Bernt Karsten Øksendal

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymochenko, Olena (2026). Optimal Stopping for Systems Driven by the Brownian Sheet. arXiv.org. ISSN 2331-8422. arXiv:2603.12853 (math). doi: https:/arxiv.org/abs/2603.12853. Fulltekst i vitenarkiv Vis sammendrag

  We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part, we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two parameter first hitting point τ = (τ1,τ2) that maximizes E e−ρτ1τ2 h B(τ1,τ2), where ρ > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed form characterization of the optimal stopping threshold. In particular, for linear rewards h(y) = y, the optimal level is y= (2ρ)−1/2 . The second problem concerns optimal stopping of the integrated discounted Brownian sheet: E τ1 0 τ2 0 e−ρtxB(t,x) dtdx . We show that the optimal first hitting level is strictly positive, and provide an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product τ1τ2.The result is surprising, in view of the (well-known) fact that the solution of the corresponding one-parameter problem is to stop at the first time t when B(t) hits the negative level y=−(2ρ)−1/2 . In the second and main part of the paper, we develop a general potential theoretic framework for two parameter optimal stopping problems associated with stochastic partial differential equations (SPDEs) driven by the Brownian sheet. For a class of Itˆo sheets and time space homogeneous SPDEs, we prove that the value function is the least superharmonic majorant of the reward and that the optimal stopping point is given by the first exit point from the continuation region. This yields a general existence theorem for optimal stopping points in the plane, extending classical one parameter theory to the two parameter Brownian sheet setting.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2025). Optimal control of SPDEs driven by time-space Brownian motion. SIAM Journal of Control and Optimization. ISSN 0363-0129. 63(1), s. 546–570. doi: 10.1137/23M1595308. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we study the optimal control of systems where the state dynamics are governed by a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion, also referred to as a Brownian sheet. These equations can, for example, model the growth of an ecosystem under uncertainty. We first explore some fundamental properties of such linear SPDEs. Next, utilizing time-space white noise calculus, we derive both Pontryagin-type necessary and sufficient conditions for the optimality of the control. Finally, we illustrate our results by solving a linear-quadratic control problem and examining an optimal harvesting problem in the plane. Potential applications to machine learning and to managing random environmental influences are also discussed.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2025). A Kalman filter for linear systems driven by time-space Brownian sheet. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. s. 1–22. doi: 10.1080/17442508.2025.2541059. Fulltekst i vitenarkiv Vis sammendrag

  We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation. This can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations of a constant and noisy observations of the Brownian sheet.

• Hachemi, Rahma Yasmina Moulay & Øksendal, Bernt Karsten (2024). The stochastic heat equation with fractional time and fractional time–space white noise. Expositiones mathematicae. ISSN 0723-0869. doi: 10.1016/j.exmath.2024.125625. Fulltekst i vitenarkiv
• Grid, Maroua; Agram, Nacira; Kebiri, Omar & Øksendal, Bernt Karsten (2024). Deep learning for solving initial path optimization of mean-field systems with memory. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 97(8), s. 1016–1037. doi: 10.1080/17442508.2024.2402741. Fulltekst i vitenarkiv Vis sammendrag

  We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean–Vlasov (mean-field) stochastic differential equation with delay, driven by Brownian motion and a pure jump Poisson random measure. The goal is to determine the optimal initial values for the system in the period [−𝛿,0], where 𝛿>0 is a delay constant, before the system starts at t = 0. Due to the delay in the dynamics, the system will, after startup, be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By utilizing this equivalence, we can find implicit expressions for the optimal investment. Moreover, we propose a deep neural network-based algorithm to solve the stochastic control problem with delay. Specifically, we employ a multi-layer feed-forward neural network for control modelling in the interval [−𝛿,0], and use back-propagation to train the feed-forward neural network. The gradient of the loss function is computed using stochastic gradient descent (SGD) with respect to the weights of the network.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2024). Fokker-Planck equations for conditional McKean-Vlasov systems driven by Brownian sheets. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2412.20572. Fulltekst i vitenarkiv
• Agram, Nacira; Øksendal, Bernt Karsten & Rems, Jan (2024). Deep learning for quadratic hedging in incomplete jump market. Digital Finance. ISSN 2524-6984. 6(3), s. 463–499. doi: 10.1007/s42521-024-00112-5. Fulltekst i vitenarkiv Vis sammendrag

  We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

• Agram, Nacira; Pucci, Giulia & Øksendal, Bernt Karsten (2024). Impulse Control of Conditional McKean–Vlasov Jump Diffusions. Journal of Optimization Theory and Applications. ISSN 0022-3239. 200, s. 1100–1130. doi: 10.1007/s10957-023-02370-6. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.

