Probability Uncertainty and Quantitative Risk最新文献

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Convergence rate of Peng’s law of large numbers under sublinear expectations 次线性期望下彭大数定律的收敛速度

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-07-06 DOI: 10.3934/puqr.2021013

Mingshang Hu, Xiaojuan Li, Xinpeng Li

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations, which improves the results presented by Song [15] and Fang et al. [3].

这篇短文为次线性期望下Peng大数定律的收敛速率提供了一种新的简单证明，改进了Song[15]和Fang等人[3]的结果。

引用次数: 7

On the speed of convergence of Picard iterations of backward stochastic differential equations 后向随机微分方程皮卡德迭代的收敛速度

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-07-05 DOI: 10.3934/puqr.2022009

Martin Hutzenthaler, T. Kruse, T. Nguyen

It is a well-established fact in the scientiﬁc literature that Picard iterations of backward stochastic diﬀerential equations with globally Lipschitz continuous nonlinearity converge at least exponentially fast to the solution. In this paper we prove that this convergence is in fact at least square-root factorially fast. We show for one example that no higher convergence speed is possible in general. Moreover, if the nonlinearity is z -independent, then the convergence is even factorially fast. Thus we reveal a phase transition in the speed of convergence of Picard iterations of backward stochastic diﬀerential equations. diﬀerential equation, Picard iteration, a priori estimate, semilinear parabolic partial diﬀerential equation

在科学文献中，具有全局Lipschitz连续非线性的倒向随机微分方程的Picard迭代至少以指数速度收敛到解。在本文中，我们证明了这种收敛实际上至少是根号阶乘快。我们通过一个例子证明，一般情况下没有更高的收敛速度。此外，如果非线性与z无关，则收敛速度甚至是阶乘快。从而揭示了倒向随机微分方程皮卡德迭代收敛速度的相变。微分方程，皮卡德迭代，先验估计，半线性抛物型偏微分方程

引用次数: 2

Stochastic maximum principle for systems driven by local martingales with spatial parameters 空间参数局部鞅驱动系统的随机极大值原理

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-06-02 DOI: 10.3934/puqr.2021011

Jian Song, M. Wang

We consider the stochastic optimal control problem for the dynamical system of the stochastic differential equation driven by a local martingale with a spatial parameter. Assuming the convexity of the control domain, we obtain the stochastic maximum principle as the necessary condition for an optimal control, and we also prove its sufficiency under proper conditions. The stochastic linear quadratic problem in this setting is also discussed.

研究了具有空间参数的局部鞅驱动的随机微分方程动力系统的随机最优控制问题。在控制域为凸性的前提下，得到了随机极大值原理作为最优控制的必要条件，并在适当条件下证明了其充分性。讨论了这种情况下的随机线性二次问题。

引用次数: 0

Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations 具有高斯创新的AR(1)和MA(1)过程中大偏差的显式二元速率函数

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-02-18 DOI: 10.3934/puqr.2023008

M. J. Karling, A. Lopes, S. Lopes

We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors $(boldsymbol{S}_n)_{n in N} = left(n^{-1}(sum_{k=1}^n X_k, sum_{k=1}^n X_k^2)right)_{n in N}$. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence $(W_n)_{n geq 2} = left(n^{-1}(sum_{k=1}^n X_k^2, sum_{k=2}^n X_k X_{k+1})right)_{n geq 2}$. Via Contraction Principle, we provide explicit rate functions for the sequences $(n^{-1} sum_{k=1}^n X_k)_{n in N}$, $(n^{-1} sum_{k=1}^n X_k^2)_{n geq 2}$ and $(n^{-1} sum_{k=2}^n X_k X_{k+1})_{n geq 2}$, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.

我们通过给出随机向量序列$(boldsymbol{S}_n)_{n in N} = left(n^{-1}(sum_{k=1}^n X_k, sum_{k=1}^n X_k^2)right)_{n in N}$的显式二元速率函数，研究了具有独立高斯创新的中心平稳AR(1)和MA(1)过程的大偏差特性。在AR(1)的情况下，我们也给出了二元随机序列$(W_n)_{n geq 2} = left(n^{-1}(sum_{k=1}^n X_k^2, sum_{k=2}^n X_k X_{k+1})right)_{n geq 2}$的显式速率函数。通过收缩原理，我们还为序列$(n^{-1} sum_{k=1}^n X_k)_{n in N}$、$(n^{-1} sum_{k=1}^n X_k^2)_{n geq 2}$和$(n^{-1} sum_{k=2}^n X_k X_{k+1})_{n geq 2}$提供了显式的速率函数。在AR(1)情况下，我们对Yule-Walker估计量的显式偏差函数的已知结果给出了新的证明。

引用次数: 1

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time 连续时间夏普比率的期限结构与无套利资产定价

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/PUQR.2021002

Patrick Beissner, E. R. Gianin

Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a time-delayed backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.

