### 统计代写|离散时间鞅理论代写martingale代考|Martingale optimal transport

statistics-lab™ 为您的留学生涯保驾护航 在代写离散时间鞅理论martingale方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写离散时间鞅理论martingale代写方面经验极为丰富，各种离散时间鞅理论martingale相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|离散时间鞅理论代写martingale代考|Formulation in R+d and multi-dimensional marginals

The MK formulation and its dual expression remain valid when $S_{1}$ and $S_{2}$ are two random variables in $\mathbb{R}{+}^{d}$. The interpretation in mathematical finance goes as follows: let us consider a payoff $c\left(s{1}, s_{2}\right)$ depending on two groups $\left(s_{1}, s_{2}\right)$, each composed of $d$ assets. The first group is $\left(s_{1}^{1}, \ldots, s_{1}^{d}\right) \in \mathbb{R}{+}^{d}$. Knowing the distribution of $S{1} \in \mathbb{R}{+}^{d}$ is equivalent to knowing (at $t=0$ ) the market values of all basket options $\mathbb{E}^{\mathbb{P}^{1}}\left[\left(S{1} \cdot \omega-K\right)^{+}\right]$for all $K \in \mathbb{R}$ and for all $\omega \in \mathbb{R}^{d}$. This equivalence can be seen by observing that basket option prices fix the Laplace transform of $S_{1}: \mathbb{E}^{P^{1}}\left[e^{\omega-S_{1}}\right]$. Although basket options are liquid only for some particular values of the weight $\omega$ (and $K)$, the values $\mathbb{E}^{\mathbb{P}^{1}}\left[\left(S_{1} \cdot \omega-K\right)^{+}\right]$can be however fixed by assuming a correlation structure (more precisely a copula, denoted co below) between the variables $\left(S_{1}^{1}, \ldots, S_{1}^{d}\right)$. For example, the first group of assets (resp. second) belongs to the same financial sector and can therefore be assumed to be strongly correlated. This is not the case for the correlation structures between $S_{1}$ and $S_{2}$ which belong to two different groups and for which the correlation information is difficult to obtain. This is found through our OT formulation. By definition of the copula co, we impose that
$$\mathbb{E}^{\mathbb{P}^{1}}\left[\lambda_{1}\left(S_{1}\right)\right] \equiv \mathbb{E}\left[\lambda_{1}\left(F_{1}^{-1}\left(U_{1}\right), \ldots F_{d}^{-1}\left(U_{d}\right)\right) \operatorname{co}\left(U_{1}, \ldots, U_{d}\right)\right]$$
where $\left(U_{i}\right){1 \leq i \leq d}$ are $d$ independent uniform random variables and $F{i}$ is the cumulative distribution of $S_{1}^{i}$ implied from $T$-Vanilla options on $S_{1}^{i}$. Note that our discussion can be extended when $S_{1} \in \mathbb{R}{+}^{d}$ and $S{2} \in \mathbb{R}_{+}^{d^{}}$ with $d \neq d^{}$.

## 统计代写|离散时间鞅理论代写martingale代考|Fréchet-Hoeffding solution

Under the so-called Spence-Mirrlees condition, $c_{12} \equiv \partial_{s_{1} s_{2}} c>0$, OT (2.6) can be solved explicitly. Let $F_{1}, F_{2}$ denote the cumulative distribution functions of $\mathbb{P}^{1}$ and $\mathbb{P}^{2}$. For the sake of simplicity, we will assume that $\mathbb{P}^{1}$ does not give mass to points and $c \in C^{2}$.
THEOREM $2.2$
Under $c_{12}>0$,
(i): The optimal measure $\mathbb{P}^{}$ has the form $$\mathbb{P}^{}\left(d s_{1}, d s_{2}\right)=\delta_{T\left(s_{1}\right)}\left(d s_{2}\right) \mathbb{P}^{1}\left(d s_{1}\right)$$
with $T$ the forward image of the measure $\mathbb{P}^{1}$ onto $\mathbb{P}^{2}: T(x)=F_{2}^{-1} \circ F_{1}(x)$.
(ii): The optimal upper bound is given by
$$\mathrm{MK}{2}=\int{0}^{1} c\left(F_{1}^{-1}(u), F_{2}^{-1}(u)\right) d u$$
This optimal bound can be attained by a static hedging strategy consisting in holding European payoffs $\lambda_{1} \in \mathrm{L}^{1}\left(\mathbb{P}^{1}\right), \lambda_{2} \in \mathrm{L}^{1}\left(\mathbb{P}^{2}\right)$ with market prices $\mathbb{E}^{\mathbb{P}^{1}}\left[\lambda_{1}\left(S_{1}\right)\right]$ and $\mathbb{E}^{\mathbb{P}^{2}}\left[\lambda_{2}\left(S_{2}\right)\right]$
$$\mathrm{MK}{2}=\mathbb{E}^{\mathbb{P}^{1}}\left[\lambda{1}\left(S_{1}\right)\right]+\mathbb{E}^{\mathbb{P}^{2}}\left[\lambda_{2}\left(S_{2}\right)\right]$$
with
$$\lambda_{2}(x)=\int_{0}^{x} c_{2}\left(T^{-1}(y), y\right) d y, \quad \lambda_{1}(x)=c(x, T(x))-\lambda_{2}(T(x))$$
The value of this static European portfolio super-replicates the payoff at maturity:
$$\lambda_{1}\left(s_{1}\right)+\lambda_{2}\left(s_{2}\right) \geq c\left(s_{1}, s_{2}\right), \quad \forall\left(s_{1}, s_{2}\right) \in \mathbb{R}{+}^{2}$$ $T$ is refereed as the Brenier map (or Fréchet-Hoeffding). Note that the above theorem requires additional conditions on $c$ in order to guarantee the integrability conditions $\lambda{1} \in L^{1}\left(\mathbb{P}^{1}\right)$ and $\lambda_{2} \in L^{1}\left(\mathbb{P}^{2}\right)$.

