We consider an optimal investment problem in which the cost of the investment decreases over time. This decrease is modelled using the negative of a non-decreasing Levy process. The decreasing cost is a way of modelling that innovations drive down the cost of the investment. We present general results on how to compute both the value of the investment, as well as the optimal time at which the investment should be done. Several explicit examples of how different Levy processes influence the value of the investment are given as illustrations of the general results. The main tools used are fluctuation theory for Levy processes and inversion of Laplace transforms. When the inversion can be done analytically, we can present analytical solutions where in some cases only numerical solution has previously been known.

Traditional models of irreversible investment problems assume that the investment starts generating cash flows immediately, i.e., at the same time as the investment is undertaken. Real-world investment situations are characterized by time-to-build or investment lags, which means that there is a time difference between when the investment is made and when the investment starts generating cash flows. We combine two existing models of investment lags to obtain a flexible, yet simple, way of modelling and analyzing the effects of investment lags. Both traditional models, and models that incorporate the effects of time-to-build, typically assume that the expected future cash flows generated by an investment are represented by a single cash flow that reflects the size of the market value of an investment. To reflect real-world cases where investments generate cash flows in several time periods, we present a framework in which cash flows are explicitly allowed to be spread out in time. Our model can be used to incorporate cases where an investment is partially sold in different time periods. Using an irreversible optimal investment timing problem case study, we show how our framework makes it possible to easily compare the effect of different cash flow timings. In this case, the value and the timing of the investment depend on a constant that in a natural way can be decomposed into three parts, thereby showing the influence of the value and timing from the respective parts of the framework.

We consider the valuation of American perpetual options with the property that they are only possible to exercise after the occurrence of a random time, which is a stopping time with respect to a given filtration. One situation where this feature is present is when making an irreversible investment, e.g. building on vacant land, while waiting for a permit to be allowed to do so. The random time in this case is the time at which the permit is given. This and the value of a version of an abandonment option are given as two applications of this modelling framework. 

Ranked events are pivotal in many important AI-applications such as Question Answering and recommendations systems. This paper studies ranked events in the setting of harness racing. For each horse there exists a probability distribution over its possible rankings. In the paper, it is shown that a set of expected positions (and more generally, higher moments) for the horses induces this probability distribution. The main contribution of the paper is a method, which extracts this induced probability distribution from a set of expected positions. An algorithm is proposed where the extraction of the induced distribution is given by the estimated expectations. MATLAB code is provided for the methodology. This approach gives freedom to model the horses in many different ways without the restrictions imposed by for instance logistic regression. To illustrate this point, we employ a neural network and ordinary ridge regression. The method is applied to predicting the distribution of the finishing positions for horses in harness racing. It outperforms both multinomial logistic regression and the market odds. The ease of use combined with fine results from the suggested approach constitutes a relevant addition to the increasingly important field of ranked events.

We consider a finite horizon optimal stopping problem with a gain function equal to the call option's. The value of the underlying process grows exponentially until a Poisson process jumps for the first time, at which the process jumps to zero and stays there forever. As applications of this model we consider valuing real options and options written on the stock of a start-up company. 

Aquaculture is a booming industry. It currently supplies almost half of all fish and shellfish eaten today, and it continues to grow faster than any other food production sector. But it is immature relative to terrestrial crop and livestock sectors, and as a consequence it lags behind in terms of the use of aquaculture specific financial risk management tools. In particular, the use of insurance instruments to manage weather related losses is little used. In the aquaculture industry there is a need for new insurance products that achieve both financial gains, in terms of reduced production and revenue risk, and environmental wins, in terms of incentivizing improved management practices. Here, we have developed a cooperative form of indemnity insurance for application to small-holder aquaculture communities in developing nations. We use and advance the theory of risk pools, applying it to an aquaculture community in Myanmar, using empirical data recently collected from a comprehensive farm survey. These data were used to parameterize numerical simulations of this aquaculture system with and without a risk pool. Results highlight the benefits and costs of a risk pool, for various combinations of key parameters. This information reveals a path forward for creating new risk management products for aquaculturalists around the world. 

In many applications of real options there is an assumption of complete capital markets. For the perpetual timing option this means that if the underlying asset does not pay out any cash flows, then there is no finite optimal time at which the investment should be undertaken. In contrast, when the market is incomplete there is a possibility of having a finite optimal stopping time even in the cases when the underlying asset does not pay out any cash flows. We discuss the incomplete case in models driven by both Brownian motion(s) and a Poisson process and connect it with the concept of an implied yield. The implied yield will in these models extend the concept of a monetary yield (i.e. a yield that represents the fraction of the value of an asset paid out as a cash flow). Several examples of incomplete market models where there could be a finite optimal time to invest are given.

We look at a characterization of elliptical distributions in the case when finiteness of moments of the random vector is not assumed. Some additional results regarding elliptical distributions are also presented.

In this thesis two dierent problems regarding real options are studied. The rst paper discusses the valuation of a timing option in an irreversible investment when the underlying model is incomplete. It is well known that in a complete model there is no nite optimal time at which to invest if the underlying asset, in our case the value of the developed project, does not pay out any strictly positive cash ows. In an incomplete model, the situation is dierent. Depending on the market price of risk in the model, there could be an optimal nite investment time even though the underlying asset does not pay out any strictly positive cash ows. Several examples of incomplete models are analyzed, and the value of the investment opportunity is calculated in each of them. The second paper concerns the valuation of random start American perpetual options. This type of perpetuate American option has the feature that it can not be exercised until a random time has occured. The reason for studying this type of option is that it provides a way of modelling the initiating of a project, e.g. the optimal time to build on a piece of land, which can not occur until a permit, or some other form of clearance, is given. The random time in the project application represents the time at which the permit is given. Two concrete examples of how to calculate the value of random start options is given.

We model a stream of cash flows as an optional stochastic process, and value the cash flows by using a continuous and strictly positive linear functional. By applying a representation theorem from the general theory of stochastic processes we are able to study this valuation principle, as well as properties of the stochastic discount factor it implies. This approach to valuation is useful in the non-presence of a financial market, as is often the case when valuing cash flows arising from insurance contracts and in the application of real options.