Nils Christian Framstad

• Framstad, N. C., Continuous-time (Ross-type) Portfolio separation, (almost) without Itô calculus, Stochastics, 89, 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), p. 38–64. doi: 10.1080/17442508.2015.1132218. Full text in Research Archive Show summary

  This paper shows how the distributions-based portfolio separation theorem – also known as the mutual fund theorem – for elliptical and stable distributions carries over from a static to a continuous-time model. Without invoking Itô stochastic calculus, only the definition of the Itô integral, we generalize and simplify an approach of Khanna and Kulldorff (http://link.springer.com/article/10.1007%2Fs007800050056 Finance Stoch. 3 (1999), pp. 167–185). In addition to (re-) covering the classical cases, this paper also gives separation results for non-symmetric stable distributions under no shorting-conditions, including a new case of one fund separation without risk-free opportunity. Applicability of the skewed cases to insurance and banking is discussed, as well as limitations. This research has been published in Stochastics: An International Journal of Probability and Stochastic Processes. © 2017 Taylor & Francis

• 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. Full text in Research Archive Show summary

  The shifted pseudoisotropic multivariate distributions are shown to satisfy Ross’ stochastic dominance criterion for two-fund monetary separation in the case with risk-free investment opportunity and furthermore to admit the Capital Asset Pricing Model under an embedding in L𝛼 condition if1 < 𝛼 ≤ 2, with the betas given in an explicit form. For the 𝛼-symmetric subclass, the market without risk-free investment opportunity admits 2𝑑-fund separation if 𝛼 = 1+1/(2𝑑−1),𝑑 ∈ N, generalizing the classical elliptical case 𝑑 = 1, and we also give the precise number of funds needed, from which it follows that we cannot, except degenerate cases, have a CAPM without risk-free opportunity. For the symmetric stable subclass, the index of stability is only of secondary interest, and several common restrictions in terms of that index can be weakened by replacing it by the (no smaller) indices of symmetry/of embedding. Finally, dynamic models with intermediate consumption inherit the separation properties of the static models.

• 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, p. 1–18. doi: 10.1016/j.reseneeco.2015.03.003. Full text in Research Archive Show summary

  Energy-intensive infrastructure may tie up fossil energy use and carbon emissions for a long time after investment, and thus be crucial for the ability to control long-run emissions. Much or most of the resulting carbon emissions can often be eliminated later, through a retrofit that may however be costly. This paper studies the joint decision to invest in such infrastructure, and retrofit it later, given that future climate damages are uncertain and follow a geometric Brownian motion process with positive drift. We find that higher climate cost volatility (for given unconditional expected costs) then delays the retrofit decision by increasing the option value of waiting to invest. The initial infrastructure is also chosen with higher energy intensity, further increasing total emissions, when volatility is higher. We provide conditions under which higher climate cost volatility increases total expected discounted climate damage from the infrastructure, which happens in a wide set of circumstances.

• 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), p. 25–45. doi: 10.1080/21606544.2013.856353. Full text in Research Archive Show summary

  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 (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), p. 223–240. doi: 10.1214/12-BJPS203. Full text in Research Archive Show summary

  This paper considers a sequence of discrete-time random walk markets with a safe and a single risky investment opportunity, and gives conditions for the existence of arbitrages or free lunches with vanishing risk, of the form of waiting to buy and selling the next period, with no shorting, and furthermore for weak convergence of the random walk to a Gaussian continuous-time stochastic process. The conditions are given in terms of the kernel representation with respect to ordinary Brownian motion and the discretisation chosen. Arbitrage and free lunch with vanishing risk examples are established where the continuous-time analogue is arbitrage-free under small transaction costs—including for the semimartingale modifications of fractional Brownian motion suggested in the seminal Rogers [Math. Finance 7 (1997) 95–105] article proving arbitrage in fBm models. © Brazilian Statistical Association, 2014

