwards Martingale Convergence (with Irina Penner)
In: Festschrift Masatoshi Fukushima, Eds. Z.-Q. Chen, N. Jacob, M. Takeda, World Scientific, Interdisciplinary Mathematical Sciences, Vol. 17, 183 - 202 (2015)
[pdf (459 KB]

•  Shifting Martingale Measures and the Birth of a Bubble as a Submartingale (with Francesca Biagini and Sorin Nedelcu)
Finance and Stochastics, Vol. 18, No. 2, 297 - 326 (2014)
[pdf (609 KB)]

•  Spatial Risk Measures and their Local Specification: The Locally Law-Invariant Case.
Statistics & Risk Modeling Vol. 31, No. 1, 79 - 101 (2014)
[pdf (344 KB)]

•  Probabilistic aspects of finance (with Alexander Schied)
Bernoulli, Vol. 19, No. 4, 1306-1326 (2013)
[ ]

•  Convex Risk Measures: Basic Facts, Law-invariance and beyond, Asymptotics for Large Portfolios (with Thomas Knispel)
Handbook of the Fundamentals of Financial Decision Making, Part II, 507 - 554, Eds. L.C. MacLean and W.T. Ziemba, World Scientific (2013)
[pdf (742 KB)]

•  Convex Capital Requirements for Large Portfolios (with Thomas Knispel)
In: Stochastic Analysis and its Applications to Mathematical Finance, Essays in Honour of Jia-an Yan, Eds. T. Zhang and X. Y. Zhou, World Scientific (2012)
[pdf (623 KB)]

•  Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles (with Beatrice Acciaio and Irina Penner)
Finance and Stochastics, Vol. 16, No. 4, 669 - 709 (2012)
[pdf (488 KB)]

•  Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations (with Thomas Knispel)
Stochastics and Dynamics, Vol. 11, Nos. 2&3 , 333 - 351 (2011)
[pdf (408 KB)]

•  Monetary valuation of cash flows under Knightian uncertainty (with Irina Penner)
Int. J. Theor. Appl. Finance, Vol. 14, No. 1, 1 - 15 (2011)
[pdf (190 KB)]

•  Local Martingales and Filtration Shrinkage (with Ph. Protter)
ESAIM Probability and Statistics, Vol. 15 (special volume in honour of Marc Yor), 25 - 38 (2011)
[pdf (393 KB)]

•  Convex and Coherent Risk Measures (with A.Schied)
Encyclopedia of Quantitative Finance, Cont, R. (Ed.). John Wiley & Sons, pp. 1200-1204 (2010)
[pdf (150 KB)]

•  The Minimal Martingale Measure (with M. Schweizer)
Encyclopedia of Quantitative Finance, Cont, R. (Ed.). Wiley & Sons, pp. 1200-1204 (2010)
[pdf (168 KB)]

•  Alles richtig und trotzdem falsch? Anmerkungen zur Finanzkrise und zur Finanzmathematik
Mitteilungen der DMV 17, 148 - 154 (2009)
[pdf (4,7 MB)]

•  Robust preferences and robust portfolio choice (with A.Schied and S. Weber)
Handbook of Numerical Analysis, XV, Bensoussan & Zhang (Editors), Mathematical Modeling and Numerical Methods in Finance, 29-89, (2009)

•  Asymptotic Arbitrage and Large Deviations (with W. Schachermayer)
Mathematics and Financial Economics 1 (3-4), 213-249 (2008)
[postscript (544K) pdf (328K)]

•  On Kiyosi Itô's work and its impact
Gauss Lecture at the ICM 2006
Proceedings of the International Congress of Mathematicians, Madrid 2006, Vol. I, 109-124
European Mathematical Society Publishing House (2007)
[postscript (2.6M) pdf (964K)]

•  A Representation of Excessive Functions as Expected Suprema (with T. Knispel)
Probability and Mathematical Statistics 26(2), 379-394, Volume in honor of Kazimierz Urbanik (2006)
[postscript (437K) pdf (233K)]

•  Potentials of a Markov Process are Expected Suprema (with T. Knispel)
ESAIM Probability and Statistics 11, 89-101
Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday (2007)
[postscript (442K) pdf (233k)]
The original publication is available at

•  Convex risk measures and the dynamics of their penalty functions (with I. Penner)
Statistics & Decisions 24(1),61-96, (2006)
[postscript (469) | pdf (262K)]

•  Robust Projections in the Class of Martingale Measures (with A. Gundel)
Illinois Journal of Mathematics; vol. 50 no. 2 (2006), 439-472
[pdf (299K)]

