Bernt Øksendal

• Agram, Nacira; Øksendal, Bernt & Hu, Yaozhong (2018). Mean-field backward stochastic differential equations and applications. arXiv.org.  ISSN 2331-8422.
• Draouil, Olfa & Øksendal, Bernt (2018). Optimal insider control of stochastic partial differential equations. Stochastics and Dynamics.  ISSN 0219-4937.  18(3) . doi: 10.1142/S0219493718500144 Fulltekst i vitenarkiv.
• Agram, Nacira & Øksendal, Bernt (2017). A Hida-Malliavin white noise calculus approach to optimal control. arXiv.org.  ISSN 2331-8422.
• Agram, Nacira & Øksendal, Bernt (2017). Stochastic control of memory mean-field processes. Applied mathematics and optimization.  ISSN 0095-4616.  s 1- 24 . doi: 10.1007/s00245-017-9425-1 Fulltekst i vitenarkiv.
• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2017). New approach to optimal control of stochastic Volterra integral equations. arXiv.org.  ISSN 2331-8422.
• Dahl, Kristina Rognlien & Øksendal, Bernt (2017). Singular recursive utility. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  89(6-7), s 994- 1014 . doi: 10.1080/17442508.2017.1303067 Fulltekst i vitenarkiv.
• Hu, Yaozhong; Øksendal, Bernt & Sulem, Agnes Biacobroda (2017). Singular mean-field control games. Stochastic Analysis and Applications.  ISSN 0736-2994.  35(5), s 823- 851 . doi: 10.1080/07362994.2017.1325745 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Røse, Elin Engen (2017). A white noise approach to insider trading, In Takeyuki Hida & Ludwig Streit (ed.),  Let Us Use White Noise.  World Scientific.  ISBN 978-981-3220-93-5.  Kapittel 6.  s 191 - 204 Fulltekst i vitenarkiv. Vis sammendrag

  We present a new approach to the optimal portfolio problem for an insider with logarithmic utility. Our method is based on white noise theory, stochastic forward integrals, Hida-Malliavin calculus and the Donsker delta function.

• Agram, Nacira & Øksendal, Bernt (2016). Model Uncertainty Stochastic Mean-Field Control. arXiv.org.  ISSN 2331-8422.
• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2016). Optimal control of forward-backward stochastic Volterra equations. arXiv.org.  ISSN 2331-8422.
• Alvarez, Luis H.; Lungu, Edward & Øksendal, Bernt (2016). Optimal multi-dimensional stochastic harvesting with density-dependent prices. Afrika Matematika.  ISSN 1012-9405.  27(3), s 427- 442 . doi: 10.1007/s13370-015-0357-0 Fulltekst i vitenarkiv.
• Dahl, Kristina Rognlien; Mohammed, Salah-Eldin; Øksendal, Bernt & Røse, Elin Engen (2016). Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives. Journal of Functional Analysis.  ISSN 0022-1236.  271(2), s 289- 329 . doi: 10.1016/j.jfa.2016.04.031 Fulltekst i vitenarkiv.
• Draouil, Olfa & Øksendal, Bernt (2016). Optimal insider control and semimartingale decompositions under enlargement of filtration. Stochastic Analysis and Applications.  ISSN 0736-2994.  34(6), s 1045- 1056 . doi: 10.1080/07362994.2016.1200989 Fulltekst i vitenarkiv.
• Draouil, Olfa & Øksendal, Bernt (2016). Stochastic differential games with inside information. Infinite Dimensional Analysis Quantum Probability and Related Topics.  ISSN 0219-0257.  19(3) . doi: 10.1142/S0219025716500168 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2016). Dynamic robust duality in utility maximization. Applied mathematics and optimization.  ISSN 0095-4616.  s 1- 31 . doi: 10.1007/s00245-016-9329-5 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2016). Optimal control of predictive mean-field equations and applications to finance. Springer Proceedings in Mathematics & statistics.  ISSN 2194-1009.  138, s 301- 320 . doi: 10.1007/978-3-319-23425-0_12 Fulltekst i vitenarkiv. Vis sammendrag

  We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process X(t) and a predictive mean-field backward SDE (BSDE) in the unknowns Y(t),Z(t),K(t,⋅). The driver of the BSDE at time t may depend not just upon the unknown processes Y(t),Z(t),K(t,⋅), but also on the predicted future value Y(t+δ), defined by the conditional expectation A(t):=E[Y(t+δ)|Ft]. We give a sufficient and a necessary maximum principle for the optimal control of such systems, and then we apply these results to the following two problems: (i) Optimal portfolio in a financial market with an insider influenced asset price process. (ii) Optimal consumption rate from a cash flow modeled as a geometric Itô-Lévy SDE, with respect to predictive recursive utility.

