Karhunen-Lo\`eve expansion for a generalization of Wiener bridge

We derive a Karhunen-Lo\`eve expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}{\pi}\sin(\pi t)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.Comment: 25 pages, 1 figure. The appendix is extende

An Explicit Time-Domain Hybrid Formulation Based on the Unified Boundary Condition

An approach to stabilize the two-surface, time domain FEM/BI hybrid by means of a unified boundary condition is presented. The first-order symplectic finite element formulation [1] is used along with a version of the unified boundary condition of Jin [2] reformulated for Maxwell's first-order equations in time to provide both stability and accuracy over the first-order ABC. Several results are presented to validate the numerical solutions. In particular the dipole in a free-space box is analyzed and compared to the Dirchlet boundary condition of Ziolkowski and Madsen [3] and to a Neuman boundary condition approach

The Future of International Investment Regulation: Towards a World Investment Organisation?

With growth in foreign investment and in the number of companies investing in foreign countries, the application of general principles of public international law has not been deemed adequate to regulate foreign investment and there is, as yet, no comprehensive international treaty on the regulation of foreign investment. Consequently, states have resorted to bilateral investment treaties (BITs), regional trade and international investment agreements (IIAs) and free trade agreements (FTAs) to supplement and complement the regime of protection for foreign investors. In the absence of an international investment court, states hosting foreign investment or investor states have opted for investor-state dispute settlement mechanism (ISDS). This mechanism has brought about its own challenges to the international law of foreign investment due to inconsistency in the application and interpretation of the key principles of international investment law by such arbitration tribunals, and further, there is no appellate mechanism to bring about some cohesion and consistency in jurisprudence. Therefore, there are various proposals mooted by scholars to address these challenges and they range from tweaks to BITs and IIAs, the creation of an appellate mechanism and the negotiation of a multilateral treaty to proposals for reform of ISDS only. After assessing the merits and demerits of such proposals, this study goes further, arguing for the creation of a World Investment Organisation (WIO) with a standing mechanism for settlement of investment disputes in order to ensure legal certainty, predictability and the promotion of the flow of foreign investment in a sustainable and responsible manner