Mathematics and Financial Economics最新文献

We study the impact of transition scenario uncertainty, namely that of future carbon price and electricity demand, on the pace of decarbonization of the electricity industry. To this end, we develop a theory of optimal stopping mean-field games with non-Markovian common noise and partial observation. For mathematical tractability, the theory is formulated in discrete time and with common noise restricted to a finite probability space. We prove the existence of Nash equilibria for this game using the linear programming approach. We then apply the general theory to build a discrete time model for the long-term dynamics of the electricity market subject to common random shocks affecting the carbon price and the electricity demand. We consider two classes of agents: conventional producers and renewable producers. The former choose an optimal moment to exit the market and the latter choose an optimal moment to enter the market by investing into renewable generation. The agents interact through the market price determined by a merit order mechanism with an exogenous stochastic demand. We illustrate our model by an example inspired by the UK electricity market, and show that scenario uncertainty leads to significant changes in the speed of replacement of conventional generators by renewable production.

We propose a model based on a large number of small competitive producers of renewable energies, to study the effect of subsidies on the aggregate level of capacity, taking into account a cannibalization effect. We first derive a model to explain how long-time equilibrium can be reached on the market of production of renewable electricity and compare this equilibrium to the case of monopoly. Then we consider the case in which other capacities of production adjust to the production of renewable energies. The analysis is based on a master equation and we get explicit formulae for the long-time equilibria. We also provide new numerical methods to simulate the master equation and the evolution of the capacities. Thus we find the optimal subsidies to be given by a central planner to the installation and the production in order to reach a desired equilibrium capacity.

We introduce a smooth decision model under ambiguity by the belief-dependent utility (BDU) proposed in Fan (Acta Math Appl Sin 37(4):682–696, 2021). Using the smooth decision model under BDU, we get the explicit optimal insurance policy for the insurer. Then the optimal insurance policy for the insured under premium constraint (the insurer is assumed to be risk neutral) is studied. The explicit results can explain some notable behaviors in insurance demand which are inconsistent with the classical insurance contracting literature. For example, if the insured is very sensitive to small losses and the insurer is not so sensitive to small losses (or the insurer is risk neutral), the insured will prefer to purchase warranties for small losses rather than buy protections against devastating losses, which is consistent with some insurance demand behaviors observed on the insurance market. If the insured is less sensitive to small losses than the insurer, insurance policy against large losses above a deductible will be popular. At last, this paper provides an example.

We provide an elementary proof of the dual representation of Expected Shortfall on the space of integrable random variables over a general probability space. Unlike the results in the extant literature, our proof only exploits basic properties of quantile functions and can thus be easily implemented in any graduate course on risk measures. As a byproduct, we obtain a new proof of the subadditivity of Expected Shortfall.