### 统计代写|离散时间鞅理论代写martingale代考|Robust quantile hedging

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• 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|Multi-marginals and infinitely-many marginals case

Most of the literature on OT focuses on the 2 -asset case with a payoff $c\left(s_{1}, s_{2}\right)$. For applications in mathematical finance, it is interesting to study the case of a multi-asset payoff $c\left(s_{1}, \ldots, s_{n}\right)$ depending on $n$ assets evaluated at the same maturity. We define the $n$-asset optimal transport problem (by duality) as
$$\mathrm{MK}{n} \equiv \sup {\mathbb{P} \in \mathcal{P}\left(\mathbb{P}^{1}, \ldots, \mathbb{P}^{n}\right)} \mathbb{E}^{\mathbb{P}}\left[c\left(S_{1}, \ldots, S_{n}\right)\right]$$
with $\mathcal{P}\left(\mathbb{P}^{1}, \ldots, \mathbb{P}^{n}\right)=\left{\mathbb{P}: S_{i} \stackrel{\mathbb{P}}{\sim} \mathbb{P}^{i}, \forall i=1, \ldots, n\right}$. This problem has been studied by Gangbo [78] and recently by Carlier [36] (see also Pass [124]) with the following requirement on the payoff:

DEFINITION $2.4$ see [36] $c \in C^{2}$ is strictly monotone of order 2 if for all $(i, j) \in{1, \ldots, n}^{2}$ with $i \neq j$, all second order derivatives $\partial_{i j} c$ are positive.
We have

THEOREM $2.4$ see $[78,36]$
If $c$ is strictly monotone of order 2 , there exists a unique optimal transference plan for the $\mathrm{MK}{n}$ transport problem, and it has the form $$\mathbb{P}^{*}\left(d s{1}, \ldots, d s_{n}\right)=\mathbb{P}^{1}\left(d s_{1}\right) \prod_{i=2}^{n} \delta_{T_{i}\left(s_{1}\right)}\left(d s_{i}\right), \quad T_{i}(s)=F_{i}^{-1} \circ F_{1}(s), i=2, \ldots, n$$
The optimal upper bound is
$$\mathrm{MK}{n}=\int c\left(x, T{2}(x), \ldots, T_{n}(x)\right) \mathbb{P}^{1}(d x)$$
An extension to the infinite many marginals case has been obtained recently by Pass $[125]$ who studies
$$\mathrm{MK}{\infty} \equiv \sup {\mathbb{P}: S_{\mathrm{t}} \sim \mathbb{P} t, \forall t \in(0, T]} \mathbb{E}\left[h\left(\int_{0}^{T} S_{t} d t\right)\right]$$
where $h$ is a convex function. Let $F_{t}$ the cumulative distribution of $\mathbb{P}^{t}$. Define the stochastic process $S_{t}^{\text {opt }}(\omega)=F_{t}^{-1}(\omega), \quad \omega \in[0,1]$. The underlying probability space is the interval $[0,1]$ with Lebesgue measure.

Here we take a cost function $c\left(s_{1}, s_{2}\right)=L\left(s_{2}-s_{1}\right)$ with $L$ a strictly concave function such that the Spence-Mirrlees condition is satisfied. From the formulation (2.8), one can link the Monge-Kantorovich formulation to the solution of a Hamilton-Jacobi equation through the Hopf-Lax formula:
PROPOSITION 2.3 see e.g. [139]
$$\mathrm{MK}{2}=\inf {u(1, \cdot)}-\mathbb{E}^{\mathbb{P}^{1}}\left[u\left(0, S_{1}\right)\right]+\mathbb{E}^{\mathbb{P}^{2}}\left[u\left(1, S_{2}\right)\right]$$
where $u(0, \cdot)$ is the (viscosity) solution at $t=0$ of the following HJ equation with terminal boundary condition $u(1,-):$
$$\partial_{t} u+H(D u)=0, \quad H(p) \equiv \inf _{q}{p q-L(q)}$$

