Nils Christian Framstad

• Framstad, N. C., Continuous-time (Ross-type) Portfolio separation, (almost) without Itô calculus,  Stochastics 89(1), 2017, 38–64.
• Framstad, N. C. and J. Strand, Energy intensive infrastructure investments with retrofits in continuous time: Effects of uncertainty on energy use and carbon emissions, Resource and Energy Economics 41, 2015, 1–18
• Framstad, N. C., On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes, Brazilian Journal of Probability and Statistics 28(2), 2014, 223–240
• Framstad, N. C., When can the environmental profile and emissions reduction be optimised independently of the pollutant level?, Journal of Environmental Economics and Policy 3(1), 2014, 25–45
• Framstad, N. C., Optimal stochastic control and non-depletion of a renewable resource under Hindy-Huang style intertemporal substitution, in B.S. Jensen and T. Palokangas (eds.), Stochastic Economic Dynamics, 2007, CBS Press
• Framstad, N. C., Arrow-Mangasarian sufficient conditions for controlled semimartingales, Stochastic Analysis and Applications 24(5), 2006, 929–938
• Framstad, N. C., A. Sulem and B. Øksendal, Sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance, Journal of Optimization Theory and Applications 121(1), 2004, 77–98 (errata corrige: vol. 124(2) p511).
• Framstad, N. C., Non-robustness with respect to intervention costs in optimal control, Stochastic Analysis and Applications 22(2), 2004, 333–340
• Framstad, N. C., Optimal harvesting of a jump diffusion population and the effect of jump uncertainty, SIAM Journal On Control and Optimization 42(4), 2003, 1451–1465
• Framstad, N. C., A. Sulem and B. Øksendal, Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs, Journal of Mathematical Economics 35(2), 2001, 233–257

• Framstad, Nils Christian (2017). Continuous-time (Ross-type) portfolio separation, (almost) without Itô calculus. Stochastics: An International Journal of Probability and Stochastic Processes.  ISSN 1744-2508.  89(1), s 38- 64 . doi: 10.1080/17442508.2015.1132218 Fulltekst i vitenarkiv.
• Framstad, Nils Christian (2015). Portfolio Theory for α-Symmetric and Pseudoisotropic Distributions: k-Fund Separation and the CAPM. Journal of Probability and Statistics.  ISSN 1687-952X.  2015 . doi: 10.1155/2015/235452 Fulltekst i vitenarkiv.
• Framstad, Nils Christian & Strand, Jon (2015). Energy intensive infrastructure investments with retrofits in continuous time: Effects of uncertainty on energy use and carbon emissions. Resource and Energy Economics.  ISSN 0928-7655.  41, s 1- 18 . doi: 10.1016/j.reseneeco.2015.03.003 Fulltekst i vitenarkiv.
• Framstad, Nils Christian (2014). On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes. Brazilian Journal of Probability and Statistics.  ISSN 0103-0752.  28(2), s 223- 240 . doi: 10.1214/12-BJPS203 Fulltekst i vitenarkiv.
• Framstad, Nils Christian (2014). When can the environmental profile and emissions reduction be optimised independently of the pollutant level?. Journal of Environmental Economics and Policy.  ISSN 2160-6544.  3(1), s 25- 45 . doi: 10.1080/21606544.2013.856353 Vis sammendrag

  Consider a model for optimal timing of a policy measure which changes the emission rate, e.g. trading off the cost of reduction against the time-additive aggregate of environmental damage, the disutility from the pollutant stock M_t the infrastructure contributes to. Intuitively, the optimal timing for an infinitesimal pollution source should reasonably not depend on its historical contribution to the stock, as this is negligible. Dropping the size assumption, we show how to reduce the minimisation problem to one not depending on the history of M, under linear evolution and suitable linearity or additivity conditions on the damage functional. We employ a functional analysis framework which allows for delay equations, non-Markovian driving noise, a choice between discrete and continuous time, and a menu of integral concepts covering stochastic calculi less frequently used in resource and environmental economics. Examples are given under the common (Markovian Itô) stochastic analysis framework.

