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We are concerned with a two-component reaction-advectiondiffusion Lotka-Volterra competition system with constant diffusion rates subject to homogeneous Neumann boundary conditions. We first prove the global existence and uniform boundedness of positive classical solutions to this system. This result complements some of the global existence results in [Y. Lou, M. Winkler and Y. Tao , SIAM J. Math. Anal., 46 (2014), 1228-1262.], where one diffusion rate is taken to be a linear function of the population density. Our second result proves that the total population of each species admits a positive lower bound, under some conditions of system parameters (e.g., when the intraspecific competition rates are large). This result of population persistence indicates that the two competing species coexist over the habitat in a long time.

The methods of complex networks have been extensively used to characterize information flow in complex systems, such as risk propagation in complex financial networks. However, network dynamics are ignored in most cases despite systems with similar topological structures exhibiting profoundly different dynamic behaviors. To observe the spatiotemporal patterns of risk propagation in complex financial networks, we combined a dynamic model with empirical networks. Our analysis revealed that hub nodes play a dominant role in risk propagation across the network and respond rapidly, thus exhibiting a degree-driven effect. The influence of key dynamic parameters, i.e., infection rate and recovery rate, was also investigated. Furthermore, the impacts of two typical characteristics of complex financial systems—the existence of community structures and frequent large fluctuations—on the spatiotemporal patterns of risk propagation were explored. About 30% of the total risk propagation flow of each community can be explained by the top 10% nodes. Thus, we can control the risk propagation flow of each community by controlling a few influential nodes in the community and, in turn, control the whole network. In extreme market states, hub nodes become more dominant, indicating better risk control.

In this paper, we study the valuation of swing options on electricity in a model where the underlying spot price is set to be the product of a deterministic seasonal pattern and Ornstein-Uhlenbeck process with Markov-modulated parameters. Under this setting, the difficulties of pricing swing options come from the various constraints embedded in contracts, e.g., the total number of rights constraint, the refraction time constraint, the local volume constraint, and the global volume constraint. Here we propose a framework for the valuation of the swing option on the condition that all the above constraints are nontrivial. To be specific, we formulate the pricing problem as an optimal stochastic control problem, which can be solved by the trinomial forest dynamic programming approach. Besides, empirical analysis is carried out on the model. We collect historical data in Nord Pool electricity market, extract the seasonal pattern, calibrate the Ornstein-Uhlenbeck process parameters in each regime, and also get market price of risk. Finally, on the basis of calibration results, a specific numerical example concerning all typical constraints is presented to demonstrate the valuation procedure.

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