• Øksendal, Bernt; Sulem, Agnes Biacobroda & Zhang, Tusheng (2016). A stochastic HJB equation for optimal control of forward-backward SDEs, In Mark Podolskij; Robert Stelzer; Steen Thorbjørnsen & Almut E. D. Veraart (ed.),  The Fascination of Probability, Statistics and their Applications. In honour of Ole E. Barndorff-Nielsen.  Springer Science+Business Media B.V..  ISBN 978-3-319-25824-9.  Kapittel 20.  s 435 - 446 Vis sammendrag

  We study optimal stochastic control problems of general coupled systems of forward-backward stochastic differential equations with jumps. By means of the Itô-Ventzell formula, the system is transformed into a controlled backward stochastic partial differential equation. Using a comparison principle for such equations we obtain a general stochastic Hamilton-Jacobi-Bellman (HJB) equation for the value function of the control problem. In the case of Markovian optimal control of jump diffusions, this equation reduces to the classical HJB equation. The results are applied to study risk minimization in financial markets.

• Agram, Nacira & Øksendal, Bernt (2015). Malliavin Calculus and Optimal Control of Stochastic Volterra Equations. Journal of Optimization Theory and Applications.  ISSN 0022-3239.  167(3), s 1070- 1094 . doi: 10.1007/s10957-015-0753-5 Fulltekst i vitenarkiv.
• Draouil, Olfa & Øksendal, Bernt (2015). A Donsker delta functional approach to optimal insider control and applications to finance. Communications in Mathematics and Statistics.  ISSN 2194-6701.  3(3), s 365- 421 . doi: 10.1007/s40304-015-0065-y Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnes Biacobroda (2015). Risk minimization in financial markets modeled by Itô-Lévy processes. Afrika Matematika.  ISSN 1012-9405.  26(5-6), s 939- 979 . doi: 10.1007/s13370-014-0248-9 Fulltekst i vitenarkiv.
• Agram, Nacira & Øksendal, Bernt (2014). Infinite horizon optimal control of forward-backward stochastic differential equations with delay. Journal of Computational and Applied Mathematics.  ISSN 0377-0427.  259, s 336- 349 . doi: 10.1016/j.cam.2013.04.048 Fulltekst i vitenarkiv.
• Corcuera, Jose Manuel; Di Nunno, Giulia; Farkas, Gergely & Øksendal, Bernt (2014). A continuous auction model with insiders and random time of information release. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (3) Fulltekst i vitenarkiv.
• Fontana, Claudio; Øksendal, Bernt & Sulem, Agnes Biacobroda (2014). Market Viability and Martingale Measures under Partial Information. Methodology and Computing in Applied Probability.  ISSN 1387-5841. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sandal, Leif Kristoffer & Ubøe, Jan (2014). Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information. Journal of Applied Probability.  ISSN 0021-9002.  51A, s 213- 226 . doi: 10.1239/jap/1417528477 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2014). Forward-Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty. Journal of Optimization Theory and Applications.  ISSN 0022-3239.  161(1), s 22- 55 . doi: 10.1007/s10957-012-0166-7 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnes Biacobroda (2014). Stochastic control of Itô-Lévy processes with applications to finance. Communications on Stochastic Analysis.  ISSN 0973-9599.  8(1), s 1- 15 Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2014). Singular control and optimal stopping of SPDEs, and backward SPDEs with reflection. Mathematics of Operations Research.  ISSN 0364-765X.  39(2), s 464- 486 . doi: 10.1287/moor.2013.0602 Fulltekst i vitenarkiv.
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). A maximum principle for infinite horizon delay equatations. SIAM Journal on Mathematical Analysis.  ISSN 0036-1410.  45(4), s 2499- 2522 . doi: 10.1137/120882809 Fulltekst i vitenarkiv.
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). A maximum principle for infinite horizon delay equations. SIAM Journal on Mathematical Analysis.  ISSN 0036-1410.  45, s 2499- 2522 . doi: 10.1137/120882809
• Agram, Nacira & Øksendal, Bernt (2013). Infinite horizon optimal control of forward-backward stochastic differential equations with delay. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439. Fulltekst i vitenarkiv.
• Baghery, Fouzia; Haadem, Sven; Øksendal, Bernt & Turpin, Isabella (2013). Optimal stopping and stochastic control differential games for jump diffusions. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  85(1), s 85- 97 . doi: 10.1080/17442508.2011.652116 Fulltekst i vitenarkiv.
• Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). Maximum principles for jump diffusion processes with infinite horizon. Automatica.  ISSN 0005-1098.  49(7), s 2267- 2275 . doi: 10.1016/j.automatica.2013.04.011 Fulltekst i vitenarkiv.
• Siegmund-Schultze, Reinhard & Øksendal, Bernt (2013). Johannes Lohne (1908-1993), den glemte norske nyoppdager av Thomas Harriot og frontkjemper for den tyske okkupasjonsmakten under 2. verdenskrig. Normat.  ISSN 0801-3500.  61(1), s 18- 32
• Øksendal, Bernt; Sandal, Leif Kristoffer & Ubøe, Jan (2013). Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models. Journal of Economic Dynamics and Control.  ISSN 0165-1889.  37(7), s 1284- 1299 . doi: 10.1016/j.jedc.2013.02.010 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2013). A stochastic control approach to robust duality in utility maximization. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2013). A stochastic HJB equation for optimal control of forward-backward SDEs. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439. Fulltekst i vitenarkiv.
• Aase, Knut; Bjuland, Terje & Øksendal, Bernt (2012). Partially informed noise traders. Mathematics and Financial Economics.  ISSN 1862-9679.  6(2), s 93- 104 . doi: 10.1007/s11579-012-0075-4 Fulltekst i vitenarkiv. Vis sammendrag