•  Incertitude financière, mesures de risque et préférences robustes
Actes du colloque "Aspects des mathématiques financières", Académie des Sciences Paris (Ed. M. Yor), Lavoisier 2006
[postscript (217K) | pdf (124K)]
English version: Financial Uncertainty, risk measures, and robust preferences.
In: "Aspects of Mathematical Finance" (Ed. M. Yor), 3-15, Springer: Berlin (2008)
[pdf (9410K)]

•  A non-linear Riesz representation in probabilistic potential theory (with N. El Karoui)
Annales de l'Institut Henri Poincare (B) Probability and Statistics 4, 41(3), pp. 269-283, 2004
[postscript (200K) | pdf (203K)]

 Equilibria in Financial Markets with Heterogeneous Agents: A Probabilistic Perspective (with U. Horst and A. Kirman )
Journal of Mathematical Economics 41 (1-2), 123-155, 2005
[postscript (644K) | pdf (337K)]

American Options, Multi-armed Bandits, and Optimal Consumption Plans: A Unifying View (with P. Bank)
In: Paris-Princeton Lecture Notes on Mathematical Finance, Lecture Notes in Mathematics, Vol. 1814, 1-42 (Eds. Carmona, R. A.; Cinlar, E.; Ekeland, I., Jouini, E.; Scheinkman, J. A., Touzi, N.), Springer Verlag 2003
[pdf (943K)]

 Stochastic Finance: An Introduction in Discrete Time (with A. Schied)
de Gruyter Series in Mathematics 27, Berlin (2002, second revised edition 2004)

Robust Preferences and Convex Measures of Risk (with A. Schied)
In: Advances in Finance and Stochastics, Essays in Honor of Dieter Sondermann, 39-56, Eds. K. Sandmann, Ph. Schönbucher. Berlin, Heidelberg, New York: Springer 2002
[pdf (239K) ]

  Convergence of locally and globally interacting Markov chains (with U. Horst)
Stochastic Processes and their Appl. 96 (1), 99-121 (2001)
[ pdf (309K)]

 Convex measures of risk and trading constraints (with A. Schied).
Finance and Stochastics 6 (4), 429-447, 2002
[ postscript (263K) | pdf (269K)]

 Probabilistic Aspects of Financial Risk.
Plenary Lecture at the Third European Congress of Mathematics.
In: Proceedings of the European Congress of Mathematics, Barcelona 2000, Birkhäuser (2001)
[ postscript (265K) | pdf (267K)]

Weak Brownian motions of arbitrary order (with C.T. Wu, M. Yor)
Ann. Inst.  Poincaré Probabilités et Statistiques; vol 36 no. 4 (2000), 447-478
[postscript zipped (157K) | pdf (331K)

Efficient Hedging: Cost versus Shortfall Risk (with P. Leukert).
Finance and Stochastics 4, 117-146 (2000)
[ postscript (276K) | pdf (292K)]

 On Itô's formula for multidimensional Brownian motion (with Ph. Protter).
Prob. Theory Relat. Fields 116, 1-20 (2000)
[ postscript (203K) | pdf (195K) ]
Der Zufall in den Wirtschaftswissenschaften: Zur Rolle der Wahrscheinlichkeitstheorie in der Theorie der Finanzmärkte.
Nova Acta Leopoldina NF 79, Nr. 308, 113-125 (1999)

 Canonical decomposition of linear transformations of two independent Brownian motions motivated by models of insider trading (with C.T. Wu, M. Yor).
Stochastic Processes Appl. 84, 137-164 (1999)
[ postscript (300 K) ]

 Quantile Hedging (with P. Leukert).
Finance and Stochastics, vol. 3, No. 3, 251-273 (1999)
[ postscript (233K) | pdf (219K) ]

 Richard von Mises (mit U. Küchler).
Mathematics in Berlin (Eds. H.G.W. Begehr, H.Koch, J. Kramer, N. Schappacher, E.-J. Thiele), 111-116. Birkhäuser, Basel (1998)

 Vom Leibniz-Kalkül zur stochastischen Analysis: Reines und Angewandtes aus der Mathematik zufälliger Schwankungen.
Leopoldina (R.3) 43, 249-257 (1998)

 Ein Nobelpreis für Mathematik ?
DMV-Mitteilungen 1/98, 4-7 (1998)

Optional decomposition and Lagrange multipliers (with Y. Kabanov).
Finance & Stochastics, vol. 2, No. 1, 69-81 (1998)