• Agram, Nacira & Øksendal, Bernt Karsten (2024). Optimal stopping of conditional McKean–Vlasov jump diffusions. Systems & control letters (Print). ISSN 0167-6911. 188. doi: 10.1016/j.sysconle.2024.105815. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study the optimal stopping problem of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula. Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2024). Fokker-Planck equation for McKean-Vlasov SPDEs driven by time-space Brownian sheet. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider a McKean-Vlasov (mean-field) stochastic partial differential equations (SPDEs) driven by a Brownian sheet. We study the propagation of chaos for a space-time Ornstein-Uhlenbeck SPDE type. Subsequently, we prove the existence and uniqueness of a nonlinear McKean-Vlasov SPDE. Finally, we establish a Fokker-Planck equation for the law of the solution of the McKean-Vlasov type SPDE driven by a time-space Brownian sheet, and we provide some examples to illustrate the results obtained.

• Hachemi, Rahma Yasmina Moulay & Øksendal, Bernt Karsten (2023). The fractional stochastic heat equation driven by time-space white noise. Fractional Calculus and Applied Analysis. ISSN 1311-0454. 26(2), s. 513–532. doi: 10.1007/s13540-023-00134-7. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt Karsten (2023). Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control. SIAM Journal of Control and Optimization. ISSN 0363-0129. 61(3), s. 1472–1493. doi: 10.1137/21M1461034. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.

• Agram, Nacira & Øksendal, Bernt Karsten (2023). The Donsker delta function and local time for McKean–Vlasov processes and applications. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2023.2286252. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean–Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon–Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean–Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

• Makhlouf, K.; Agram, Nacira; Hilbert, A. & Øksendal, Bernt Karsten (2023). SPDEs with space interactions and application to population modelling. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV). ISSN 1292-8119. 29. doi: 10.1051/cocv/2023010. Fulltekst i vitenarkiv Vis sammendrag

  We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions , in the sense that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of strong, smooth solutions of a class of SPDEs with space interactions, and we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a space-mean stochastic reaction-diffusion equation.

• Øksendal, Bernt Karsten (2023). Space-time stochastic calculus and white noise. I Morel, Jean-Michel & Teissier, Bernard (Red.), Mathematics Going Forward - Collected Mathematical Brushstrokes. Springer. ISSN 9783031122439. s. 629–649. doi: 10.1007/978-3-031-12244-6_44. Fulltekst i vitenarkiv Vis sammendrag

  In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida–Malliavin calculus, which are useful tools for studying such equations. As an illustration I apply white noise calculus to find an explicit solution of an SPDE describing population growth in an environment subject to time-space white noise. Dedicated to Catriona Byrne, in gratitude for her support and encouragement through 40 years.

• Agram, Nacira; Labed, Saloua; Øksendal, Bernt Karsten & Yakhlef, Samia (2022). Singular Control of Stochastic Volterra Integral Equations. Acta Mathematica Scientia. ISSN 0252-9602. 42(3), s. 1003–1017. doi: 10.1007/s10473-022-0311-9. Fulltekst i vitenarkiv
• Agram, Nacira; Hu, Yaozhong & Øksendal, Bernt Karsten (2022). Mean-field backward stochastic differential equations and applications. Systems & control letters (Print). ISSN 0167-6911. 162. doi: 10.1016/j.sysconle.2022.105196. Fulltekst i vitenarkiv
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt Karsten & Proske, Frank Norbert (2021). Optimal stopping, randomized stopping and singular control with general information flow. Theory of Probability and its Applications. ISSN 0040-585X. 66(4), s. 760–773. doi: 10.4213/tvp5514. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt (2020). A financial market with singular drift and no arbitrage. Mathematics and Financial Economics. ISSN 1862-9679. doi: 10.1007/s11579-020-00284-9. Fulltekst i vitenarkiv Vis sammendrag

  We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay θ>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as θ>0. This implies that there is no arbitrage in the market in that case. However, when θ goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

