This paper introduces a mathematical model for a currently popular financial product called a stock loan. Quantitative analysis is carried out to establish explicitly the value of such a loan, as well as the ranges of fair values of the loan size and interest, and the fee for providing such a service.

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We provide conditions on a one-period-two-date pure exchange economy with rankdependent utility agents under which Arrow-Debreu equilibria exist. When such an equilibrium exists, we show that the state-price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.

The comparative statics of the optimal portfolios across individuals is carried out for the Black-Scholes market model. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent would invest less wealth (in absolute value) in the risky asset. Introduction.Portfolio selection is one of the classical problems in the economics of uncertainty. The optimal portfolios depend on agents' characters (preference and wealth level) and on the market's structure (the risk-free return, the return and risk of the risky assets). Various agents would have different allocations of wealth between the risk-free asset and the risky assets, due to the differences in preference and/or the differences in wealth level. The comparative statics of the optimal portfolios w.r.t. (with respect to) preference and/or wealth level was first carried out by Arrow [1] and Pratt [18] for a static model with a risk-free asset and a risky asset. For this model, if the excess return of the risky asset is positive, then (i) the more risk-averse an agent is, the less wealth is invested in the risky asset; and (ii) if an agent displays decreasing absolute (relative) risk aversion, then the amount (proportion) of wealth invested in the risky asset is increasing in the total wealth.Since then, decades have passed. Except for some specific cases such as constant absolute (relative) risk aversion, in which the solutions can be explicitly worked out, few works have been reported for dynamic models, as far as we know, until Borell [3]. For a continuous-time complete market model, where the risky assets price process follows a joint geometric Brownian motion, and for an agent who consumes only at the terminal time, Borell [3] analyzed the changes of the optimal portfolios across the wealth levels. The similar conclusions that hold for the static models have been obtained there by showing that the indirect utility function inherits the decreasing absolute (relative) risk aversion from the vNM (von Neumann-Morgenstern) utility function. 1 The purpose of this paper is to investigate how the agents' preferences impact the optimal portfolios for the Black-Scholes market model. Here we compare the optimal portfolios across individuals instead of across wealth levels. As a result (see

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In this paper, for a process S, we establish a duality relation between K p , the L p (P)-closure of the space of claims in L p (P), which are attainable by "simple" strategies, and M q,s , all signed martingale measures Q with dQ dP ∈ L q (P), where p ≥ 1, q ≥ 1 and 1 p + 1 q = 1. If there exists a Q ∈ M q,s with dQ dP > 0 a.s., then K p consists precisely of the random variables T 0 ϑ dS such that ϑ is predictable S-integrable and E[ dQ dP T 0 ϑ dS ] = 0 for all Q ∈ M q,s . The duality relation corresponding to the case p = q = 2 is used to investigate the Markowitz's problem of mean-variance portfolio optimization in an incomplete market of semimartingale model via martingale/convex duality method. The duality relationship between the mean-variance efficient portfolios and the variance-optimal signed martingale measure (VSMM) is established. It turns out that the so-called market price of risk is just the standard deviation of the VSMM. An illustrative example of application to a geometric Lévy processes model is also given.

In this paper we investigate the problem of mean-variance portfolio choice with bankruptcy prohibition. For incomplete markets with continuous assets' price processes and for complete markets, it is shown that the mean-variance efficient portfolios can be expressed as the optimal strategies of partial hedging for quadratic loss function. Thus, mean-variance portfolio choice, in these cases, can be viewed as expected utility maximization with non-negative marginal utility.

BackgroundValvular heart disease is a leading cause of cardiovascular mortality, especially in China. More than a half of valvular heart diseases are caused by acute rheumatic fever. microRNA is involved in many physiological and pathological processes. However, the miRNA profile of the rheumatic valvular heart disease is unknown. This research is to discuss microRNAs and their target gene pathways involved in rheumatic heart valve disease.MethodsSerum miRNA from one healthy individual and four rheumatic heart disease patients were sequenced. Specific differentially expressed miRNAs were quantified by Q-PCR in 40 patients, with 20 low-to-moderate rheumatic mitral valve stenosis patients and 20 severe mitral valve stenosis patients. The target relationship between certain miRNA and predicted target genes were analysis by Luciferase reporter assay. The IL-1β and IL1R1 expression levels were analyzed by immunohistochemistry and western blot in the mitral valve from surgery of mitral valve replacement.ResultsThe results showed that 13 and 91 miRNAs were commonly upregulated or downregulated in all four patients. Nine miRNAs, 1 upregulated and 8 downregulated, that had a similar fold change in all 4 patients were selected for quantitative PCR verification. The results showed similar results from miRNA sequencing. Within these 9 tested miRNAs, hsa-miR-205-3p and hsa-miR-3909 showed a low degree of dispersion between the members of each group. Hsa miR-205-3p and hsa-miR-3909 were predicted to target the 3’UTR of IL-1β and IL1R1 respectively. This was verified by luciferase reporter assays. Immunohistochemistry and Western blot results showed that the mitral valve from rheumatic valve heart disease showed higher levels of IL- 1β and IL1R1 expression compared with congenital heart valve disease. This suggested a difference between rheumatic heart valve disease and other types of heart valve diseases, with more inflammatory responses in the former.ConclusionIn the present study, by next generation sequencing of miRNAs, it was revealed that interleukin 1β and interleukin 1 receptor 1 was involved in rheumatic heart diseases. And this is useful for diagnosis and understanding of mechanism of rheumatic heart disease.Electronic supplementary materialThe online version of this article (10.1186/s12872-018-0788-2) contains supplementary material, which is available to authorized users.