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A Review of Reinforcement Learning in Financial Applications 金融应用中的强化学习回顾

IF 7.9 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Annual Review of Statistics and Its Application

Pub Date : 2024-11-15 DOI: 10.1146/annurev-statistics-112723-034423

Yahui Bai, Yuhe Gao, Runzhe Wan, Sheng Zhang, Rui Song

In recent years, there has been a growing trend of applying reinforcement learning (RL) in financial applications. This approach has shown great potential for decision-making tasks in finance. In this review, we present a comprehensive study of the applications of RL in finance and conduct a series of meta-analyses to investigate the common themes in the literature, such as the factors that most significantly affect RL's performance compared with traditional methods. Moreover, we identify challenges, including explainability, Markov decision process modeling, and robustness, that hinder the broader utilization of RL in the financial industry and discuss recent advancements in overcoming these challenges. Finally, we propose future research directions, such as benchmarking, contextual RL, multi-agent RL, and model-based RL to address these challenges and to further enhance the implementation of RL in finance.

近年来，在金融应用中应用强化学习（RL）的趋势越来越明显。这种方法在金融决策任务中显示出巨大的潜力。在这篇综述中，我们对强化学习在金融领域的应用进行了全面研究，并进行了一系列元分析，以探讨文献中的共同主题，例如与传统方法相比，哪些因素对强化学习的性能影响最大。此外，我们还发现了一些挑战，包括可解释性、马尔可夫决策过程建模和稳健性，这些挑战阻碍了 RL 在金融业的广泛应用，并讨论了在克服这些挑战方面的最新进展。最后，我们提出了未来的研究方向，如基准测试、情境 RL、多代理 RL 和基于模型的 RL，以应对这些挑战并进一步加强 RL 在金融领域的应用。

引用次数: 0

Global existence, uniqueness and $$L^{infty }$$ -bound of weak solutions of fractional time-space Keller-Segel system 分数时空凯勒-西格尔系统弱解的全局存在性、唯一性和 $$L^{infty }$ -bound

IF 3 2区数学 Q1 MATHEMATICS

Fractional Calculus and Applied Analysis

Pub Date : 2024-11-15 DOI: 10.1007/s13540-024-00353-6

Fei Gao, Liujie Guo, Xinyi Xie, Hui Zhan

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in ({mathbb {R}}^{n}), (nge 2). The global existence and (L^{infty })-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) (b>1-frac{alpha }{n}), for any initial value and birth rate; (ii) (0<ble 1-frac{alpha }{n}), for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the (L^{infty })-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong.

本文研究了一类在 ({mathbb {R}}^{n}), (nge 2) 中具有对数源项的时空分式抛物-椭圆 Keller-Segel 方程的弱解的性质。建立了弱解的全局存在性和（L^{infty } ）边界。我们主要将阻尼系数分为两种情况：（i）(b>1-frac{alpha }{n})，适用于任意初值和出生率；（ii）(0<ble 1-frac{alpha }{n})，适用于小初值和小出生率。通过验证所构造的正则化方程的解的存在性，并结合时间分式偏微分方程的广义紧凑性准则，得到了存在性结果。同时，我们通过建立分式微分不等式并使用 Moser 迭代方法得到了弱解的 (L^{infty })-bound 。此外，当阻尼系数较强时，我们利用超收缩估计证明了弱解的唯一性。

引用次数: 0

Global existence versus finite time blowup dichotomy for the dispersion managed NLS 分散管理 NLS 的全局存在与有限时间爆炸二分法

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-11-15 DOI: 10.1016/j.na.2024.113696

Mi-Ran Choi , Younghun Hong , Young-Ran Lee

We consider the Gabitov–Turitsyn equation or the dispersion managed nonlinear Schrödinger equation of a power-type nonlinearity

i \partial_{t} u + d_{av} \partial_{x}^{2} u + \int_{0}^{1} e^{- i r \partial_{x}^{2}} ({| e^{i r \partial_{x}^{2}} u |}^{p - 1} e^{i r \partial_{x}^{2}} u) d r = 0

and prove the global existence versus finite time blowup dichotomy for the mass-supercritical cases, that is,

p > 9

我们考虑了功率型非线性 i∂tu+dav∂x2u+∫01e-ir∂x2(|eir∂x2u|p-1eir∂x2u)dr=0 的加比托夫-图里岑方程或分散管理非线性薛定谔方程，并证明了质量超临界情况（即 p>9）下的全局存在与有限时间炸毁二分法。

引用次数: 0

Long time behaviour of solutions to non-local and non-linear dispersal problems 非局部和非线性分散问题解决方案的长时间特性

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-15 DOI: 10.1016/j.jde.2024.10.046

Maciej Tadej

This paper explores a non-linear, non-local model describing the evolution of a single species. We investigate scenarios where the spatial domain is either an arbitrary bounded and open subset of the n-dimensional Euclidean space or a periodic environment modelled by n-dimensional torus. The analysis includes the study of spectrum of the linear, bounded operator in the considered equation, which is a scaled, non-local analogue of classical Laplacian with Neumann boundaries. In particular we show the explicit formulas for eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of eigenvalues. Within the context of the non-linear evolution problem, we establish the existence of an invariant region, give a criterion for convergence to the mean mass, and construct spatially heterogeneous steady states.