• Framstad, Nils Christian (2013). Ross-type Dynamic Portfolio Separation (almost) without Ito Stochastic Calculus. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  2013(21)
• Framstad, Nils Christian (2013). When can environmental profile and emissions reductions be optimized independently of the pollutant level?. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  2013(12)
• Framstad, Nils Christian & Strand, Jon (2013). Energy Intensive Infrastructure Investments with Retrofits in Continuous Time: Effects of Uncertainty on Energy Use and Carbon Emissions. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  2013(11)
• Framstad, Nils Christian (2011). A remark on R.S. Pindyck: "Irreversibilities and the timing of environmental policy". Resource and Energy Economics.  ISSN 0928-7655.  33(3), s 756- 760 . doi: 10.1016/j.reseneeco.2011.01.004
• Framstad, Nils Christian (2011). On free lunches in random walk markets with short-sale constraints and small transaction costs, and weak convergence to Gaussian continuous-time processes. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  20
• Framstad, Nils Christian (2011). Portfolio Separation Properties of the Skew-Elliptical Distributions. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  (02)
• Framstad, Nils Christian (2011). Portfolio Separation with α-symmetric and Psuedo-isotropic Distributions. Memorandum from Department of Economics, University of Oslo.  ISSN 0809-8786.  (12)
• Framstad, Nils Christian (2011). Portfolio separation properties of the skew-elliptical distributions, with generalizations. Statistics and Probability Letters.  ISSN 0167-7152.  81(12), s 1862- 1866 . doi: 10.1016/j.spl.2011.07.006
• Framstad, Nils Christian (2007). Optimal stochastic control and non-depletion of a renewable resource under Hindy-Huang style intertemporal substitution, In Bjarne S Jensen & Tapio Palokangas (ed.),  Stochastic Economic Dynamics.  Copenhagen Business School Press.  ISBN 9788763001854.  Kapittel 12.  s 361 - 372
• Framstad, Nils Christian (2006). Arrow-Mangasarian sufficient conditions for controlled semimartingales. Stochastic Analysis and Applications.  ISSN 0736-2994.  24, s 929- 938
• Framstad, Nils Christian (2004). Coherent portfolio separation – inherent systemic risk?. International Journal of Theoretical and Applied Finance.  ISSN 0219-0249.  07, s 909- 917 . doi: 10.1142/S0219024904002712 Vis sammendrag

  A stylized market risk model is studied. It turns out that quantifying risk by quantile-VaR, coherent risk measures or other functionals that are positively homogeneous, has a consequence akin to assuming multi-normal returns, namely a two fund separation property. Heuristic arguments indicate that this may be a source of systemic risk to the financial industry.

• Framstad, Nils Christian (2004). Non-robustness with respect to intervention costs in optimal control. Stochastic Analysis and Applications.  ISSN 0736-2994.  22, s 333- 340
• Framstad, Nils Christian (2004). On portfolio separtion in the Merton problem with bankruptcy or default, In Sergio Albeverio; Anne boutet de monvel & Habib Ouerdiane (ed.),  Proceedings of the International Conference on Stochastic Analysis and Applications.  Springer.  ISBN 978-1-4020-2467-2.  Kapittel.  s 249 - 265
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (2004). A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. Journal of Optimization Theory and Applications.  ISSN 0022-3239.  121(1), s 77- 98 . doi: 10.1023/B:JOTA.0000026132.62934.96 Vis sammendrag

  We give a verification theorem by employing Arrow’s generalization of the Mangasarian sufficient condition to a general jump diffusion setting and show the connections of adjoint processes to dynamic programming. The result is applied to financial optimization problems.

• Framstad, Nils Christian (2003). Optimal Harvesting of a Jump Diffusion Population and the Effect of Jump Uncertainty. SIAM Journal of Control and Optimization.  ISSN 0363-0129.  42(4), s 1451- 1465 . doi: 10.1137/S0363012902385910
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (2001). Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs. Journal of Mathematical Economics.  ISSN 0304-4068.  35(2), s 233- 257 . doi: 10.1016/S0304-4068(00)00067-7

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• Framstad, Nils Christian (2018). Dynamic conservation contracts. Vis sammendrag

  In order to preserve the environment, a conservation-minded principal offers owners of a stock (forests, wildlife, fossile fuels ...) state-contingent contracts which provide a running income as compensation for leaving conserving the stock rather than extracting and selling in a market with linear demand. We formulate a stylized mathematical model for such dynamic contracts. Depending on extraction cost structure, the resulting Markov differential game could yield a stable or unstable dynamic system. In the simplest unstable case, the smallest resource stocks will be depleted and only the largest conserved; furthermore, a small increase in one out of two initial stocks could reduce terminal stock by 50 %.

