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Author: OptHuang

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content/papers.md

@@ -14,7 +14,7 @@ Below is a list of my research papers.
 
 ## Published
 
-- C. Huang, H. Song, J. Yang, and B. Zhou. [Error analysis of finite difference scheme for american option pricing under regime-switching with jumps](https://www.sciencedirect.com/science/article/abs/pii/S0377042723004284). *J. Comput. Appl. Math.*, accepted for publication, 2023. [[Note]](/posts/american-option/)
+- C. Huang, H. Song, J. Yang, and B. Zhou. [Error analysis of finite difference scheme for american option pricing under regime-switching with jumps](/documents/option_pricing.pdf). *J. Comput. Appl. Math.*, accepted for publication, 2023. [[Note]](/posts/american-option/)
 - Z. Gao, C. Huang, H. Song, and B. Zhou. Projection and contraction method for pricing american option under regime-switching model. *Journal of Jilin University (Science Edition)*, 60:1090-1096, 2022.
 
 ## Work in Progress

content/papers.zh-cn.md

@@ -14,7 +14,7 @@ draft: false
 
 ## 已发表
 
-- C. Huang, H. Song, J. Yang, and B. Zhou. [Error analysis of finite difference scheme for american option pricing under regime-switching with jumps](https://www.sciencedirect.com/science/article/abs/pii/S0377042723004284). *J. Comput. Appl. Math.*, accepted for publication, 2023. [[笔记]](/zh-cn/posts/american-option/)
+- C. Huang, H. Song, J. Yang, and B. Zhou. [Error analysis of finite difference scheme for american option pricing under regime-switching with jumps](/documents/option_pricing.pdf). *J. Comput. Appl. Math.*, accepted for publication, 2023. [[笔记]](/zh-cn/posts/american-option/)
 - 高子涵, 黄存昕, 宋海明, 周搏成. 求解体制转换模型下美式期权定价问题的投影收缩算法. *吉林大学学报 (理学版)*, 60:1090-1096, 2022.
 
 ## 进行中的工作

content/posts/american-option.md

@@ -2,30 +2,71 @@
 title: "Error analysis of finite difference scheme for American option pricing under regime-switching with jumps"
 date: 2023-09-01
 description: "We provide a rigorous error analysis for the finite difference scheme applied to American option pricing under regime-switching models with jumps, establishing convergence rates and stability conditions."
-tags: ["Numerical Analysis", "Finance", "Option Pricing"]
+tags: ["Numerical Analysis", "Mathematical Finance", "Option Pricing"]
+coverImage: "/images/blog/american-option.jpg"
 cardGradient: "linear-gradient(135deg, #fa709a 0%, #fee140 100%)"
 math: true
 ---
 
 > **Paper**: C. Huang, H. Song, J. Yang, and B. Zhou. *Error analysis of finite difference scheme for american option pricing under regime-switching with jumps*. J. Comput. Appl. Math., accepted for publication, 2023.
 >
-> [Journal](https://www.sciencedirect.com/science/article/abs/pii/S0377042723004284)
+> [PDF](/documents/option_pricing.pdf)
 
 ## Overview
 
-*(Brief overview of the paper -- to be filled in.)*
+This paper develops an efficient numerical method for pricing American options under regime-switching jump-diffusion models (Merton's and Kou's models). The problem is formulated as a free-boundary problem or complementarity problem with integral and differential terms on an unbounded domain. We propose a novel truncation technique, apply composite trapezoidal formula and finite difference schemes, and use a projection and contraction method (PCM) to solve the resulting linear complementarity problem (LCP).
 