  The single auction equilibrium of Kyle’s (1985) is studied, in which noise traders may be partially informed, or alternatively they can be manipulated. Unlike Kyle’s assumption that the quantity traded by the noise traders is independent of the asset value, we assume that the noise traders are able to correlate their trade with the true price. This has several implications for the equilibrium, one being that the informed trader’s expected profits decrease as the noise traders’ ability to correlate positively improve. In the limit, the noise traders do not lose on average, and the informed trader makes zero expected profits. When the correlation is negative, we interpret this as manipulation. In this case the insider makes the highest expected profits, and the informativeness of prices is at its minimum.

• Aase, Knut; Bjuland, Terje & Øksendal, Bernt (2012). Strategic insider trading equilibrium: A filter theory approach. Afrika Matematika.  ISSN 1012-9405.  23(2), s 145- 162 . doi: 10.1007/s13370-011-0026-x Fulltekst i vitenarkiv. Vis sammendrag

  The continuous-time version of Kyle’s (Econometrica 53(6):1315–1336, 1985) model of asset pricing with asymmetric information is studied, and generalized in various directions, i.e., by allowing time-varying liquidity trading, and by having weaker a priori assumptions on the model. This extension is made possible by the use of filtering theory. We derive the optimal trade for an insider and the corresponding price of the risky asset; the insider’s trading intensity satisfies a deterministic integral equation, given perfect inside information.

• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2012). A MAXIMUM PRINCIPLE FOR INFINITE HORIZON DELAY EQUATIONS. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  4 Fulltekst i vitenarkiv.
• Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2012). Maximum principles for jump diffusion processes with infinite horizon. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  April(3) Fulltekst i vitenarkiv.
• Meyer-Brandis, Thilo; Hu, Yaozhong; Øksendal, Bernt & Biagini, Francesca (2012). Insider trading equilibrium in a market with memory. Mathematics and Financial Economics.  ISSN 1862-9679.  6, s 229- 247
• Meyer-Brandis, Thilo; Øksendal, Bernt & Zhou, Xunyu (2012). A mean-field stochastic maximum principle via Malliavin calculus. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  84(5-6), s 643- 666 . doi: 10.1080/17442508.2011.651619
• Øksendal, Bernt & Sulem, Agnes (2012). Singular stochastic control and optimal stopping with partial information of itô-lévy processes. SIAM Journal of Control and Optimization.  ISSN 0363-0129.  50(4), s 2254- 2287 . doi: 10.1137/100793931 Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2012). Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs, In Tusheng Zhang & Xunyu Zhou (ed.),  Stochastic Analysis and Applications to Finance.  World Scientific.  ISBN 978-981-4383-57-8.  Artikkel.
• Øksendal, Bernt & Ta, An Thi Kieu (2012). A maximum principle for stochastic differential games with g-expectations and partial information. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  84(2-3), s 137- 155 . doi: 10.1080/17442508.2010.532875 Fulltekst i vitenarkiv.
• Øksendal, Bernt & Zhang, Tusheng (2012). Backward stochastic differential equations with respect to general filtrations and applications to insider finance. Communications on Stochastic Analysis.  ISSN 0973-9599.  6(4), s 703- 722 Fulltekst i vitenarkiv.
• Aase, Knut Kristian; Bjuland, Terje & Øksendal, Bernt (2011). An anticipative linear filtering equation. Systems & control letters (Print).  ISSN 0167-6911.  60(7), s 468- 471 . doi: 10.1016/j.sysconle.2011.04.001
• Aase, Knut Kristian; Bjuland, Terje & Øksendal, Bernt (2011). Insider trading with partially informed traders. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (16) Fulltekst i vitenarkiv.
• Baghery, Fouzia; Haadem, Sven; Turpin, Isabella & Øksendal, Bernt (2011). Optimal stopping and stochastic control differential games for jump diffusions. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508. . doi: 10.1080/17442508.2011.652116
• Di Nunno, Giulia; Pamen, Olivier Menoukeu; Øksendal, Bernt & Proske, Frank Norbert (2011). A general maximum principle for anticipative stochastic control and applications to insider trading, In Giulia Di Nunno & Bernt Øksendal (ed.),  Advanced Mathematical Methods for Finance.  Springer.  ISBN 978-3-642-18411-6.  Chapter.  s 181 - 221
• Di Nunno, Giulia; Øksendal, Bernt; Menoukeu Pamen, Olivier & Proske, Frank Norbert (2011). Uniqueness of Decompositions of Skorohod-Semimartingales. Infinite Dimensional Analysis Quantum Probability and Related Topics.  ISSN 0219-0257.  14(1), s 15- 24 . doi: 10.1142/S0219025711004274
• Federico, Salvatore & Øksendal, Bernt (2011). Optimal stopping of stochastic differential equations with delay driven by Lévy noise. Potential Analysis.  ISSN 0926-2601.  34(2), s 181- 198 . doi: 10.1007/s11118-010-9187-8
• Øksendal, Bernt; Aase, Knut & Bjuland, Terje (2011). Strategic insider trading equilibrium: A filter theory approach. Afrika Matematika.  ISSN 1012-9405.  23(2), s 145- 162 . doi: 10.1007/s13370-011-0026-x Vis sammendrag

  The continuous-time version of Kyle’s (Econometrica 53(6):1315–1336, 1985) model of asset pricing with asymmetric information is studied, and generalized in various directions, i.e., by allowing time-varying liquidity trading, and by having weaker a priori assumptions on the model. This extension is made possible by the use of filtering theory. We derive the optimal trade for an insider and the corresponding price of the risky asset; the insider’s trading intensity satisfies a deterministic integral equation, given perfect inside information.

