Dynkin Game of Stochastic Differential Equations with Random Coefficients, and Associated Backward Stochastic Partial Differential Variational Inequality

A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tang's maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a Markov process to that of a non-Markov process, and establish a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be a random field of It\^o's type which takes values in a suitable Sobolev space. We then prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a backward stochastic partial differential variational inequality (BSPDVI, for short) with two obstacles. We obtain the existence and uniqueness result and a comparison theorem for strong solution of the BSPDVI. Moreover, we study the monotonicity on the strong solution of the BSPDVI by the comparison theorem for BSPDVI and define the free boundaries. Finally, we identify the counterparts for an optimal stopping time problem as a special Dynkin game.Comment: 40 page

Efficient routing strategies in scale-free networks with limited bandwidth

We study the traffic dynamics in complex networks where each link is assigned a limited and identical bandwidth. Although the first-in-first-out (FIFO) queuing rule is widely applied in the routing protocol of information packets, here we argue that if we drop this rule, the overall throughput of the network can be remarkably enhanced. We proposed some efficient routing strategies that do not strictly obey the FIFO rule. Comparing with the routine shortest path strategy, the throughput for both Barab\'asi-Albert (BA) networks and the real Internet, the throughput can be improved more than five times. We calculate the theoretical limitation of the throughput. In BA networks, our proposed strategy can achieve 88% of the theoretical optimum, yet for the real Internet, it is about 12%, implying that we have a huge space to further improve the routing strategy for the real Internet. Finally we discuss possibly promising ways to design more efficient routing strategies for the Internet.Comment: 5 pages, 4 figure

Analytical vectorial structure of non-paraxial four-petal Gaussian beams in the far field

The analytical vectorial structure of non-paraxial four-petal Gaussian beams(FPGBs) in the far field has been studied based on vector angular spectrum method and stationary phase method. In terms of analytical electromagnetic representations of the TE and TM terms, the energy flux distributions of the TE term, the TM term, and the whole beam are derived in the far field, respectively. According to our investigation, the FPGBs can evolve into a number of small petals in the far field. The number of the petals is determined by the order of input beam. The physical pictures of the FPGBs are well illustrated from the vectorial structure, which is beneficial to strengthen the understanding of vectorial properties of the FPGBs