 ## Motivation
 
-*(What problem does this work address? -- to be filled in.)*
+Classical option pricing models face several limitations:
+- The Black-Scholes model cannot explain asset price jumps caused by external factors
+- Standard models fail to capture cyclical changes in option prices due to short-term political or economic uncertainty
+- Combining regime-switching and jump-diffusion models leads to a system of coupled partial integro-differential equations (PIDEs) without closed-form solutions
+
+Two main challenges arise:
+1. The problem is a partial integro-differential equation defined on an infinite domain with integral terms over unbounded regions, requiring reasonable truncation
+2. Designing a numerical scheme with complete theoretical analysis and efficient solution methods is difficult
 
 ## Main Contributions
 
-*(Summarize the key findings -- to be filled in.)*
+- **Novel truncation technique**: We analyze the relation of optimal exercise boundaries among several options to localize the infinite domain problem. The left boundary condition is exact (based on permanent American option bounds), while the right boundary is relaxed.
+
+- **Efficient discretization**: We apply a composite trapezoidal formula for integral terms (ensuring the discretized integral matrix is Toeplitz) and finite difference methods for differential terms, resulting in an LCP with numerically friendly structure.
+
+- **Complete theoretical analysis**: We prove stability, monotonicity, and consistency of the discretization scheme, and establish error estimates of order $O(\Delta\tau + \Delta x^2)$.
+
+- **Efficient solver**: A tailored projection and contraction method (PCM) is proposed to solve the discretized LCP, exploiting the special structure of the matrices.
+
+- **Numerical validation**: Extensive experiments verify the theoretical results and demonstrate efficiency for both Merton's and Kou's models.
 
 ## Key Ideas
 
-*(Core methodology or approach -- to be filled in.)*
+**Truncation Strategy**
+
+We establish inequalities for option prices and optimal exercise boundaries between the original problem and two related problems (permanent American options and standard American options without regime-switching). This allows us to:
+- Use the optimal exercise boundary of permanent American options as the left truncation point
+- Apply an empirical estimate ($\ln 3K$) for the right truncation
+- Obtain exact boundary conditions on the left and zero boundary conditions on the right
+
+**Discretization Approach**
+
+- **Integral terms**: Composite trapezoidal formula creates a Toeplitz structure, enabling potential FFT acceleration
+- **Differential terms**: Backward Euler method in time, central difference for second-order spatial derivatives
+- **Result**: An LCP in finite dimension with a positive definite coefficient matrix $B$
+
+**Theoretical Guarantees**
+
+- **Stability**: Solution norms are uniformly bounded for all time steps
+- **Monotonicity**: The scheme is monotone and independent of mesh partitions  
+- **Consistency**: Truncation error is $O(\Delta\tau + \Delta x^2)$
+- **Convergence**: The numerical scheme converges to the viscosity solution
+
+**Solution Method**
+
+The projection and contraction method (PCM) efficiently solves the LCP at each time step by:
+- Projecting onto the feasible region defined by the complementarity conditions
+- Using contraction steps with adaptive step sizes
+- Exploiting the structure of matrix $B$ (symmetric, positive definite, tridiagonal-block)
 
 ---
 

content/posts/american-option.zh-cn.md

@@ -2,30 +2,71 @@
 title: "带跳跃的体制转换下美式期权定价有限差分格式的误差分析"
 date: 2023-09-01
 description: "我们对体制转换带跳跃模型下美式期权定价的有限差分格式进行了严格的误差分析，建立了收敛速率和稳定性条件。"
-tags: ["数值分析", "金融", "期权定价"]
+tags: ["数值分析", "数学金融", "期权定价"]
+coverImage: "/images/blog/american-option.jpg"
 cardGradient: "linear-gradient(135deg, #fa709a 0%, #fee140 100%)"
 math: true
 ---
 
 > **论文**: C. Huang, H. Song, J. Yang, and B. Zhou. *Error analysis of finite difference scheme for american option pricing under regime-switching with jumps*. J. Comput. Appl. Math., accepted for publication, 2023.
 >
-> [期刊链接](https://www.sciencedirect.com/science/article/abs/pii/S0377042723004284)
+> [PDF](/documents/option_pricing.pdf)
 