• Øksendal, Bernt; Sandal, Leif Kristoffer & Ubøe, Jan (2011). Stochastic Stackelberg equilibria with applications to time dependent newsvendor models. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439. Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2011). Portfolio optimization under model uncertainty and BSDE games. Quantitative finance (Print).  ISSN 1469-7688.  11(11), s 1665- 1674 . doi: 10.1080/14697688.2011.615219
• Øksendal, Bernt & Sulem, Agnès (2011). Robust stochastic control and equivalent martingale measures, In Arturo Kohatsu-Higa; Nicolas Privault & Shuenn-Jyi Sheu (ed.),  Stochastic Analysis with Financial Applications Hong Kong 2009.  Springer.  ISBN 978-3-0348-0096-9.  part 1 - Bidrag til Vol.65 i serien "Progress in Probability".  s 179 - 189
• Øksendal, Bernt; Sulem, Agnes & Zhang, Tusheng (2011). OPTIMAL CONTROL OF STOCHASTIC DELAY EQUATIONS AND TIME-ADVANCED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. Advances in Applied Probability.  ISSN 0001-8678.  43(2), s 572- 596
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2011). Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs. Prepint Series - Pure Mathematics.  ISSN 0806-2439.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2011). Singular control of SPDEs and backward SPDEs with reflection. Prepint Series - Pure Mathematics.  ISSN 0806-2439.
• Corcuera, José Manuel; Di Nunno, Giulia; Farkas, Gergely & Øksendal, Bernt (2010). Kyle-Back's model with Lévy noise. Prepint Series - Pure Mathematics.  ISSN 0806-2439.  26
• Øksendal, Bernt & Sulem, Agnès (2010). An anticipative stochastic calculus approach to pricing in markets driven by Lévy processes. Communications on Stochastic Analysis.  ISSN 0973-9599.  4(2), s 179- 199
• Øksendal, Bernt & Sulem, Agnès (2010). Singular stochastic control and optimal stopping with partial information of Itô-Lévy processes. Preprint series (Universitetet i Oslo. Matematisk institutt).  ISSN 0806-2439.  (9)
• Øksendal, Bernt & Zhang, Tusheng (2010). Optimal control with partial information for stochastic Volterra equations. International Journal of Stochastic Analysis.  ISSN 2090-3332.  2010 . doi: 10.1155/2010/329185 Fulltekst i vitenarkiv.
• Di Nunno, Giulia & Øksendal, Bernt (2009). Optimal portfolio, partial information and Malliavin calculus. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  81(3-4), s 303- 322 . doi: 10.1080/17442500902917979
• Di Nunno, Giulia; Øksendal, Bernt; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2009). A general maximum principle for anticipative stochastic control and applications to insider trading. Prepint Series - Pure Mathematics.  ISSN 0806-2439.  (21)
• Di Nunno, Giulia; Øksendal, Bernt; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2009). Uniqueness of Decompositions of Skorohod-Semimartingales. Prepint Series - Pure Mathematics.  ISSN 0806-2439.  (10)
• Di Nunno, Giulia; Øksendal, Bernt; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2009). Uniqueness of Decompositions of Skorohod-Semimartingales. Prepint Series - Pure Mathematics.  ISSN 0806-2439.  (10)
• Øksendal, Bernt & Sulem, Agnès (2009). MAXIMUM PRINCIPLES FOR OPTIMAL CONTROL OF FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS. SIAM Journal of Control and Optimization.  ISSN 0363-0129.  48(5), s 2945- 2976 Vis sammendrag

  We present various versions of the maximum principle for optimal control of forward-backward stochastic differential equations (SDE) with jumps. Our study is motivated by risk minimization via g-expectations. We first prove a general sufficient maximum principle for optimal control with partial information of a stochastic system consisting of a forward and a backward SDE driven by Levy processes. We then present a Malliavin calculus approach which allows us to handle non-Markovian systems. Finally, we give examples of applications.

