Probability, Uncertainty and Quantitative Risk

March 2022 , Volume 7 , Issue 1

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On the laws of the iterated logarithm with mean-uncertainty under sublinear expectations
Xiaofan Guo, Shan Li and Xinpeng Li
2022, 7(1): 1-12 doi: 10.3934/puqr.2022001 +[Abstract](630) +[HTML](165) +[PDF](446.9KB)

A new Hartman–Wintner-type law of the iterated logarithm for independent random variables with mean-uncertainty under sublinear expectations is established by the martingale analogue of the Kolmogorov law of the iterated logarithm in classical probability theory.

Quantitative stability and numerical analysis of Markovian quadratic BSDEs with reflection
Dingqian Sun, Gechun Liang and Shanjian Tang
2022, 7(1): 13-30 doi: 10.3934/puqr.2022002 +[Abstract](359) +[HTML](120) +[PDF](563.45KB)

We study the quantitative stability of solutions to Markovian quadratic reflected backward stochastic differential equations (BSDEs) with bounded terminal data. By virtue of bounded mean oscillation martingale and change of measure techniques, we obtain stability estimates for the variation of the solutions with different underlying forward processes. In addition, we propose a truncated discrete-time numerical scheme for quadratic reflected BSDEs and obtain the explicit rate of convergence by applying the quantitative stability result.

Threshold reweighted Nadaraya–Watson estimation of jump-diffusion models
Kunyang Song, Yuping Song and Hanchao Wang
2022, 7(1): 31-44 doi: 10.3934/puqr.2022003 +[Abstract](423) +[HTML](133) +[PDF](578.42KB)

In this paper, we propose a new method to estimate the diffusion function in the jump-diffusion model. First, a threshold reweighted Nadaraya–Watson-type estimator is introduced. Then, we establish asymptotic normality for the estimator and conduct Monte Carlo simulations through two examples to verify the better finite-sampling properties. Finally, our estimator is demonstrated through the actual data of the Shanghai Interbank Offered Rate in China.

Lower and upper pricing of financial assets
Robert Elliott, Dilip B. Madan and Tak Kuen Siu
2022, 7(1): 45-66 doi: 10.3934/puqr.2022004 +[Abstract](420) +[HTML](135) +[PDF](600.17KB)

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability. When financial valuation recognizes the uncertainty of probability, the best the market may offer is a two price framework of a lower and upper valuation. The martingale theory of asset prices is then replaced by the theory of nonlinear martingales. When dealing with pure jump compensators describing probability, the uncertainty in probability is captured by introducing parametric measure distortions. The two price framework then alters asset pricing theory by requiring two required return equations, one each for the lower upper valuation. Proxying lower and upper valuations by daily lows and highs, the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.



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