# American Institute of Mathematical Sciences

ISSN:
2095-9672

eISSN:
2367-0126

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## Probability, Uncertainty and Quantitative Risk

December 2021 , Volume 6 , Issue 4

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2021, 6(4): 267-300 doi: 10.3934/puqr.2021014 +[Abstract](565) +[HTML](115) +[PDF](1209.26KB)
Abstract:

This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.

2021, 6(4): 301-318 doi: 10.3934/puqr.2021015 +[Abstract](168) +[HTML](94) +[PDF](711.03KB)
Abstract:

This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator \begin{document}$g$\end{document} satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable \begin{document}$y$\end{document} , and a stochastic-Lipschitz condition in the state variable \begin{document}$z$\end{document} . This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.

2021, 6(4): 319-342 doi: 10.3934/puqr.2021016 +[Abstract](142) +[HTML](88) +[PDF](891.78KB)
Abstract:

The existence, uniqueness, and strict comparison for solutions to a BSDE driven by a multi-dimensional RCLL martingale are developed. The goal is to develop a general multi-asset framework encompassing a wide spectrum of non-linear financial models with jumps, including as particular cases, the setups studied by Peng and Xu [27, 28] and Dumitrescu et al. [7] who dealt with BSDEs driven by a one-dimensional Brownian motion and a purely discontinuous martingale with a single jump.

2021, 6(4): 343-368 doi: 10.3934/puqr.2021017 +[Abstract](107) +[HTML](65) +[PDF](648.17KB)
Abstract:

This paper analyzes Conditional Value-at-Risk (CVaR) based partial hedging and its applications on equity-linked life insurance contracts in a Jump-Diffusion market model with transaction costs. A nonlinear partial differential equation (PDE) that an option value process inclusive of transaction costs should satisfy is provided. In particular, the closed-form expression of a European call option price is given. Meanwhile, the CVaR-based partial hedging strategy for a call option is derived explicitly. Both the CVaR hedging price and the weights of the hedging portfolio are based on an adjusted volatility. We obtain estimated values of expected total hedging errors and total transaction costs by a simulation method. Furthermore,our results are implemented to derive target clients’ survival probabilities and age of equity-linked life insurance contracts.

2021, 6(4): 369-390 doi: 10.3934/puqr.2021018 +[Abstract](118) +[HTML](51) +[PDF](1065.06KB)
Abstract:

In this paper, we extend the definition of conditional \begin{document}$G{\text{-}}{\rm{expectation}}$\end{document} to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional \begin{document}$G{\text{-}}{\rm{expectation}}$\end{document} for each random variable \begin{document}$X$\end{document} , which is the downward limit (respectively, upward limit) of a monotone sequence \begin{document}$\{X_{i}\}$\end{document} in \begin{document}$L_{G}^{1}(\Omega)$\end{document} . To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional \begin{document}$G{\text{-}}{\rm{expectation}}$\end{document} .

2021, 6(4): 391-408 doi: 10.3934/puqr.2021019 +[Abstract](200) +[HTML](64) +[PDF](859.9KB)
Abstract:

This paper is dedicated to solving high-dimensional coupled FBSDEs with non-Lipschitz diffusion coefficients numerically. Under mild conditions, we provided a posterior estimate of the numerical solution that holds for any time duration. This posterior estimate validates the convergence of the recently proposed Deep BSDE method. In addition, we developed a numerical scheme based on the Deep BSDE method and presented numerical examples in financial markets to demonstrate the high performance.

2021, 6(4): 409-460 doi: 10.3934/puqr.2021020 +[Abstract](136) +[HTML](52) +[PDF](1487.49KB)
Abstract:

In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov’s converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of the iterated logarithm for independent and identically distributed random variables under the sub-linear expectation are obtained. In the paper, it is also shown that if the sub-linear expectation space is rich enough, it will have no continuous capacity. The laws of the iterated logarithm are established without the assumption on the continuity of capacities.