• Øksendal, Bernt & Sulem, Agnès (2009). Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM Journal of Control and Optimization.  ISSN 0363-0129.  48(5), s 2945- 2976 . doi: 10.1137/080739781
• Øksendal, Bernt & Sulem, Agnès (2009). RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS. Mathematical Finance.  ISSN 0960-1627.  19(4), s 619- 637
• Ta, An Thi Kieu & Øksendal, Bernt (2008). Maximum Principle for Stochastic Differential Games with Partial Information. Journal of Optimization Theory and Applications.  ISSN 0022-3239.  139(3), s 463- 483 . doi: 10.1007/s10957-008-9398-y

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• Di Nunno, Giulia & Øksendal, Bernt (ed.) (2011). Advanced Mathematical Methods for Finance. Springer.  ISBN 978-3-642-18411-6.  536 s.
• Di Nunno, Giulia & Øksendal, Bernt (ed.) (2011). Advanced Mathematical Methods for Finance. Springer.  ISBN 978-3-642-18411-6.  536 s.
• Holden, Helge; Øksendal, Bernt; Ubøe, Jan & Zhang, Tusheng (2010). Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Springer Science+Business Media B.V..  ISBN 978-0-387-89487-4.  305 s.
• Holden, Helge; Øksendal, Bernt; Ubøe, Jan & Zhang, Tusheng (2009). Stochastic Partial Differential Equations. Springer.  ISBN 978-0-387-89487-4.  305 s.
• Di Nunno, Giulia; Øksendal, Bernt & Proske, Frank (2009). Malliavin Calculus for Lévy Processes with Applications to Finance. Springer.  ISBN 9783540785712.  418 s. Vis sammendrag

  While the original works on Malliavin calculus aimed to study the smoothness of densities of solutions to stochastic differential equations, this book has another goal. It portrays the most important and innovative applications in stochastic control and finance, such as hedging in complete and incomplete markets, optimisation in the presence of asymmetric information and also pricing and sensitivity analysis. In a self-contained fashion, both the Malliavin calculus with respect to Brownian motion and general Lévy type of noise are treated. Besides, forward integration is included and indeed extended to general Lévy processes. The forward integration is a recent development within anticipative stochastic calculus that, together with the Malliavin calculus, provides new methods for the study of insider trading problems. To allow more flexibility in the treatment of the mathematical tools, the generalization of Malliavin calculus to the white noise framework is also discussed. This book is a valuable resource for graduate students, lecturers in stochastic analysis and applied researchers.

• Di Nunno, Giulia; Øksendal, Bernt & Proske, Frank Norbert (2009). Malliavin Calculus for Lévy Processes and Applications to Finance. Springer.  ISBN 978-3-540-78571-2.  413 s.

Se alle arbeider i Cristin

• Dahl, Kristina Rognlien & Øksendal, Bernt (2015). Singular recursive utility.
• Øksendal, Bernt & Sulem, Agnès (2015). Applications of stochastic analysis, In Nicholas J. Higham (ed.),  The Princeton Companion to Applied Mathematics.  Princeton University Press.  ISBN 978-0-691-15039-0.  IV.14.  s 319 - 327
• Dahl, Kristina Rognlien; Øksendal, Bernt; Røse, Elin Engen & Mohammed, Salah-Eldin (2014). Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2014). A comparison theorem for backward SPDEs with jumps.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2014). A comparison theorem for backward SPDEs with jumps.
• Aase, Knut; Bjuland, Terje & Øksendal, Bernt (2011). Insider trading with partially informed traders. Norges Handelshoeyskole. Institutt for Foretaksoekonomi. Discussion Paper. 21. Vis sammendrag

  The single auction equilibrium of Kyle's (1985) is studied, in which noise traders may be partially informed, or alternatively they can be manipulated. Unlike Kyle's assumption that the quantity traded by the noise traders is independent of the asset value, we assume that the noise traders are able to correlate their trade with the true price. This has several implications for the equilibrium, one being that the insider's expected profits decrease as the noise traders' ability to correlate positively improve. In the limit, the noise traders do not lose on average, and the insider makes zero expected profits. When the correlation is negative, we interpret this as manipulation. In this case the insider makes the highest expected profits, and the informativeness of prices is at its minimum.