## 统计代写|离散时间鞅理论代写martingale代考|Brenier’s solution

The Fréchet-Hoeffding solution has been generalized in $\mathbb{R}^{d}$ by Brenier [29] first in the case of a quadratic cost function and then extended to concave payoff $c=c\left(s_{1}-s_{2}\right)$ by Gangbo and McCann $[79]$ and others:
THEOREM 2.3 Brenier $=c\left(s_{1}, s_{2}\right)=-\left|s_{1}-s_{2}\right|^{2} / 2$
(i): If $\mathbb{P}^{1}$ has no atoms, then there is a unique optimal $\mathbb{P}^{}$, which is a Monge solution: $$\mathbb{P}^{}=\delta_{T\left(s_{1}\right)}\left(d s_{2}\right) \mathbb{P}^{1}\left(s_{1}\right)$$
with $T=\nabla \lambda_{1} . \nabla \lambda_{1}$ is the unique gradient of a convex function $\lambda_{1}$.
(ii): The optimal bound is attained by a static hedging strategy with $\lambda_{2}(x)=$ $c(x, T(x))-\lambda_{1}(x)$ and $\lambda_{1}$ uniquely specified by
$$\left(\nabla \lambda_{1}\right) # \mathbb{P}^{1}=\mathbb{P}^{2}$$
The notation $T # \mathbb{P}^{1}=\mathbb{P}^{2}$ means that for all $U \in \mathrm{L}^{1}\left(\mathbb{P}^{2}\right)$ :
$$\mathbb{E}^{\mathbb{P}^{1}}\left[U\left(T\left(S_{1}\right)\right)\right]=\mathbb{E}^{\mathbb{P}^{2}}\left[U\left(S_{2}\right)\right]$$
If $T$ is differentiable, this condition reads
$$|\operatorname{det} \nabla T| \mathbb{P}^{2}(T(x))=\mathbb{P}^{1}(x)$$

This theorem has been generalized to a strictly concave, superlinear ${ }^{2}$ cost function $c\left(s_{1}, s_{2}\right)=c\left(s_{1}-s_{2}\right)$. The Brenier map is then
$$T(x)=x-\nabla c^{}\left(\nabla \lambda_{1}(x)\right)$$ for some $c$-concave function $\lambda_{1}$ which is uniquely fixed by the requirement $T_{#} \mathbb{P}^{1}=\mathbb{P}^{2}$. Here $c^{} \equiv \inf _{x}{p . x-c(x)}$ is the Legendre transform of $c .$

## 统计代写|离散时间鞅理论代写martingale代考|Formulation in R+d and multi-dimensional marginals

MK 公式及其对偶表达式在以下情况下仍然有效小号1和小号2是两个随机变量R+d. 数学金融中的解释如下：让我们考虑一个回报C(s1,s2)取决于两组(s1,s2), 每个由d资产。第一组是(s11,…,s1d)∈R+d. 了解分布情况小号1∈R+d相当于知道（在吨=0) 所有篮子期权的市值和磷1[(小号1⋅ω−ķ)+]对所有人ķ∈R并为所有人ω∈Rd. 这种等价性可以通过观察篮子期权价格固定拉普拉斯变换来看出小号1:和磷1[和ω−小号1]. 尽管篮子期权仅对某些特定的权重值具有流动性ω（和ķ)， 价值和磷1[(小号1⋅ω−ķ)+]然而，可以通过假设变量之间的相关结构（更准确地说是一个 copula，在下面表示为 co）来固定(小号11,…,小号1d). 例如，第一组资产（分别是第二组）属于同一金融部门，因此可以假设它们是强相关的。之间的相关结构并非如此小号1和小号2属于两个不同的组，并且很难获得相关信息。这是通过我们的 OT 公式发现的。根据 copula co 的定义，我们强加

## 统计代写|离散时间鞅理论代写martingale代考|Fréchet-Hoeffding solution

(i): 最优度量磷有形式

(ii)：最优上限由下式给出

λ2(X)=∫0XC2(吨−1(是),是)d是,λ1(X)=C(X,吨(X))−λ2(吨(X))

λ1(s1)+λ2(s2)≥C(s1,s2),∀(s1,s2)∈R+2吨被称为 Brenier 地图（或 Fréchet-Hoeffding）。请注意，上述定理需要附加条件C为了保证可积性条件λ1∈大号1(磷1)和λ2∈大号1(磷2).

## 统计代写|离散时间鞅理论代写martingale代考|Brenier’s solution

Fréchet-Hoeffding 解已被推广到RdBrenier [29] 首先在二次成本函数的情况下，然后扩展到凹支付C=C(s1−s2)通过 Gangbo 和 McCann[79]和其他人：

(i): 如果磷1没有原子，则存在唯一最优磷，这是一个 Monge 解决方案：

(ii)：通过静态对冲策略获得最优界限λ2(X)= C(X,吨(X))−λ1(X)和λ1唯一指定的

\left(\例如 \lambda_{1}\right)#\mathbb{P}^{1}=\mathbb{P}^{2}\left(\例如 \lambda_{1}\right)#\mathbb{P}^{1}=\mathbb{P}^{2}

|这⁡∇吨|磷2(吨(X))=磷1(X)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。