• Draouil, Olfa & Øksendal, Bernt (2019). A white noise approach to optimal insider control of systems with delay. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 476(1), s. 101–119. doi: 10.1016/j.jmaa.2019.02.065. Fulltekst i vitenarkiv Vis sammendrag

  We use a white noise approach to study the problem of optimal insider control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system. These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the logarithmic utility optimal insider portfolio in a generalized Black-Scholes type financial market. A classical result of Pikovski and Karatzas shows that when the inside information is , where T is the terminal time of the trading period, then the market is not viable, i.e. the maximal utility is infinite. We consider two extensions to delay of this result and prove the following: If the risky asset price is given by a stochastic delay equation, the resulting insider market is still not viable. If, however, there is delay in the information flow to the insider, the market becomes viable.

• Agram, Nacira & Øksendal, Bernt (2019). Model uncertainty stochastic mean-field control. Stochastic Analysis and Applications. ISSN 0736-2994. 37(1), s. 36–56. doi: 10.1080/07362994.2018.1499036. Fulltekst i vitenarkiv
• Draouil, Olfa & Øksendal, Bernt (2019). Viable insider markets. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 91(7), s. 959–973. doi: 10.1080/17442508.2019.1612895. Fulltekst i vitenarkiv
• Agram, Nacira; Bachouch, Achref; Øksendal, Bernt & Proske, Frank Norbert (2019). Singular control and optimal stopping of memory mean-field processes. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 51(1), s. 450–468. doi: 10.1137/18M1174787. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory; (ii) reflected advanced mean-field backward stochastic differential equations; and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: (1) We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations; (2) we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information; and (3) we deduce a relation between the optimal singular control of an MMSDE and the optimal stopping of such processes.

• Agram, Nacira; Hilbert, Astrid & Øksendal, Bernt (2019). Singular control of SPDEs with space-mean dynamics. Mathematical Control and Related Fields. ISSN 2156-8472. doi: 10.3934/mcrf.2020004. Fulltekst i vitenarkiv Vis sammendrag

  We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for these control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

• Agram, Nacira & Øksendal, Bernt (2019). Mean-field stochastic control with elephant memory in infinite time horizon. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 91(7), s. 1041–1066. doi: 10.1080/17442508.2019.1635600. Fulltekst i vitenarkiv Vis sammendrag

  Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2019). New approach to optimal control of stochastic Volterra integral equations. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 91(6), s. 873–894. doi: 10.1080/17442508.2018.1557186. Fulltekst i vitenarkiv Vis sammendrag

  We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus. We give conditions under which there exist unique solutions of such equations. Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus. As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

• Dumitrescu, Roxana; Øksendal, Bernt & Sulem, Agnès (2018). Stochastic Control for Mean-Field Stochastic Partial Differential Equations with Jumps. Journal of Optimization Theory and Applications. ISSN 0022-3239. 176(3), s. 559–584. doi: 10.1007/s10957-018-1243-3. Fulltekst i vitenarkiv
• Agram, Nacira; Hu, Yaozhong & Øksendal, Bernt (2018). Mean-field backward stochastic differential equations and applications. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Draouil, Olfa & Øksendal, Bernt (2018). Optimal insider control of stochastic partial differential equations. Stochastics and Dynamics. ISSN 0219-4937. 18(1). doi: 10.1142/S0219493718500144. Fulltekst i vitenarkiv Vis sammendrag

  We study the problem of optimal insider control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: - The controller has access to inside information, i.e. access to information about a future state of the system. - The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and necessary maximum principle for this type of control problem, in two cases: - The control is allowed to depend both on time t and on the space variable x. - The control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from non-linear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples. The final version of this research is published by Stochastics and Dynamics. © World Scientific Publishing

• Hu, Yaozhong & Øksendal, Bernt (2018). Linear Volterra backward stochastic integral equations. Stochastic Processes and their Applications. ISSN 0304-4149. 129(2), s. 626–633. doi: 10.1016/j.spa.2018.03.016. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt (2018). A Hida-Malliavin white noise calculus approach to optimal control. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 21(3), s. 1850014-1–1850014-21. doi: 10.1142/S0219025718500145. Fulltekst i vitenarkiv Vis sammendrag