• 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). doi: 10.1016/j.reseneeco.2015.03.003. Full text in Research Archive
• 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). doi: 10.1080/17442508.2015.1132218. Full text in Research Archive
• 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). doi: 10.1080/21606544.2013.856353. Full text in Research Archive
• Framstad, Nils Christian (2011). Portfolio Separation Properties of the Skew-Elliptical Distributions. Memorandum from Department of Economics, University of Oslo. ISSN 0809-8786. Full text in Research Archive
• Framstad, Nils Christian (2011). Portfolio Separation with α-symmetric and Psuedo-isotropic Distributions. Memorandum from Department of Economics, University of Oslo. ISSN 0809-8786. doi: 10.1155/2017/9594547. Full text in Research Archive
• 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. doi: 10.1214/12-bjps203. Full text in Research Archive
• 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), p. 756–760. doi: 10.1016/j.reseneeco.2011.01.004. Full text in Research Archive
• Framstad, Nils Christian (2011). Portfolio separation properties of the skew-elliptical distributions, with generalizations. Statistics and Probability Letters. ISSN 0167-7152. 81(12), p. 1862–1866. doi: 10.1016/j.spl.2011.07.006. Full text in Research Archive
• Framstad, Nils Christian (2007). Optimal stochastic control and non-depletion of a renewable resource under Hindy-Huang style intertemporal substitution. In Jensen, Bjarne S & Palokangas, Tapio (Ed.), Stochastic Economic Dynamics. Copenhagen Business School Press. ISSN 9788763001854. p. 361–372. Full text in Research Archive
• Framstad, Nils Christian (2006). Arrow-Mangasarian sufficient conditions for controlled semimartingales. Stochastic Analysis and Applications. ISSN 0736-2994. 24, p. 929–938. Full text in Research Archive
• 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), p. 77–98. doi: 10.1023/JOTA.0000026132.62934.96. Full text in Research Archive Show summary

  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 (2004). On portfolio separtion in the Merton problem with bankruptcy or default. In Albeverio, Sergio; monvel, Anne boutet de & Ouerdiane, Habib (Ed.), Proceedings of the International Conference on Stochastic Analysis and Applications. Springer. ISSN 9781402024672. p. 249–265. Full text in Research Archive
• Framstad, Nils Christian (2004). Coherent portfolio separation – inherent systemic risk? International Journal of Theoretical and Applied Finance. ISSN 0219-0249. 07, p. 909–917. doi: 10.1142/S0219024904002712. Full text in Research Archive Show summary

  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, p. 333–340. Full text in Research Archive
• 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), p. 1451–1465. doi: 10.1137/S0363012902385910. Full text in Research Archive
• 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), p. 233–257. doi: 10.1016/S0304-4068(00)00067-7. Full text in Research Archive

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• Framstad, Nils Christian (2025). Norm-symmetries in nonlinear expectation spaces with implications for portfolio choice and the CAPM. Full text in Research Archive Show summary

  Ambiguous probabilities and nonlinear expectation spaces offer a much richer class of distributional properties. We review some attempts at justifying a “CAPM”-type asset pricing model and propose more distributions that do admit the crucial property of two-fund separation. We point out symmetries—and thus portfolio choices—that may well arise on probability spaces, and symmetries which cannot or have implications often considered objectionable.

• Framstad, Nils Christian (2019). Introduction to stochastic control theory with applications to finance. Full text in Research Archive
• Framstad, Nils Christian (2018). Dynamic conservation contracts. Full text in Research Archive Show summary

  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: 10.1155/2017/9594547. Full text in Research Archive
• Framstad, Nils Christian & Harstad, Bård Gjul (2017). Conservation Contracts for Exhaustible Resources. Full text in Research Archive Show summary

  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. Full text in Research Archive Show summary

  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. Full text in Research Archive
• Framstad, Nils Christian (2014). The effect of small intervention costs on the optimal extraction of dividends or of renewable resources. Full text in Research Archive
• Framstad, Nils Christian (2013). Portfolio theory for a class of non-symmetric heavy-tailed distributions, and applicability to insurance. Full text in Research Archive
• Framstad, Nils Christian (2011). Some cases of Ross-type portfolio separation – α-stable, α-symmetric and pseudo-isotropic distributions, with generalizations. Full text in Research Archive
• 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), p. 511–512. doi: 10.1007/s10957-004-0949-6. Full text in Research Archive
• 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. Full text in Research Archive Show summary

  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 (2002). NON-ROBUSTNESS WITH RESPECT TO INTERVENTION COSTS IN OPTIMAL CONTROL. Universitetet i Oslo. Full text in Research Archive Show summary

  It is well known that a wide range of impulse control problems are continuous, but not differentiable, with respect to intervention cost as it approaches zero. We show a similar non-C1 result, assuming that the problem and the value function are sufficiently nice. The result is that if the value function is locally C1 in the state variable, then it does not admit first order approximation in the intervention cost. Our analysis also covers problems with reflection (i.e. vanishing impulse).