 Zur Versicherungsmathematik von Derivaten.
Der Aktuar 3/4, 158-161 (1997)

 Optional decompositions under constraints (with D. Kramkov).
Prob. Theory Relat. Fields, vol. 109, 1-25 (1997)

 Entropy minimization and Schrödinger processes in infinite dimensions (with N. Gantert).
Annals of Probability, vol. 25, 901-926 (1997)

 Quadratic covariation and an extension of Itô's formula (with Ph. Protter and A. N. Shiryaev).
Bernoulli, vol. 1, 149 -169 (1995)

 Stock price fluctuation as a diffusion in a random environment.
Phil. Trans. R. Soc. Lond. A (1994) 347, 471-483

 A Microeconomic approach to diffusion models for stock prices (with M. Schweizer).
Mathematical Finance, Vol.3, No.1 (1993), 1-23

 Cancelling anticipation by a Girsanov transformation: a paradox on Wiener space (with P. Imkeller).
Annales Inst. H. Poincaré, vol.29, no. 4 (1993), 569-586

 Anticipating Girsanov transformations: a conditional approach (with R. Buckdahn).
Probability. Theory Relat. Fields 95, (1993), 311-330

 Probabilistic Aspects of Options.
Rolf Nevanlinna Institute Reports B6, Helsinki (1991)
[ pdf (3,7MB)]

 Orthogonal Martingale Representation (with R. J. Elliott).
Stochastic Analysis, in honor of Moshe Zakai, eds. E. Mayer-Wolf, E. Merzbach, A. Schwartz, Academic Press (1991)

 Martin Boundaries on Wiener Space.
Diffusion Processes and Related Problems In Analysis, vol. I. Ed. M. Pinsky, Progress in Probability 22, 3 - 16, Birkhäuser (1991)

 Hedging of Contingent Claims under Incomplete Information (with M. Schweizer).
Applied Stochastic Analysis, eds. M.H.A. Davis and R.J. Elliott, 389-414, Gordon and Breach, London (1990)

 Hedging by Sequential Regression: An Introduction to the Mathematics of Option Trading (with M. Schweizer).
ASTIN Bulletin 18, No.2 (1989), 147-160

 Large Deviations and Surface Entropy (with M. Ort).
Colloque Paul Lévy sur les Processus Stochastiques. Astérisque 157/158 (1988), 173-190

 Random Fields and Diffusion Processes.
Ecole d' Eté de Probabilités de St. Flour XVI. Lecture Notes in Mathematics 1362, Springer (1988), 101-203

 Large Deviations for the Empirical Field of a Gibbs Measure (with S. Orey).
Annals of Probability 16, No.3 (1988), 961-977

 On Large Deviations and Relative Entropy of Markov Random Fields.
Stochastic Differential Systems (Eds. H.J. Engelbert, W.Schmidt), Lecture Notes in Control and Information Sciences 96, Springer (1987), 27-32

 A Minimal Fluctuation Property for Coin Tossing and Locally Symmetric Martingales (with J.M. Clark).
Stochastic Differential Systems (Eds. H.J. Engelbert, W. Schmidt), Lecture Notes in Control and Information Sciences 96, Springer (1987), 333-337

 Time Reversal and Smoothing of Infinite-dimensional Diffusion Processes (with J.D. Deuschel).
Proc. 1. Int. Ascona-Como meeting, Stochastic Processes in Classical and Quantum Systems, (Eds. S. Albeverio, G. Casati, D. Merlini), Lecture Notes in Physics 262, Springer (1986), 179-186

 Hedging of Non-redundant Contingent Claims (with D. Sondermann).
Contributions to Mathematical Economics. In Honor of G. Debreu (Eds. W. Hildenbrand and A. Mas-Colell), Elsevier Science Publ., North-Holland (1986), 205-223
[ pdf (1.1 MB)]

 Time Reversal of Infinite-Dimensional Diffusions (with A. Wakolbinger).
Stochastic Processes and their Appl. 22 (1986), 59-77

 Time Reversal on Wiener Space.
Stochastic Processes - Mathematic and Physics, Lecture Notes in Mathematics 1158, Springer (1986), 119-129

 An Entropy Approach to the Time Reversal of Diffusion Processes.
Stochastic Differential Systems, Proceedings of the IFIP-WG 7/1 Working Conference, Marseille-Luminy (Eds. M. Métivier and E. Pardoux), Lecture Notes in Control and Information Sciences 69, Springer (1985), 156-163

