Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing $N$-peakon weak solutions, for all $N\geq 1$, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for $2$-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for $N$-peakon weak solutions for all $N\geq 2$. When $N$ is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when $N$ is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH $N$-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions for all $N\geq 2$. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.Comment: Minor errata in Eqns. (32) to (34) and Lemma 1 have been fixe

When do we have borrower or credit volume rationing in competitive credit market with imperfect information?

This paper examines the conditions for credit volume or borrower rationing in a competitive credit market in which the project characteristics are private information of the borrowers. There can only be credit volume rationing if the higher-risk credit applicants have a higher return in the event of a project success than the lower-risk credit applicants. Then the higher-risk borrowers are not rationed and obtain the social efficient credit volume. If the incentive compatibility constraint of the higher risk borrowers is binding, the lower-risk borrowers are credit volume rationed such that the constraint holds as an equation. If credit volume rationing is not sufficient to separate the borrower types, there is additionally a rationing of the low-risk borrowers. If the low-risk borrowers prefer a pooling to a separating contract, then there will not be a Cournot-Nash separating equilibrium, but a Wilson and a Grossmann pooling equilibrium

Does borrowers' impatience disclose their hidden information about default risk?

This chapter provides new evidence on borrowers' hidden information about their riskiness and its link to their impatience. To do so, I analyze consumer loans on the German platform Smava, which has a unique peer-to-peer lending process. Observationally identical but unobservably riskier borrowers offer investors a higher interest rate. This helps them to obtain their loan faster and with a higher probability. Very impatient borrowers who use Smava's instant loan service pay a higher interest rate and have a higher default risk than less impatient borrowers. These findings suggest that borrowers' impatience can be used to screen their riskiness

On the existence of credit rationing and screening with loan size in competitive markets with imperfect information

Although credit rationing has been a stylized fact since the groundbreaking papers by Stiglitz and Weiss (1981, hereinafter S-W) and Besanko and Thakor (1987a, hereinafter B-T), Arnold and Riley (2009) note that credit rationing is unlikely in the S-W model, and Clemenz (1993) shows that it does not exist in the B-T model. In this chapter, I explain why credit rationing, more specifically rationing of loan applicants, does exist in a competitive market with imperfect information, and occurs only for low-risk loan applicants. In cases of indivisible investment technologies, low-risk applicants are rationed. In cases of divisible investment technologies, rationing of loan size is restricted to rationing of loan applicants. In the event that the difference in the marginal return between the investment technologies is sufficiently small relative to the difference in their riskiness, rationing of loan size alone yields high opportunity costs; in addition, low-risk loan applicants are rationed in this case

Stress Testing German Industry Sectors: Results from a Vine Copula Based Quantile Regression

Measuring interdependence between probabilities of default (PDs) in different industry sectors of an economy plays a crucial role in financial stress testing. Thereby, regression approaches may be employed to model the impact of stressed industry sectors as covariates on other response sectors. We identify vine copula based quantile regression as an eligible tool for conducting such stress tests as this method has good robustness properties, takes into account potential nonlinearities of conditional quantile functions and ensures that no quantile crossing effects occur. We illustrate its performance by a data set of sector specific PDs for the German economy. Empirical results are provided for a rough and a fine-grained industry sector classification scheme. Amongst others, we confirm that a stressed automobile industry has a severe impact on the German economy as a whole at different quantile levels whereas e.g., for a stressed financial sector the impact is rather moderate. Moreover, the vine copula based quantile regression approach is benchmarked against both classical linear quantile regression and expectile regression in order to illustrate its methodological effectiveness in the scenarios evaluated.Comment: 12 page