 ## 概述
 
-*（论文简要概述 -- 待填写。）*
+本文针对体制转换跳跃-扩散模型（Merton 模型和 Kou 模型）下美式期权定价问题，开发了一种高效的数值方法。该问题可以描述为具有积分项和微分项的自由边界问题或互补问题，定义在无界域上。我们提出了一种新颖的截断技术，应用复合梯形公式和有限差分格式进行离散化，并使用投影收缩方法（PCM）求解得到的线性互补问题（LCP）。
 
 ## 研究动机
 
-*（这项工作解决了什么问题？ -- 待填写。）*
+经典期权定价模型面临几个局限性：
+- Black-Scholes 模型无法解释由外部因素引起的资产价格跳跃现象
+- 标准模型无法捕捉由短期政治或经济不确定性导致的期权价格周期性变化
+- 结合体制转换和跳跃扩散模型会导致一个耦合的偏积分微分方程（PIDEs）系统，无法获得闭式解
+
+主要面临两大挑战：
+1. 问题是一个定义在无穷域上的偏积分微分方程，包含无界区域上的积分项，需要合理的截断方法
+2. 设计具有完整理论分析和高效求解方法的数值格式较为困难
 
 ## 主要贡献
 
-*（总结关键发现 -- 待填写。）*
+- **新颖的截断技术**：通过分析不同期权之间最优执行边界的关系来局域化无穷域问题。左边界条件是精确的（基于永久美式期权的界），而右边界是放松的。
+
+- **高效离散化**：对积分项应用复合梯形公式（确保离散积分矩阵是 Toeplitz 矩阵），对微分项应用有限差分方法，得到具有数值友好结构的 LCP。
+
+- **完整理论分析**：证明了离散格式的稳定性、单调性和一致性，并建立了 $O(\Delta\tau + \Delta x^2)$ 阶的误差估计。
+
+- **高效求解器**：提出了一种定制的投影收缩方法（PCM）来求解离散化 LCP，充分利用了矩阵的特殊结构。
+
+- **数值验证**：大量实验验证了理论结果，并证明了该方法对 Merton 模型和 Kou 模型的有效性。
 
 ## 核心思想
 
-*（核心方法或途径 -- 待填写。）*
+**截断策略**
+
+我们建立了原问题与两个相关问题（永久美式期权和无体制转换的标准美式期权）之间期权价格和最优执行边界的不等式关系。这使我们能够：
+- 使用永久美式期权的最优执行边界作为左截断点
+- 对右截断应用经验估计（$\ln 3K$）
+- 在左边获得精确边界条件，在右边获得零边界条件
+
+**离散化方法**
+
+- **积分项**：复合梯形公式创建 Toeplitz 结构，可以使用 FFT 加速
+- **微分项**：时间方向使用后向 Euler 方法，空间二阶导数使用中心差分
+- **结果**：有限维 LCP，系数矩阵 $B$ 是正定的
+
+**理论保证**
+
+- **稳定性**：解的范数在所有时间步上一致有界
+- **单调性**：格式是单调的且与网格划分无关
+- **一致性**：截断误差为 $O(\Delta\tau + \Delta x^2)$
+- **收敛性**：数值格式收敛到粘性解
+
+**求解方法**
+
+投影收缩方法（PCM）通过以下方式高效求解每个时间步的 LCP：
+- 投影到由互补条件定义的可行域
+- 使用自适应步长的收缩步骤
+- 利用矩阵 $B$ 的结构（对称、正定、三对角块状）
 
 ---
 

content/posts/exact-lower-bounds.md

@@ -1,6 +1,7 @@
 ---
 title: "Exact lower bounds on the non-convergence probability of probabilistic direct search"
 date: 2026-02-01
+draft: true
 description: "We derive exact lower bounds on the non-convergence probability of probabilistic direct search methods by studying the distribution of a random series, providing sharp theoretical limits."
 tags: ["Optimization", "Probability", "Work in Progress"]
 cardGradient: "linear-gradient(135deg, #0c3483 0%, #a2b6df 100%)"