 Von der Brownschen Bewegung zum Brownschen Blatt: Einige neuere Richtungen in der Theorie der stochastischen Prozesse.
Perspectives in Mathematics, Anniversary of Oberwolfach 1984, Birkhäuser, Basel (1984), 159-190

 Almost sure Convergence of Multiparameter Martingales for Markov Random Fields.
Annals of Probability, vol. 12, No.1 (1984) 133-140

 A Covariance Estimate for Gibbs Measures.
J. of Functional Analysis, vol. 46, No.3 (1982), 387-395

 Dirichlet Processes.
Stochastic Integrals, Lecture Notes in Mathematics 851, Springer (1981), 476-778

 Calcul d' Itô sans Probabilités. Sém.
Probabilités Strasbourg XV, Lecture Notes in Mathematics 850, Springer (1981), 143-15

 Macroscopic Convergence of Markov Chains on Infinite Product Spaces.
In Colloquia Mathematica Soc. Janos Bolyai 27, Random Fields (1979), 363-371

 Local Interactions with a Global Signal: A Voter Model.
Biological Growth and Spread, Mathematical Theories and Applications. Lecture Notes in Biomathematics 38, Springer (1980), 141-144

 On the Global Markov Property. Quantum Fields - Algebras,
Processes (Ed. L.Streit), Springer-Verlag Wien (1980), 293-302

 Tail Structure of Markov Chains on infinite Product Spaces.
Z. Wahrscheinlichkeitstheorie verw. Geb. 50 (1979), 273-285

 Martingale Criteria for Stochastic Stability.
Probability Theory, Banach Center Publications vol. 5, Warsaw (1979), 89-96

 Quasimartingales à deux indices.
C. R. Acad. Sc. Paris, t. 288, Sr. A (1979), 61-64

 On the Asymptotic Behavior of Stochastic Economic Processes (with M. Majumdar).
J. of Mathematical Economics 5 (1978), 275-287

 Zur Dynamik interdependenter Präferenzen.
Quantitative Wirtschaftsforschung (Ed. H. Albach et al.), Mohr-Verlag Tübingen (1977), 199-206

 An "Inner" Variational Principle for Markov Fields on a Graph (with J.L. Snell).
Z. Wahrscheinlichkeitstheorie verw. Geb. 39 (1977), 187-195

 Martingales and Potential Theory.
Proceedings of the Eight Brazilian Mathematical Colloquium, Pocos de Caldas 1971. Inst. Mat. Pura Apl., Rio de Janeiro (1977), 131-14

 The Bernoulli Principle and the Dirichlet Problem.
Mathematical Economics and Game Theory, Essays in Honor of O. Morgenstern. Lecture Notes in Economics and Mathematical Systems 141, Springer (1977), 208-217

 A Liapunov Principle for Semimartingales.
Proceedings of the AMS Symposia in Pure Mathematics, vol. 31 (1977), 15-21

 On the Potential Theory of Stochastic Fields.
Bulletin of the International Statistical Institute, vol. 46, Invited Papers (1975), 362-370

 Stochastische Bewegungen und ihre ersten Integrale.
Jahresber. Deutsch. Math. Verein. 76 (1975), 199-216

 Phase Transition and Martin Boundary.
Sém. Prob. Strasbourg IX, Lecture Notes in Mathematics 465 (1975), 305-317

 Relative Densities of Semimartingales (with H. Airault).
Inventiones Math. 27 (1974), 299-327

 Random Economies with many Interacting Agents.
J. of Mathematical Economics 1 (1974), 51-6

 Stochastic Holomorphy.
Mathematische Annalen 207 (1974),245-255

 On Entropy and Information Gain in Random Fields.
Z. Wahrscheinlichkeitstheorie verw. Geb. 26 (1973), 207-217

 On the Representation of Semimartingales.
Annals of Probability 1, No. 4 (1973), 580-58

 Optimal Stopping of Constrained Brownian Motion.
J. of Applied Probability 9 (1972), 557-571

 The Exit Measure of a Supermartingale.
Z. Wahrscheinlichkeitstheorie verw. Geb. 21 (1972), 154-166

 Ein Littlewood-Kriterium für Martingale und insbesondere für Dirichlet-Lösungen.
Elliptische Differentialgleichungen II (Ed. G. Anger), Berlin:Akademie-Verlag (1971), 113-118

 Feine Topologie am Martinrand eines Standardprozesses.
Z. Wahrscheinlichkeitstheorie verw. Geb. 12 (1969), 127-144

Last Update: 20.04.2006