  The classical maximum principle for optimal stochastic control states that if a control û is optimal, then the corresponding Hamiltonian has a maximum at u=û. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2018). Optimal control of forward-backward stochastic Volterra equations, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis: The Helge Holden Anniversary Volume. European Mathematical Society (EMS) Press. ISSN 9783037191866. s. 3–36. Fulltekst i vitenarkiv
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2018). Optimal stopping, randomized stopping and singular control with partial information flow. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt (2018). Mean-Field Stochastic Control with Elephant Memory in Finite and Infinite Time Horizon. arXiv.org. ISSN 2331-8422. doi: 10.1080/17442508.2019.1635600. Fulltekst i vitenarkiv
• Agram, Nacira; Hilbert, Astrid & Øksendal, Bernt (2018). SPDEs with Space-Mean Dynamics. arXiv.org. ISSN 2331-8422. doi: 10.3934/mcrf.2020004. Fulltekst i vitenarkiv
• Dumitrescu, Roxana; Øksendal, Bernt & Sulem, Agnès (2018). Stochastic control of general mean-field SPDEs with jumps. Journal of Optimization Theory and Applications. ISSN 0022-3239. 176(3), s. 559–584. doi: 10.1007/s10957-018-1243-3. Fulltekst i vitenarkiv Vis sammendrag

  We study optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove the existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We apply our results to find the explicit optimal control for an optimal harvesting problem.

• Biagini, Francesca; Meyer-Brandis, Thilo; Øksendal, Bernt & Paczka, Krzysztof Jaroslaw (2018). Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk (PUQR). ISSN 2095-9672. 3. doi: 10.1186/s41546-018-0033-z. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we study strongly robust optimal control problems under volatility uncertainty. In the G-framework, we adapt the stochastic maximum principle to find necessary and sufficient conditions for the existence of a strongly robust optimal control.

• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2017). New approach to optimal control of stochastic Volterra integral equations. arXiv.org. ISSN 2331-8422. s. 1–22. doi: 10.1080/17442508.2018.1557186. Fulltekst i vitenarkiv
• Øksendal, Bernt & Røse, Elin Engen (2017). A white noise approach to insider trading. I Hida, Takeyuki & Streit, Ludwig (Red.), Let Us Use White Noise. World Scientific. ISSN 9789813220935. s. 191–204. doi: 10.1142/9789813220942_0006. 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 (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 Vis sammendrag

  By a memory mean-field process we mean the solution X(\cdot ) of a stochastic mean-field equation involving not just the current state X(t) and its law \mathcal {L}(X(t)) at time t, but also the state values X(s) and its law \mathcal {L}(X(s)) at some previous times s < t. Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space \mathcal {M} of measures on \mathbb {R} with the norm || \cdot ||_{\mathcal {M}} introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-advanced backward stochastic differential equations (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process. The final version of this research has been published in Applied Mathematics and Optimization. © Springer Verlag

• 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 Vis sammendrag

  We introduce the concept of singular recursive utility. This leads to a kind of singular backward stochastic differential equation (BSDE) which, to the best of our knowledge, has not been studied before. We show conditions for existence and uniqueness of a solution for this kind of singular BSDE. Furthermore, we analyze the problem of maximizing the singular recursive utility. We derive sufficient and necessary maximum principles for this problem, and connect it to the Skorohod reflection problem. Finally, we apply our results to a specific cash flow. In this case, we find that the optimal consumption rate is given by the solution to the corresponding Skorohod reflection problem. This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics on 24 Mar 2017, available online: http://www.tandfonline.com/10.1080/17442508.2017.1303067

• 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 Vis sammendrag

  This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod problem, whose solution is proved to exist uniquely. Motivations are given as optimal harvesting of stochastic mean-field systems, optimal irreversible investments under uncertainty and mean-field singular investment games. In particular, a simple singular mean-field investment game is studied, where the Nash equilibrium exists but is not unique. The final version of this research has been published in Stochastic Analysis and Applications. © 2017 Taylor & Francis

• Agram, Nacira & Øksendal, Bernt (2016). Model Uncertainty Stochastic Mean-Field Control. arXiv.org. ISSN 2331-8422. doi: 10.1080/07362994.2018.1499036. 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 Vis sammendrag

  We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida–Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.