content/posts/exact-lower-bounds.zh-cn.md

@@ -1,4 +1,5 @@
 ---
+draft: true
 title: "概率直接搜索非收敛概率的精确下界：随机级数的分布"
 date: 2026-02-01
 description: "我们通过研究随机级数的分布，推导出概率直接搜索方法非收敛概率的精确下界，提供了尖锐的理论极限。"

content/posts/gradient-convergence.md

@@ -1,8 +1,9 @@
 ---
 title: "Gradient convergence of direct search based on sufficient decrease"
 date: 2025-12-01
-description: "We study the gradient convergence properties of direct search methods that employ sufficient decrease conditions, providing new convergence guarantees for this class of algorithms."
+description: "We prove that direct search achieves full gradient convergence, strengthening the classical liminf result to the full limit."
 tags: ["Optimization", "Direct Search", "Convergence"]
+coverImage: "/images/blog/grad_converge.jpg"
 cardGradient: "linear-gradient(135deg, #43e97b 0%, #38f9d7 100%)"
 math: true
 ---
@@ -13,20 +14,36 @@ math: true
 
 ## Overview
 
-*(Brief overview of the paper -- to be filled in.)*
+This paper studies a direct-search method based on sufficient decrease for unconstrained smooth optimization. Under standard assumptions, we improve the classical guarantee $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$ to the full limit $\lim_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$. This stronger result means the gradient norm converges to zero along the entire sequence, not just along a subsequence.
 
 ## Motivation
 
-*(What problem does this work address? -- to be filled in.)*
+- **A long-standing open question**: The classical theory for direct search only guarantees that small gradients appear infinitely often ($\liminf = 0$). This does not rule out oscillation—the iterates could wander between stationary and non-stationary regions
+- **Simplicity**: Previous results achieving full gradient convergence require additional mechanisms (complete polling or line-search globalization). Can we achieve the same with just sufficient decrease?
+- **Theoretical completeness**: Upgrading $\liminf$ to $\lim$ provides a cleaner convergence guarantee
 
 ## Main Contributions
 
-*(Summarize the key findings -- to be filled in.)*
+- **Full gradient convergence**: Under standard assumptions (Lipschitz continuous gradient, bounded below objective, positive cosine measure), we prove $\lim_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$
+
+- **Minimal mechanism**: No additional mechanism beyond the sufficient decrease condition and step-size dynamics ($\gamma > 1$) is needed
+
+- **Extension to locally Lipschitz gradients**: If $\nabla f$ is only locally Lipschitz continuous, we show that every accumulation point of the iterates is stationary
 
 ## Key Ideas
 
-*(Core methodology or approach -- to be filled in.)*
+**Proof by contradiction**: Suppose $\limsup_{k\to\infty} \lVert\nabla f(x_k)\rVert > 0$. Define two regions based on gradient norm:
+- Small gradient region: $S_\leq^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert \leq \epsilon\rbrace$
+- Large gradient region: $S_>^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert > 2\epsilon\rbrace$
 
----
+**Crossing analysis**: Track when iterates enter/exit these regions via stopping times $m_j^\epsilon$ and $n_j^\epsilon$. Key observations:
+
+1. **Geometric separation**: Lipschitz continuity implies $\text{dist}(S_\leq^\epsilon, S_>^\epsilon) \geq \epsilon/L$, so each crossing covers at least distance $D_j^\epsilon \geq \epsilon/L$
+
+2. **Step-size dynamics**: Inside the large gradient region, sufficient decrease is always achieved, so $\alpha_{k+1} = \gamma \alpha_k$ (step sizes grow geometrically)
+
+3. **Scaling law**: The function decrease during crossing $j$ satisfies
+$$\Delta f_j^\epsilon \geq C \cdot (D_j^\epsilon)^p$$
+for some constant $C > 0$ (where $p > 1$ is the forcing function exponent)
 