• Framstad, Nils Christian (2017). Corrigendum to “Portfolio Theory for α-Symmetric and Pseudoisotropic Distributions: k-Fund Separation and the CAPM”. Journal of Probability and Statistics.  ISSN 1687-952X.  2017 . doi: https://doi.org/10.1155/2017/9594547
• Framstad, Nils Christian & Harstad, Bård Gjul (2017). Conservation Contracts for Exhaustible Resources. Vis sammendrag

  This paper studies how to best incentivize owners to conserve rather than deplete exhaustible resources. This is an important issue when it comes to forest conservation agreements, but it may also become important for other environmental problems, such as climate change. We present a dynamic model where each resource owner benefits from extracting and selling the resource over time. A third party, or principal, is harmed by the extracted amount or she benefits from conservation. The principal can set up payment schedules that incentivize the owners to conserve. We show that the best contract induces the smallest resource stocks to be depleted first, while the largest stock will be extracted from later. To little is conserved permanently and the speed of extraction is too high. These three results are reversed if and only if it is very costly to protect the resource. By comparison, the first best would require that more is conserved, and that extraction begins where the extraction cost is lowest. The difference to the first best is magnified if some buyers boycott the products.

• Framstad, Nils Christian (2016). Spectrally negative stable vectors, their covariations on the positive orthant, and the Capital Asset Pricing Model. Vis sammendrag

  Non-Gaussian spectrally negative multivariate stable distributions have some properties which could be considered appealing for financial modelling, and are shown to admit a CAPM when the risky opportunities cannot be shorted. We introduce a distribution class which unifies and generalizes the spectrally negative stable class and the pseudo-isotropic class (itself generalizing the elliptical and symmetric-stables) and which admit monetary two-fund separation under no short sale and a CAPM under an additional geometric condition of embedding in L^p(R^d) for p > 1.We give the betas in term of the dispersion measure, and propose an extended definition of the covariation measure of association commonly restricted to symmetric stables with index exceeding one.

• Framstad, Nils Christian (2015). Revisiting some results and counterexamples in stochastic portfolio optimization.
• Framstad, Nils Christian (2014). The Effect of Small Intervention Costs on the Optimal Extraction of Dividends and Renewable Resources in a Jump-Diffusion Model. Memorandum from Department of Economics, University of Oslo. 25. Vis sammendrag

  A risk-neutral agent optimizes extraction of dividends or renewable natural resources modelled by a jump-diffusion stock process, where the optimal strategy is characterized as the minimal intervention required to keep the stock process inside a given region. The introduction of a small fixed cost per intervention, is shown to induce a loss at worst of order ϰ⅔, corresponding to a minimal intervention size of order ϰ⅓, under suitable conditions; there are degenerate cases if purely discontinuous harvesting is optimal for the frictionless problem. If extraction is reversible, at cost between half and twice the extraction cost, the exponents are 1/2 and 1/4, agreeing with the effect of fixed costs in a consumption–portfolio optimization problem for a risk-averse agent.

• Framstad, Nils Christian (2014). The effect of small intervention costs on the optimal extraction of dividends or of renewable resources.
• Framstad, Nils Christian (2013). Portfolio theory for a class of non-symmetric heavy-tailed distributions, and applicability to insurance.
• Framstad, Nils Christian (2011). Some cases of Ross-type portfolio separation – α-stable, α-symmetric and pseudo-isotropic distributions, with generalizations.
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (2005). Errata corrige: "Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance". Journal of Optimization Theory and Applications.  ISSN 0022-3239.  124(2), s 511- 512 . doi: 10.1007/s10957-004-0949-6
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (2001). A Sufficient Stochastic Maximum Principle for Optimal Control of Jump Diffusions and Applications to Finance. Vis sammendrag

  We give a verification theorem by employing Arrow's generalization of the Mangasarian sufficient condition to a general jump diffusion setting, and show the adjoint processes' connections to dynamic programming. The result is applied to a portfolio optimization problem.

• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (1999). Optimal Consumption and Portfolio in a Jump Diffusion Market. Vis sammendrag

  We consider the problem of optimal consumption and portfolio in a jump diffusion market consisting of a bank account and a stock, whose price is modelled by a geometric Lévy process. We show that in the absence of transaction costs, the solution in the jump diffusion case has the same form as in the pure diusion case solved by Merton [M]. In particular, the optimal portfolio is to keep a constant fraction of wealth invested in the stock. This constant is smaller than the corresponding optimal fraction in the pure diffusion case.

• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (1999). Optimal Consumption and Portfolio in a Jump Diffusion Market with proportional transaction costs.

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