• 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 Vis sammendrag

  We combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples. This is the Author’s Original Manuscript of an article published by Taylor & Francis in Stochastic Analysis and Applications on 01 Sep 2016, available online: http://www.tandfonline.com/10.1080/07362994.2016.1200989

• Agram, Nacira; Øksendal, Bernt & Yakhlef, Samia (2016). Optimal control of forward-backward stochastic Volterra equations. arXiv.org. ISSN 2331-8422. doi: 10.4171/186-1/1. 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 Vis sammendrag

  A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth X^*(T) : = X_{\varphi ^*}(T) of the problem to maximize the expected U-utility of the terminal wealth X_{\varphi }(T) generated by admissible portfolios \varphi (t); 0 \le t \le T in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario \frac{dQ^*}{dP} of the dual problem to minimize the expected V-value of \frac{dQ}{dP} over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all t \in [0,T]. We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process; 0 \le t \le T. In the terminal time case t=T we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \varphi ^* and the optimal measure Q^*. We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples. The final version of this research has been published in Applied Mathematics and Optimization. © Springer Verlag

• 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 Vis sammendrag

  We prove a verification theorem for a class of singular control problems which model optimal harvesting with density-dependent prices or optimal dividend policy with capitaldependent utilities. The result is applied to solve explicitly some examples of such optimal harvesting/optimal dividend problems. In particular, we show that if the unit price decreases with population density, then the optimal harvesting policy may not exist in the ordinary sense, but can be expressed as a ”chattering policy”, i.e. the limit as ∆x and ∆t go to 0 of taking out a sequence of small quantities of size ∆x within small time periods of size ∆t.

• Øksendal, Bernt & Sulem, Agnès (2016). Optimal control of predictive mean-field equations and applications to finance. Springer Proceedings in Mathematics & Statistics. ISSN 2194-1017. 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. I Podolskij, Mark; Stelzer, Robert; Thorbjørnsen, Steen & Veraart, Almut E. D. (Red.), The Fascination of Probability, Statistics and their Applications. In honour of Ole E. Barndorff-Nielsen. Springer Science+Business Media B.V.. ISSN 9783319258249. s. 435–446. doi: 10.1007/978-3-319-25826-3_20. Fulltekst i vitenarkiv 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.

• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2016). A stochastic HJBI equation for optimal control of forward-backward SDEs. I Podolskij, Mark; Stelzer, Robert; Thorbjørnsen, Steen & Veraart, Almut E. D. (Red.), The Fascination of Probability, Statistics and their Applications. In honour of Ole E. Barndorff-Nielsen. Springer Science+Business Media B.V.. ISSN 9783319258249. Fulltekst i vitenarkiv
• Øksendal, Bernt & Sulem, Agnès (2016). Optimal control of predictive mean-field equations and applications in finance. I Benth, Fred Espen & Nunno, Giulia Di (Red.), Stochastics of Environmental and Financial Economics. Springer Science+Business Media B.V.. ISSN 9783319234243. doi: 10.1007/978-3-319-23425-0_12. 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 Vis sammendrag

  In this article we consider a stochastic optimal control problem where the dynamics of the state process, X(t), is a controlled stochastic differential equation with jumps, delay and noisy memory. The term noisy memory is, to the best of our knowledge, new. By this we mean that the dynamics of X(t) depend on R t t−δ X(s)dB(s) (where B(t) is a Brownian motion). Hence, the dependence is noisy because of the Brownian motion, and it involves memory due to the influence from the previous values of the state process. We derive necessary and sufficient maximum principles for this stochastic control problem in two different ways, resulting in two sets of maximum principles. The first set of maximum principles is derived using Malliavin calculus techniques, while the second set comes from reduction to a discrete delay optimal control problem, and application of previously known results by Øksendal, Sulem and Zhang. The maximum principles also apply to the case where the controller has only partial information, in the sense that the admissible controls are adapted to a sub-σ-algebra of the natural filtration.

• 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 Vis sammendrag

  This paper is mainly a survey of recent research developments regarding methods for risk minimization in financial markets modeled by Itô-Lévy processes, but it also contains some new results on the underlying stochastic maximum principle. The concept of a convex risk measure is introduced, and two representations of such measures are given, namely: (i) the dual representation and (ii) the representation by means of backward stochastic differential equations (BSDEs) with jumps. Depending on the representation, the corresponding risk minimal portfolio problem is studied, either in the context of stochastic differential games or optimal control of forward-backward SDEs. The related concept of recursive utility is also introduced, and corresponding recursive utility maximization problems are studied. In either case the maximum principle for optimal stochastic control plays a crucial role, and in the paper we prove a version of this principle which is stronger than what was previously known. The theory is illustrated by examples, showing explicitly the risk minimizing portfolio in some cases. Afrika Matematika May 2014, Available online on 18 May 2014 The final publication is available at Springer