PROOF From the dynamic programming principle, $u$, satisfying HamiltonJacobi equation (2.21), can be written as
$$u(t, x)=\inf {\zeta} u\left(1, x+\int{t}^{1} \dot{\zeta}(s) d s\right)-\int_{t}^{1} L(\dot{\zeta}(s)) d s$$
The minimisation over $\dot{\zeta}$ gives that $\dot{\zeta}$ is a constant $q$ (Fréchet derivative with respect to $\dot{\zeta}$ gives the critical equation $\left.\frac{d^{2} \zeta(t)}{d t^{2}}=0\right)$.
$$u(t, x)=\inf {q} u(1, x+q(1-t))-L(q)(1-t)$$ By setting $y=x+q(1-t)$, we get Hopf-Lax’s formula: $$u(t, x)=\inf {y} u(1, y)-L\left(\frac{y-x}{1-t}\right)(1-t)$$
For $t=0$, this gives that $-u(0, \cdot)$ is the L-transform of $u(1, \cdot):-u(0, x)=$ $\sup _{y} L(y-x)-u(1, y)$. We conclude with Proposition 2.1.

In the next section, we introduce a martingale version of OT, first developed in $[17,77]$ where we have obtained a Monge-Kantorovich duality result.

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

We consider a payoff $c\left(s_{1}, s_{2}\right)$ depending on a single asset evaluated at two dates $t_{1}{2}$ : DEFINITION $2.5$ $$\widetilde{\mathrm{MK}}{2} \equiv \inf {\mathcal{M}^{}\left(\mathbb{P}^{1}, \mathbb{P}^{2}\right)} \mathbb{E}^{\mathbb{P}^{1}}\left[\lambda{1}\left(S_{1}\right)\right]+\mathbb{E}^{\mathbb{P}^{2}}\left[\lambda_{2}\left(S_{2}\right)\right]$$ where $\mathcal{M}^{}\left(\mathbb{P}^{1}, \mathbb{P}^{2}\right)$ is the set of functions $\lambda_{1} \in \mathrm{L}^{1}\left(\mathbb{P}^{1}\right), \lambda_{2} \in \mathrm{L}^{1}\left(\mathbb{P}^{2}\right)$ and $H$ a bounded continuous function on $\mathbb{R}{+}$such that $$\lambda{1}\left(s_{1}\right)+\lambda_{2}\left(s_{2}\right)+H\left(s_{1}\right)\left(s_{2}-s_{1}\right) \geq c\left(s_{1}, s_{2}\right), \quad \forall\left(s_{1}, s_{2}\right) \in \mathbb{R}{+}^{2}$$ This corresponds to a semi-static hedging strategy which consists in holding European payoffs $\lambda{1}$ and $\lambda_{2}$ and applying a delta strategy at $t_{1}$, generating a

$\mathrm{P} \& \mathrm{~L} H\left(s_{1}\right)\left(s_{2}-s_{1}\right)$ at $t_{2}$ with zero cost. We could also add a term $H_{0}\left(S_{0}\right)\left(s_{1}-\right.$ $\left.S_{0}\right)$ corresponding to performing a delta-hedging at $t=0$. As this term can be incorporated into $\lambda_{1}\left(s_{1}\right)$, it is not included. Similarly, an intermediate deltahedging term $H_{i}\left(S_{0}, \ldots, s_{t_{i}}\right)\left(s_{t_{i+1}}-s_{t_{i}}\right)$ where $0<t_{i}<t_{i+1} \leq t_{2}$ can be added but it can be shown that the optimal solution is attained for $H_{i}=0$. These terms are therefore not needed and will be disregarded next (see Corollary $2.1)$

Note that in comparison with the OT MK $\mathrm{MK}{2}$ previously reported, we have $\overline{\mathrm{MK}}{2} \leq \mathrm{MK}_{2}$ due to the appearance of the function $H$.

At this point, a natural question is how the classical results in OT generalize in the present martingale version. We follow closely our introduction of OT and explain how the various concepts previously explained extend to the present setting. Our research partly originates from a systematic derivation of Skorokhod embedding solutions and understanding of particle methods for non-linear McKean stochastic differential equations appearing in the calibration of financial models (see Section 4.2.4). From a practical point of view, the derivation of these optimal bounds allows to better understand the risk of exotic options as illustrated in the next example.

## 统计代写|离散时间鞅理论代写martingale代考|Multi-marginals and infinitely-many marginals case

Pass 最近获得了对无限多边际情况的扩展[125]谁学习

∂吨在+H(D在)=0,H(p)≡信息qpq−大号(q)

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

λ1(s1)+λ2(s2)+H(s1)(s2−s1)≥C(s1,s2),∀(s1,s2)∈R+2这对应于一种半静态对冲策略，包括持有欧洲收益λ1和λ2并在吨1，生成一个

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