• Framstad, Nils Christian (2002). ARROW-MANGASARIAN SUFFICIENT CONDITIONS FOR CONTROLLED SEMIMARTINGALES. Universitetet i Oslo. Full text in Research Archive Show summary

  This paper shows a version of Arrow's generalization of Mangasarian's sufficient conditions valid for controlled stochastic differential equations driven by semimartingales. The infinite case is covered. An example is given.

• Framstad, Nils Christian (2001). A REMARK ON BANKRUPTCY AND DEFAULT IN THE MERTON PROBLEM. Universitetet i Oslo. Full text in Research Archive Show summary

  When portfolio choice in discontinuous asset price models is considered, the possibility of a jump to 0 is often overlooked. We solve the Merton problem for a (modified) HARA agent in a geometric Lévy market where the market coefficients can change simultanously with the jumps in the prices. Two fund separation does to some extent carry over.

• Framstad, Nils Christian (2001). POTFOLIO SEPARATION WITHOUT STOCHASTIC CALCULUS (ALMOST). Universitetet i Oslo. Full text in Research Archive Show summary

  While it is common knowledge that portfolio separation in a continuous-time lognormal market is due to the basic properties of the normal distribution, the usual exposition found in text books relies on dynamic programming and therefore invokes Itô stochastic calculus. Khanna & Kulldorff (1999) gives a rigorous proof which essentially reduces to the elementary properties assuming a risk free asset exists, an assumption we drop. Further simplifications are given, and generalizations to (symmetric and non-symmetric) alpha-stable driving noise.

• Framstad, Nils Christian (2001). A REMARK ON NON-DEPLETION OF A NATURAL RESOURCE UNDER INTERTEMPORAL PREFERENCES. Universitetet i Oslo. Full text in Research Archive Show summary

  For a wide class of models concerning the optimal harvesting of a natural resource, an expected profit maximizer will not deplete the resource completely if its relative growth rate is strictly greater than the interest rate. This well-known principle is extended to preferences with durability in consumption, and which are risk averse (or linear) sufficiently close to zero, as long as immediate depletion yields finite utility.

• Framstad, Nils Christian (2001). JUMP UNCERTAINTY VERSUS BROWNIAN NOISE IN STOCHASTIC OPTIMAL HARVESTING MODELS. Universitetet i Oslo. Full text in Research Archive Show summary

  The problem of irreversibly harvesting from a general one dimensional (Wiener-Poisson) jump diffusion population model is studied. For a wide class of models, including stochastic generalizations of the logistic model earlier studied by Lungu & Øksendal (1997) and Alvarez & Shepp (1998), the optimal strategy is a downwards local time reflection at a trigger level x*. Both these works find that this trigger level is higher than of the corresponding deterministic problem; we show that this property depends crucially upon the uncertainty being Brownian. Furthermore, we give conditions under which jump uncertainty also increases x* compared to the deterministic model.

• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (2001). A Sufficient Stochastic Maximum Principle for Optimal Control of Jump Diffusions and Applications to Finance. Universitetet i Oslo. ISSN 9788255313120. Full text in Research Archive Show summary

  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 (2000). PORTFOLIO SEPARATION IN MARKETS DRIVEN BY INDEPENDENT INCREMENT PROCESSES. Universitetet i Oslo. Full text in Research Archive
• Framstad, Nils Christian; Øksendal, Bernt Karsten & Sulem, Agnès (2000). Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs. Universitetet i Oslo. Full text in Research Archive
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (1999). Optimal Consumption and Portfolio in a Jump Diffusion Market with proportional transaction costs. INRIA. Full text in Research Archive
• Framstad, Nils Christian; Øksendal, Bernt & Sulem, Agnès (1999). Optimal Consumption and Portfolio in a Jump Diffusion Market. Norges Handelshøyskole. Full text in Research Archive Show summary

  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.

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