• Øksendal, Bernt & Sulem, Agnès (2015). Applications of stochastic analysis. I Higham, Nicholas J. (Red.), The Princeton Companion to Applied Mathematics. Princeton University Press. ISSN 9780691150390. Fulltekst i vitenarkiv
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2015). A comparison theorem for backward SPDEs with jumps. I Chen, Zhen-Qing (Red.), Festscrift Masatoshi Fukushima. World Scientific. ISSN 9789814596527. doi: 10.1142/9789814596534_0023. 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 Vis sammendrag

  We study optimal insider control problems, i.e. optimal control problems of stochastic systems where the controller at any time t, in addition to knowledge about the history of the system up to this time, also has additional information related to a future value of the system. Since this puts the associated controlled systems outside the context of semimartingales, we apply anticipative white noise analysis, including forward integration and Hida-Malliavin calculus to study the problem. Combining this with Donsker delta functionals we transform the insider control problem into a classical (but parametrised) adapted control system, albeit with a non-classical performance functional. We establish a sufficient and a necessary maximum principle for such systems. Then we apply the results to obtain explicit solutions for some optimal insider portfolio problems in financial markets described by Itˆo-L´evy processes. Finally, in the Appendix we give a brief survey of the concepts and results we need from the theory of white noise, forward integrals and Hida-Malliavin calculus.

• 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 Vis sammendrag

  Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory. The final version of this research has been published in Journal of Optimization Theory and Applications. © 2015 Springer Verlag

• 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 Vis sammendrag

  We consider a problem of optimal control of an infinite horizon system governed by forward–backward stochastic differential equations with delay. Sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived. We illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay. NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics

• 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 Vis sammendrag

  We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index. Published online: 01 Sep 2012 The final publication is available at Springer

• 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 Vis sammendrag

  We give a short introduction to the stochastic calculus for Itô-Lévy processes and review briefly the two main methods of optimal control of systems described by such processes: (i) Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation (ii) The stochastic maximum principle and its associated backward stochastic differential equation (BSDE). The two methods are illustrated by application to the classical portfolio optimization problem in finance. A second application is the problem of risk minimization in a financial market. Using a dual representation of risk, we arrive at a stochastic differential game, which is solved by using the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, which is an extension of the HJB equation to stochastic differential games. Communications on Stochastic Analysis Vol. 8, No. 1, March 2014 s. 1-15 https://www.math.lsu.edu/cosa/ Posted here with permission.

• 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 & 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 Vis sammendrag

  We consider general singular control problems for random fields given by a stochastic partial differential equation (SPDE). We show that under some conditions the optimal singular control can be identified with the solution of a coupled system of SPDE and a reflected backward SPDE (RBSPDE). As an illustration we apply the result to a singular optimal harvesting problem from a population whose density is modeled as a stochastic reaction-diffusion equation. Existence and uniqueness of solutions of RBSPDEs are established, as well as comparison theorems. We then establish a relation between RBSPDEs and optimal stopping of SPDEs, and apply the result to a risk-minimizing stopping problem.

• 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. doi: 10.1007/s11009-014-9397-4. Fulltekst i vitenarkiv Vis sammendrag

  We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the optimal portfolio problem has a solution up to a stopping time, if and only if the (normalised) marginal utility of the terminal wealth generates a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control problems under partial information. We then characterize a global notion of market viability in terms of partial information local martingale deflators (PILMDs). We illustrate our results by means of a simple example. Methodology and Computing in Applied Probability February 2014 . Published online: 22 Feb 2014 The final publication is available at Springer

• 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. Fulltekst i vitenarkiv
• 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 Vis sammendrag

  In this paper, we prove a 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 Itô–Lévy process, and to increase realism information is delayed, e.g., due to production time. A special feature of our time-continuous model is that it allows for a price-dependent demand, thereby opening for strategies where pricing is used to manipulate the demand.

• 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 Vis sammendrag

  We study stochastic differential games of jump diffusions driven by Brownian motions and compensated Poisson random measures, where one of the players can choose the stochastic control and the other player can decide when to stop the system. We prove a verification theorem for such games in terms of a Hamilton–Jacobi–Bellman variational inequality. The results are applied to study some specific examples, including optimal resource extraction in a worst-case scenario, and risk minimizing optimal portfolio and stopping. This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports

• 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 Vis sammendrag

  We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results with an application to the optimal consumption rate from an economic quantity.