• Biagini, Francesca; Meyer-Brandis, Thilo; Hu, Yaozhong & Øksendal, Bernt (2011). Insider trading equilibrium in a market with memory. Preprint series (Universitetet i Oslo. Matematisk institutt). 2011/7. Fulltekst i vitenarkiv.
• Øksendal, Bernt (2011). Mathematics and Finance: The Black-Scholes Option Pricing Formula and Beyond. Preprint series: Pure mathematics. 11. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sandal, Leif Kristoffer & Ubøe, Jan (2011). Stochastic Stackelberg equilibria with applications to time dependent newsvendor models. Norges Handelshoeyskole. Institutt for Foretaksoekonomi. Discussion Paper. 9. Vis sammendrag

  In this paper we prove a sufficient maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is an Ito-Levy process, and to increase realism information is delayed, e.g., due to production time. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time variable coefficients. Moreover, these results are operational because we are able to offer explicit solution formulas. An interesting finding is that more precise information may be a considerable disadvantage for the retailer.

• Øksendal, Bernt & Sulem, Agnès (2011). Forward-backward SDE games and stochastic control under model uncertainty. Preprint series (Universitetet i Oslo. Matematisk institutt). August 2011. Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2011). Portfolio optimization under model uncertainty and BSDE games. Preprint series: Pure mathematics. 1. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2011). Optimal partial information control of SPDEs with delay and time-advanced backward SPDEs. Preprint series (Universitetet i Oslo. Matematisk institutt). 7. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2011). Singular control of SPDEs and backward SPDEs with reflection. Preprint series (Universitetet i Oslo. Matematisk institutt). 8. Fulltekst i vitenarkiv.
• Aase, Knut K.; Bjuland, Terje & Øksendal, Bernt (2010). An anticipative linear filtering equation. Vis sammendrag

  In the classical Kalman-Bucy filter and in the subsequent literature so far, it has been assumed that the initial value of the signal process is independent of both the noise of the signal and of the noise of the observations.The purpose of this paper is to prove a filtering equation for a linear system where the (normally distributed) initial value X0 of the signal process Xt has a given correlation function with the noise (Brownian motion Bt) of the observation process Zt. This situation is of interest in applications to insider trading in finance. We prove a Riccati type equation for the mean square error S(t):= E[(Xt - ^Xt)**2]; 0 <= t <= T; where ^Xt is the filtered estimate for Xt. Moreover, we establish a stochastic differential equation for ^Xt based on S(t). Our method is based on an enlargement of filtration technique, which allows us to put the anticipative linear filter problem into the context of a non-anticipative two-dimensional linear filter problem with a correlation between the signal noise and the observation noise.

• Aase, Knut K.; Bjuland, Terje & Øksendal, Bernt (2010). Strategic Insider Trading Equilibrium: A Filter Theory Approach. Vis sammendrag

  The continuous-time version of Kyle's (1985) model of asset pricing with asymmetric information is studied, and generalized in various directions, i.e., by allowing time-varying liquidity trading, and by having weaker a priori assumptions on the model. This extension is made possible by the use of filtering theory. We derive the optimal trade for an insider and the corresponding price of the risky asset; the insider's trading intensity satisfies a deterministic integral equation, given perfect inside information.

• Aase, Knut; Bjuland, Terje & Øksendal, Bernt (2010). An anticipative linear filtering equation. Preprint series: Pure mathematics. 13. Fulltekst i vitenarkiv.
• Aase, Knut; Bjuland, Terje & Øksendal, Bernt (2010). Strategic Insider Trading Equilibrium: A Filter Theory Approach. Preprint series: Pure mathematics. 14. Fulltekst i vitenarkiv.
• Ta, An Thi Kieu & Øksendal, Bernt (2010). A Maximum Principle for Stochastic Differential Games with g–expectations and partial information. Preprint series: Pure mathematics. 4. Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2010). Robust stochastic control and equivalent martingale measures. Preprint series: Pure mathematics. 8. Fulltekst i vitenarkiv.
• Øksendal, Bernt & Sulem, Agnès (2010). Singular stochastic control and optimal stopping with partial information of Itô-Lévy processes. Preprint series: Pure mathematics. 9. Fulltekst i vitenarkiv.
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2010). Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Preprint series: Pure mathematics. 16. Fulltekst i vitenarkiv.
• Øksendal, Bernt (2009). Optimal control of PDEs and forward-backward SDEs, with applications to risk minimization.
• Øksendal, Bernt (2009). The Black-Scholes option pricing formula and beyond. Matilde, Dansk Matematisk Forening.  37, s 9- 14

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