本文探讨了描述单一物种进化的非线性、非局部模型。我们研究了空间域是 n 维欧几里得空间的任意有界开放子集或以 n 维环状体为模型的周期性环境的情形。分析包括对所考虑方程中的线性有界算子谱的研究，该算子是具有诺伊曼边界的经典拉普拉斯算子的缩放非局部类似物。我们特别展示了特征值和特征函数的明确公式。此外，我们还展示了特征值的渐近行为。在非线性演化问题的背景下，我们确定了不变区域的存在，给出了向平均质量收敛的标准，并构建了空间异质稳态。

引用次数: 0

Solving Riemann problems with a topological tool 用拓扑工具解决黎曼问题

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-15 DOI: 10.1016/j.jde.2024.11.002

Cesar S. Eschenazi , Wanderson J. Lambert , Marlon M. López-Flores , Dan Marchesin , Carlos F.B. Palmeira , Bradley J. Plohr

In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.

在之前的工作中，我们开发了一个拓扑框架，用于求解守恒定律系统的黎曼初值问题。其核心是一个称为波流形的可变流形，其点代表冲击波和稀释波。在本文中，我们详细构建了具有二次通量函数的两个守恒定律系统的三维波流形。我们使用适应坐标，推导出波流形内重要曲面和曲线的明确公式，并以图形显示。曲面根据冲击类型将流形细分为多个区域，例如与拉克斯可接受性准则相对应的区域。曲线参数化稀释波、冲击波和复合波，以连续的波形出现。虽然波形曲线在状态空间中重叠，但它们在波形流形中是分离的。我们通过构建与慢特征速度族相关的波曲线，利用冲击曲线生成一个曲面，并将该曲面与快速族波曲线相交，从而求解黎曼问题。这种构造被应用于解决几个示例的黎曼问题。

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引用次数: 0

A characterization of parallel surfaces in Minkowski space via minimal and maximal surfaces 通过最小和最大曲面表征闵科夫斯基空间中的平行曲面

IF 0.6 4区数学 Q3 MATHEMATICS

Differential Geometry and its Applications

Pub Date : 2024-11-15 DOI: 10.1016/j.difgeo.2024.102204

José Eduardo Núñez Ortiz, Gabriel Ruiz-Hernández

We give a characterization of parallel surfaces in the three dimensional Minkowski space. We consider the following construction on a non degenerate surface M. Given a non degenerate curve in the surface we have the ruled surface orthogonal to M along the curve. We prove that if this orthogonal surface is either maximal or minimal then the curve is a geodesic of M. Moreover such geodesic is either a planar line of curvature of M or it has both constant curvature and constant no zero torsion. A first result says that if M is a surface such that through every point pass two non degenerate geodesics, both with constant curvature and torsion, then the surface is parallel. Our main result says that if M is a surface then through every point pass three non degenerate curves whose associated ruled orthogonal surfaces are either maximal or minimal if and only if M is a parallel surface.

我们给出了三维闵科夫斯基空间中平行曲面的特征。给定曲面中的一条非退化曲线，我们就有了沿该曲线与 M 正交的规则曲面。我们证明，如果这个正交曲面是最大或最小的，那么这条曲线就是 M 的一条大地线。此外，这条大地线要么是 M 的一条平面曲率线，要么具有恒定曲率和恒定无零扭。第一个结果表明，如果 M 是一个曲面，且每一点都经过两条非退化的大地线，且这两条大地线都具有恒定的曲率和扭转，那么这个曲面是平行的。我们的主要结果表明，如果 M 是一个曲面，那么通过每一点的三条非退化曲线，其相关的规则正交曲面要么是最大的，要么是最小的，当且仅当 M 是一个平行曲面。

引用次数: 0

Up to the first two order Melnikov analysis for the exact cyclicity of planar piecewise linear vector fields with nonlinear switching curve 具有非线性切换曲线的平面片断线性矢量场精确周期性的梅利尼科夫分析（最高一阶二阶

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations

Pub Date : 2024-11-15 DOI: 10.1016/j.jde.2024.11.007

Liqin Zhao, Zheng Si, Ranran Jia

In this paper, we focus on providing the exact bounds for the maximum number of limit cycles

Z (3, n)

that planar piecewise linear differential systems with two zones separated by the curve

y = x^{3}

under perturbation of arbitrary polynomials of

x, y

with degree n can have, where

n \in N

. By the first two order Melnikov functions, we achieve that

Z (3, 2) = 12

Z (3, n) = 2 n + 1

for

3 \leq n \leq 88

and

Z (3, n) \geq 2 n + 1

for any n. The results are novel and improve the previous results in the literature.

