Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of ℝn → ℝn associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of ℝnu → ℝ nx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

Asset Prices Under Knightian Uncertainty

We extend Lucas’s classic asset-price model by opening the stochastic process driving dividends to Knightian uncertainty arising from unforeseeable change. Implementing Muth’s hypothesis, we represent participants’ expectations as being consistent with our model’s predictions and formalize their ambiguity-averse decisions with maximization of intertemporal multiple-priors utility. We characterize the asset-price function with a stochastic Euler equation and derive a novel prediction that the relationship between prices and dividends undergoes unforeseeable change. Our approach accords participants’ expectations, driven by both fundamental and psychological factors, an autonomous role in driving the asset price over time, without presuming that participants are irrational.

Anti-symmetric clustering signals in the observed power spectrum

In this paper, we study how to directly measure the effect of peculiar velocities in the observed angular power spectra. We do this by constructing a new anti-symmetric estimator of Large Scale Structure using different dark matter tracers. We show that the Doppler term is the major component of our estimator and we show that we can measure it with a signal-to-noise ratio up to ∼ 50 using a futuristic SKAO HI galaxy survey. We demonstrate the utility of this estimator by using it to provide constraints on the Euler equation.

Do Investment-Based Models Explain Equity Returns? Evidence from Euler Equations

We investigate the empirical implications of the investment-based model of asset pricing for the Hansen-Jagannathan and Kozak-Nagel-Santosh discount factors in the linear span of equity returns. We find that the stochastic discount factors satisfying the Euler equation for equity returns cannot satisfy the Euler equation for investment returns because returns on corporate investment covary inversely with the sources of equity risk relative to returns on equity. As a result, the model fails to replicate the level of the risk premium. Our results suggest that joint restrictions on the optimality of investment and consumption pose stringent conditions for candidate production models.