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• Øksendal, Bernt (2020). A financial market with singular drift and no arbitrage. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to White Noise Theory and Applications. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to Stochastic Control Theory for Jump Diffusions. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to Stochastic Control of Jump Diffusions, with Applications to Mathematical Finance. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). A financial market with a local time drift term and delay. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to White Noise Theory and Applications. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to Stochastic Control Theory for Jump Diffusions. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). Introduction to Stochastic Control of Jump Diffusions, with Applications to Mathematical Finance. Fulltekst i vitenarkiv
• Øksendal, Bernt (2019). A financial market with a local time drift term and delay. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Optimal control of stochastic Volterra equations. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Optimal control of mean-field stochastic differential equations. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Introduction to Stochastic Control and applications. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Introduction to Stochastic Control with Applications. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). An Introduction to Stochastic Control, with Applications to Mathematical Finance. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). SPDEs with space - mean dynamics. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt (2018). Correction to: Stochastic Control of Memory Mean-Field Processes. Applied Mathematics and Optimization. ISSN 0095-4616. 79(1), s. 205–206. doi: 10.1007/s00245-018-9483-z. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Introduction to stochastic control with jump diffusions and applications to finance. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). New approach to optimal control of stochastic Volterra equations. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Introduction to optimal stochastic control with inside information. Fulltekst i vitenarkiv
• Øksendal, Bernt (2018). Model uncertainty stochastic mean-field control. Fulltekst i vitenarkiv
• Øksendal, Bernt (2017). Optimal control of mean-field systems with memory. Fulltekst i vitenarkiv
• Øksendal, Bernt (2017). Optimal control of mean-field systems with memory. Fulltekst i vitenarkiv
• Dahl, Kristina Rognlien & Øksendal, Bernt (2015). Singular recursive utility. Fulltekst i vitenarkiv
• Øksendal, Bernt & Sulem, Agnès (2015). Applications of stochastic analysis. I Higham, Nicholas J. (Red.), The Princeton Companion to Applied Mathematics. Princeton University Press. ISSN 9780691150390. s. 319–327. Fulltekst i vitenarkiv
• Dahl, Kristina Rognlien; Øksendal, Bernt; Røse, Elin Engen & Mohammed, Salah-Eldin (2014). Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives. Fulltekst i vitenarkiv
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2014). A comparison theorem for backward SPDEs with jumps. Fulltekst i vitenarkiv
• Øksendal, Bernt; Sulem, Agnès & Zhang, Tusheng (2014). A comparison theorem for backward SPDEs with jumps. Fulltekst i vitenarkiv
• Aase, Knut Kristian & Øksendal, Bernt (2019). Strategic Insider Trading Equilibrium with a non-fiduciary market maker. Norges Handelshøyskole. Fulltekst i vitenarkiv Vis sammendrag

  The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of the order ow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.

• Aase, Knut Kristian & Øksendal, Bernt (2019). Strategic Insider Trading in Continuous Time: A New Approach. Norges Handelshøyskole. Fulltekst i vitenarkiv Vis sammendrag

  The continuous-time version of Kyle's (1985) model of asset pricing with asymmetric information is studied, and generalized by allowing time-varying noise trading. From rather simple assumptions we are able to derive the optimal trade for an insider; the trading intensity satisfies a deterministic integral equation, given perfect inside information, which we give a closed form solution to. We use a new technique called forward integration in order to find the optimal trading strategy. This is an extension of the stochastic integral which takes account of the informational asymmetry inherent in this problem. The market makers' price response is found by the use of filtering theory. The novelty is our approach, which could be extended in scope.

• Corcuera, Jose Manuel; Nunno, Giulia Di; Farkas, Gergely & Øksendal, Bernt (2014). A continuous auction model with insiders and random time of information release. Universitetet i Oslo. Fulltekst i vitenarkiv
• Agram, Nacira & Øksendal, Bernt (2013). Infinite horizon optimal control of forward-backward stochastic differential equations with delay. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  We consider a problem of optimal control of an infinite horizon system governed by forward-backward stochastic differential equations with delay. Sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived. We illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay.

• Øksendal, Bernt & Sulem, Agnès (2013). A stochastic control approach to robust duality in utility maximization. Universitetet i Oslo. Fulltekst i vitenarkiv

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