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Rural subjects, the agricultural industrial structure, public services and rural governance are fully empowered by digital villages. This empowerment effectively compensates for the urban-rural digital divide and promotes the equalization of urban-rural income, consumption, education, medical care, and governance. Based on the three-stage data envelopment analysis (DEA) model and Malmquist index, this article conducts an in-depth study of the static and dynamic efficiency trends of digital villages that empower urban-rural balanced development in 31 provinces in China from 2015 to 2020. The results show that comprehensive technical efficiency of 31 provinces is weak DEA effective, and that the scale efficiency is the main factor affecting comprehensive technical efficiency. The educational level, local finance and industrial structure optimization have a significant positive impact on efficiency evaluation, but technological innovation and the urbanization level have a significant negative impact. Total factor productivity shows diminishing marginal utility based on the Malmquist index and its decomposition change. Restricted by the change in technological progress, the efficiency of digital villages in China in enabling urban-rural equilibrium needs to be further improved.

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This paper investigates the valuation of vulnerable European options considering the market prices of common systematic jump risks under regime-switching jump-diffusion models. The way of regime-switching Esscher transform is adopted to identify an equivalent martingale measure for pricing vulnerable European options. Explicit analytical pricing formulae for vulnerable European options are derived by risk-neutral pricing theory. For comparison, the other two cases are also considered separately. The first case considers all jump risks as unsystematic risks while the second one assumes all jumps risks to be systematic risks. Numerical examples for the valuation of vulnerable European options are provided to illustrate our results and indicate the influence of the market prices of jump risks on the valuation of vulnerable European options.