Low correlation between morning spot and 24-hour urine samples for estimating sodium intake in an Iranian population: Isfahan Salt Study

Introduction: Although difficult, the 24-hour urine sodium excretion is still considered as the gold standard method to estimate salt intake. The current study aimed to assess the validity of using spot urine samples in comparison with the standard 24-hour urine collection to estimate sodium and potassium intake in healthy Iranian adults. Methods and subjects: This cross-sectional study was performed on 1099 healthy Iranians aged 18–69 years. Samples of 24-hour and fasting morning spot urine were collected to measure sodium and potassium excretions. Tanaka’s formula was utilized to predict the 24-hour sodium and potassium urinary excretions based on the spot values. Results: The difference between measured and estimated sodium excretion values was 4265 mg/day (95% CI: 4106–4424; P < 0.001) and 2242 mg/day in case of potassium excretion (95% CI: 2140–2344; P < 0.001). There was a weak significant correlation between the 24-hour urine sodium and potassium excretion and the predicted values (intraclass correlations: 0.22 and 0.28, respectively; both P < 0.001). Conclusion: The weak association between the predicted and measured values of sodium and potassium along with the marked overestimation of daily sodium and potassium excretions based on the spot urine and using Tanaka formula indicates that Tanaka formula is not practical for the prediction of sodium and potassium or salt intake in Iranian adults. Using other spot urine sampling times and/or adopting a formula designed based on the characteristics of the Iranian population may increase the validity of spot urine tests.

Tanaka formula for strictly stable processes

For symmetric Levy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor 2007. In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ∈ 1, 2, including spectrally positive and negative cases in a framework of Ito’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin 1996.

The Tanaka formula for symmetric stable processes with index $\alpha$, $0<\alpha<2$

Для симметричного $\alpha$-устойчивого процесса $Z=(Z_t)_{t\ge0}$, $0<\alpha<2$, любого $a\in\mathbf{R}$ и $\gamma\in(0,2)$ такого, что $\alpha-1<\gamma<\alpha$, мы приводим в явном виде разложение Дуба-Мейера для субмартингала $|Z-a|^\gamma=(|Z_t-a|^{\gamma})_{t\ge0}$, состоящее из константы $|a|^{\gamma}$, стохастического интеграла по компенсированной пуассоновской случайной мере, ассоциированной с $Z$, и предсказуемого возрастающего процесса. Для $1<\alpha<2$ мы рассматриваем также случай $\gamma=\alpha-1$, соответствующий знаменитой формуле Танака. Это распространяет результаты Салминена и Йора [11] на общий случай $0<\alpha<2$ с использованием альтернативного подхода. Работы по близкой проблематике: Танака [13], Фитцсиммонс и Гетур [4], Т. Ямада [16] и К. Ямада [15].

The functional Meyer–Tanaka formula

The functional Itô formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer–Tanaka formula. Following the idea of the proof of the classical time-dependent Meyer–Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obłój.