在本文中，我们重点给出了平面片断线性微分系统的最大极限循环数 Z(3,n)的精确边界，在 n∈N 时，该系统在 x,y 的度数为 n 的任意多项式的扰动下，有两个区域被曲线 y=x3 分隔。通过一阶二阶梅利尼科夫函数，我们得到了 3≤n≤88 时 Z(3,2)=12, Z(3,n)=2n+1 和任意 n 时 Z(3,n)≥2n+1 的结果。

引用次数: 0

Regularity and symmetry results for the vectorial p-Laplacian 矢量 p 拉普拉卡方的正则性和对称性结果

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-11-15 DOI: 10.1016/j.na.2024.113700

Luigi Montoro, Luigi Muglia, Berardino Sciunzi, Domenico Vuono

We obtain some regularity results for solutions to vectorial

p

-Laplace equations

- Δ_{p} u = - div ({| D u |}^{p - 2} D u) = f (x, u) in Ω .

More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.

我们获得了矢量 p-Laplace 方程 -Δpu=-div(|Du|p-2Du)=f(x,u)inΩ 的解的一些正则性结果。作为正则性结果的一个结果，我们推导出了一个加权索波列夫不等式，它导致了弱比较原则。作为推论，在适当的假设条件下，我们通过移动平面技术推导出解的对称性和单调性结果。

引用次数: 0

Sobolev spaces for singular perturbation of 2D Laplace operator 二维拉普拉斯算子奇异扰动的索波列夫空间

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-11-15 DOI: 10.1016/j.na.2024.113710

Vladimir Georgiev , Mario Rastrelli

We study the perturbed Sobolev space

H_{α}^{1, r}

r \in (1, \infty),

associated with singular perturbation

Δ_{α}

of Laplace operator in Euclidean space of dimension

2 .

The main results give the possibility to extend the

L^{2}

theory of perturbed Sobolev space to the

L^{r}

case. When

r \in (2, \infty)

we have appropriate representation of the functions in

H_{α}^{1, r}

in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.

我们研究了扰动索波列夫空间 Hα1,r, r∈(1,∞)，它与 2 维欧几里得空间中拉普拉斯算子的奇异扰动 Δα 相关联。主要结果提供了将扰动索波列夫空间的 L2 理论扩展到 Lr 情况的可能性。当 r∈(2,∞)时，我们在 Hα1,r 中得到了函数在规则和奇异部分的适当表示。在质量临界和质量超临界情况下，我们还建立了与这种奇异扰动相关的 NLS 的局部良好拟合应用。

引用次数: 0

Deep mixed residual method for solving PDE-constrained optimization problems 解决 PDE 受限优化问题的深度混合残差法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-15 DOI: 10.1016/j.camwa.2024.11.009

Jinjun Yong , Xianbing Luo , Shuyu Sun , Changlun Ye

The deep mixed residual method (DeepMRM) is a technique to solve partial differential equation. In this paper, it is applied to tackle PDE-constrained optimization problems (PDE-COPs). For a PDE-COP, we transform it into an optimality system, and then employ mixed residual method (MRM) on this system. By implementing the DeepMRM with three different network structures (fully connected neural network, residual network, and attention fully connected neural network), we successfully solve PDE-COPs including elliptic, semi-linear elliptic, and Navier-Stokes (NS) equation constrained optimization problems. Compared with the exact or high-fidelity solutions, the DeepMRM provides an effective approach for solving PDE-COPs using the three different network structures.

深度混合残差法（DeepMRM）是一种求解偏微分方程的技术。本文将其应用于解决 PDE 约束优化问题（PDE-COPs）。对于 PDE-COP，我们将其转化为一个优化系统，然后在该系统上采用混合残差法（MRM）。通过使用三种不同的网络结构（全连接神经网络、残差网络和关注全连接神经网络）实现 DeepMRM，我们成功地解决了包括椭圆、半线性椭圆和纳维-斯托克斯（NS）方程约束优化问题在内的 PDE-COPs 问题。与精确解或高保真解相比，DeepMRM 提供了一种利用三种不同网络结构求解 PDE-COP 的有效方法。

引用次数: 0

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