-*This note is being updated. Stay tuned!*
+4. **Contradiction**: Since $f$ is bounded below, $\Delta f_j^\epsilon \to 0$. But by the scaling law, this forces $D_j^\epsilon \to 0$, contradicting $D_j^\epsilon \geq \epsilon/L > 0$

content/posts/gradient-convergence.zh-cn.md

@@ -1,8 +1,9 @@
 ---
 title: "基于充分下降的直接搜索的梯度收敛性"
 date: 2025-12-01
-description: "我们研究了采用充分下降条件的直接搜索方法的梯度收敛性质，为这类算法提供了新的收敛保证。"
+description: "我们证明直接搜索实现完整的梯度收敛，将经典的 liminf 结果加强为完整的极限。"
 tags: ["优化", "直接搜索", "收敛性"]
+coverImage: "/images/blog/grad_converge.jpg"
 cardGradient: "linear-gradient(135deg, #43e97b 0%, #38f9d7 100%)"
 math: true
 ---
@@ -13,20 +14,36 @@ math: true
 
 ## 概述
 
-*（论文简要概述 -- 待填写。）*
+本文研究了基于充分下降条件（sufficient decrease）的直接搜索方法。在标准假设下，我们将经典的收敛保证 $\liminf_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$ 提升为完整的极限 $\lim_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$。这个更强的结果意味着梯度范数沿整个序列收敛到零，而不仅仅是沿某个子序列。
 
 ## 研究动机
 
-*（这项工作解决了什么问题？ -- 待填写。）*
+- **一个长期的开放问题**：直接搜索的经典理论只保证小梯度无穷多次出现（$\liminf = 0$）。这不能排除振荡——迭代点可能在平稳区域和非平稳区域之间徘徊
+- **简洁性**：以往实现完整梯度收敛的结果需要额外机制（完全轮询或线搜索全局化）。我们能否仅用充分下降条件达到同样效果？
+- **理论完整性**：将 $\liminf$ 提升为 $\lim$ 提供了更简洁的收敛保证
 
 ## 主要贡献
 
-*（总结关键发现 -- 待填写。）*
+- **完整的梯度收敛**：在标准假设下（Lipschitz 连续梯度、目标函数有下界、正余弦测度），我们证明 $\lim_{k\to\infty} \lVert\nabla f(x_k)\rVert = 0$
+
+- **最小机制**：除了充分下降条件和步长动态（$\gamma > 1$）之外，不需要额外机制
+
+- **推广到局部 Lipschitz 梯度**：如果 $\nabla f$ 仅是局部 Lipschitz 连续的，我们证明迭代点的每个聚点都是平稳点
 
 ## 核心思想
 
-*（核心方法或途径 -- 待填写。）*
+**反证法**：假设 $\limsup_{k\to\infty} \lVert\nabla f(x_k)\rVert > 0$。基于梯度范数定义两个区域：
+- 小梯度区域：$S_\leq^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert \leq \epsilon\rbrace$
+- 大梯度区域：$S_>^\epsilon = \lbrace x : \lVert\nabla f(x)\rVert > 2\epsilon\rbrace$
 
----
+**穿越分析**：通过停时 $m_j^\epsilon$ 和 $n_j^\epsilon$ 追踪迭代点进入/离开这些区域的时刻。关键观察：
+
+1. **几何分离**：Lipschitz 连续性保证 $\text{dist}(S_\leq^\epsilon, S_>^\epsilon) \geq \epsilon/L$，因此每次穿越至少覆盖距离 $D_j^\epsilon \geq \epsilon/L$
+
+2. **步长动态**：在大梯度区域内，充分下降条件总是满足，因此 $\alpha_{k+1} = \gamma \alpha_k$（步长几何增长）
+
+3. **缩放律**：第 $j$ 次穿越期间的函数下降满足
+$$\Delta f_j^\epsilon \geq C \cdot (D_j^\epsilon)^p$$
+其中 $C > 0$ 是某个常数（$p > 1$ 是强制函数的指数）
 
-*本笔记正在更新中，敬请期待！*
+4. **矛盾**：由于 $f$ 有下界，$\Delta f_j^\epsilon \to 0$。但根据缩放律，这迫使 $D_j^\epsilon \to 0$，与 $D_j^\epsilon \geq \epsilon/L > 0$ 矛盾