在金融和实证论证的推动下，为了引入更灵活的定价方法，我们提供了一种基于后向沃尔泰拉方程的资产定价新方法。该方法依赖于连续时间内无套利和不完全的市场设置，通过根据评估时间或支付期限选择非唯一的定价措施。我们表明，在后一种情况下，动力学可以通过本文介绍的时间延迟倒向随机Volterra积分方程来捕获，据我们所知，该方程尚未被研究过。然后证明了时滞倒向随机Volterra积分方程的存在唯一性。最后，我们提出了一个卢卡斯式的基于消费的资产定价模型，该模型证明了与夏普比率期限结构相匹配的随机贴现因子的出现。

引用次数: 3

Reduced-form setting under model uncertainty with non-linear affine intensities 具有非线性仿射强度的模型不确定性下的简化形式设置

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021008

F. Biagini, Katharina Oberpriller

In this paper we extend the reduced-form setting under model uncertainty introduced in [5] to include intensities following an affine process under parameter uncertainty, as defined in [15]. This framework allows us to introduce a longevity bond under model uncertainty in a way consistent with the classical case under one prior and to compute its valuation numerically. Moreover, we price a contingent claim with the sublinear conditional operator such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind” as in [6].

在本文中，我们扩展了[5]中引入的模型不确定性下的简化形式设置，以包括参数不确定性下仿射过程后的强度，如[15]中定义的那样。该框架允许我们在模型不确定性下以一种与经典情况一致的方式引入长寿债券，并以数值方式计算其估值。此外，我们使用次线性条件算子对或有债权进行定价，使得扩展市场在[6]中“第一类无套利”的意义上仍然是无套利的。

引用次数: 5

Existence, uniqueness and strict comparison theorems for BSDEs driven by RCLL martingales RCLL鞅驱动的BSDEs的存在唯一性及严格比较定理

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021016

Tianyang Nie, M. Rutkowski

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.

给出了由多维RCLL鞅驱动的BSDE解的存在性、唯一性和严格比较性。目标是开发一个通用的多资产框架，包括广泛的具有跳跃的非线性金融模型，包括彭和徐[27,28]和Dumitrescu等人[7]研究的设置，他们处理由一维布朗运动和具有单跳的纯不连续鞅驱动的BSDEs。

引用次数: 1

Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables 依赖有界或亚高斯随机变量的改进Hoeffding不等式

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/PUQR.2021003

Y. Tanoue

When addressing various financial problems, such as estimating stock portfolio risk, it is necessary to derive the distribution of the sum of the dependent random variables. Although deriving this distribution requires identifying the joint distribution of these random variables, exact estimation of the joint distribution of dependent random variables is difficult. Therefore, in recent years, studies have been conducted on the bound of the sum of dependent random variables with dependence uncertainty. In this study, we obtain an improved Hoeffding inequality for dependent bounded variables. Further, we expand the above result to the case of sub-Gaussian random variables.

在处理各种金融问题时，例如估计股票投资组合风险，有必要推导出相关随机变量和的分布。虽然导出这种分布需要识别这些随机变量的联合分布，但准确估计相关随机变量的联合分布是困难的。因此，近年来人们对具有相关不确定性的相关随机变量和的界进行了研究。在本研究中，我们得到了一个改进的有界变量的Hoeffding不等式。进一步，我们将上述结果推广到亚高斯随机变量的情况。

引用次数: 2

On the laws of the iterated logarithm under sub-linear expectations 次线性期望下的迭代对数规律

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021020

Li-Xin Zhang

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.

本文建立了次线性期望空间中独立随机变量迭代对数律的一般形式，其中随机变量不一定是同分布的。建立了独立随机变量最大和的指数不等式和Kolmogorov逆指数不等式，作为证明迭代对数定律的工具。作为应用，得到了独立同分布随机变量在次线性期望下的迭代对数律的充要条件。本文还证明了当亚线性期望空间足够丰富时，它将不具有连续容量。在不假设容量连续性的情况下，建立了迭代对数定律。

引用次数: 5

CVaR-hedging and its applications to equity-linked life insurance contracts with transaction costs cvar套期保值及其在具有交易成本的股票挂钩人寿保险合同中的应用

IF 1.5 2区数学 Q3 STATISTICS & PROBABILITY

Probability Uncertainty and Quantitative Risk

Pub Date : 2021-01-01 DOI: 10.3934/puqr.2021017

Alexander Melnikov,Hongxi Wan

<p style='text-indent:20px;'>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.</p>

<p style='text-indent:20px;'>本文在具有交易成本的跳跃-扩散市场模型中，分析了基于条件风险价值(CVaR)的部分套期保值及其在股票挂钩人寿保险合同中的应用。给出了包含交易费用的期权价值过程应满足的非线性偏微分方程。特别地，给出了欧式看涨期权价格的封闭表达式。同时，明确推导了基于cvar的看涨期权部分套期保值策略。CVaR套期保值价格和套期保值组合权重均基于调整后的波动率。通过仿真方法得到了预期总套期误差和总交易成本的估计值。此外，将我们的结果应用于获得目标客户的股票挂钩人寿保险合同的生存概率和年龄。

引用次数: 0

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