### 金融代写|随机偏微分方程代写Stochastic partialResults for Quadratic Nonlinearities

$$\partial_{t} u=\Delta u+\xi,$$

statistics-lab™ 为您的留学生涯保驾护航 在代写随机偏微分方程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。我们的专家在代写随机偏微分方程代写方面经验极为丰富，各种代写随机偏微分方程相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Results for Quadratic Nonlinearities

This section states rigorous results for the approximation via amplitude equations for quadratic nonlinearities. We focus only on the interesting case, where $P_{c} B(a, a)=0$, which was discussed for additive noise on a formal level in Section 1.1.3. The case with $P_{c} B(a, a) \neq 0$ is similar to the cubic case. The formal result for our case is completely analogous to the one stated in Section 1.1.2, we summarise details below. Nevertheless, in this case in general we cannot bound moments of solutions. We have to use cut-off techniques in order to use moments.

Here we present a somewhat simpler model with multiplicative noise, in order to simplify the presentation. We review the results of [Blö05a] for additive noise in

Section 2.6. In [Blö05a] also fractional (i.e. smoother) additive noise was used, but we do not focus on that.
Consider
$$\partial_{t} u=L u+\varepsilon^{2} A u+B(u, u)+\varepsilon u \dot{\beta},$$
with $L$ and $A$ as in Assumption $2.1$ and $2.2$, and $B$ some bilinear mapping defined later on in Assumption 2.4.

Let us recall the formal derivation of the amplitude equation, which is similar to Section 1.1.3. Plugging the ansatz
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\varepsilon^{2} \psi_{o}\left(\varepsilon^{2} t\right)$$
with $a \in \mathcal{N}$ and $\psi_{o} \in P_{s} X$ into (2.40), we derive in lowest order of $\varepsilon>0$
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $\mathcal{N}: \quad 0=B_{c}(a, a)$,
$\mathcal{O}\left(\varepsilon^{3}\right)$ in $\mathcal{N}: \quad \partial_{T} a=A_{c} a+2 B_{c}\left(a, \psi_{o}\right)+a \partial_{T} \tilde{\beta}$,
$\mathcal{O}\left(\varepsilon^{2}\right)$ in $P_{s} X: \quad 0=L \psi_{o}+B_{s}(a, a) .$
Note that $\tilde{\beta}(T)=\varepsilon \beta\left(T \varepsilon^{-2}\right)$ is again a rescaled Brownian motion. From (2.41) we see that $B_{c}(a, a)=0\left(B_{c}:=P_{c} B\right.$, as usual $)$ is necessary for the approach presented. Finally, projecting (2.43) to $P_{s} X$ and solving for $\psi_{o}$ yields
$$\partial_{T} a=A_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)+a \partial_{T} \tilde{\beta}$$
or in integrated form
$$a(T)=a(0)+\int_{0}^{T}\leftA_{c} a-2 B_{c}\left(a, L_{s}^{-1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$
where we consider as before Itô-differentials. Nevertheless, as discussed before in Section 2.1, we could also consider Stratonovič-differentials everywhere, and still obtain the same result. An interesting feature of (2.45) is that the amplitude equation involves a cubic nonlinearity. Therefore, we can expect nonlinear stability of the amplitude equation, which is in general not present for the SPDE.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Attractivity

We use a cut-off technique, as in general we cannot control moments of solutions. There are some special cases like for instance one-dimensional Burgers, surface growth, or Kuramoto-Sivashinsky equation (see [BGR02; DPDT94; DE01]), where we actually can derive bounds for moments. But for our results it is enough to cut off the nonlinearity for large solutions, in order to keep it small for solutions that get too large.

This technique is well known for SDEs with blow-ups. See for example [McK69]. For a detailed discussion see Section $6.3$ of [HT94]. The idea is always to cut off the nonlinearities, in order to derive bounds for moments and to compute probabilities. But solutions of the modified equation with cut-off and the original equation coincide, as long as both are small. Note that for the local attractivity result we are anyway only interested in solutions that are small. To be more precise we look at solutions of order $\mathcal{O}(\varepsilon)$.

The main result is a local attractivity result for solutions of order $\mathcal{O}(\varepsilon)$. It shows that if $u(0)$ is of order $\mathcal{O}(\varepsilon)$, then at some time $t_{\varepsilon}=\mathcal{O}\left(\ln \left(\varepsilon^{-1}\right)\right)$ the probability is almost 1 that $u\left(t_{\varepsilon}\right)$ is still of order $\mathcal{O}(\varepsilon)$, but $P_{s} u\left(t_{e}\right)$ decreased to order $\mathcal{O}\left(\varepsilon^{2}\right)$.
Theorem 2.4 (Attractivity) Let Assumptions 2.1, 2.3, and 2.4 be true.
For all small $\kappa>0$, all $\delta>0$ and $p>0$ there are constants $C>0, \delta_{1}, \delta_{2}>0$ such that for $t_{e}=\frac{2}{\omega} \ln \left(\varepsilon^{-1}\right)$ and all mild solutions in the sense of Definition $2.5$
$$\mathbb{P}\left(\left|u\left(t_{\varepsilon}\right)\right| \leq \delta_{1} \varepsilon,\left|P_{s} u\left(t_{\varepsilon}\right)\right| \leq \delta_{2} \varepsilon^{2}\right) \geq \mathbb{P}(|u(0)| \leq \delta \varepsilon)-C \varepsilon^{p}$$
for all $\varepsilon \in(0,1)$.
The proof relies on the linear stability of (2.40) and cut-off techniques. We postpone the proof to Section $2.4 .4$ as it is not difficult but technical. Let us first discuss the results for the residual and the approximation.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

For a solution of the amplitude equation $(2.45)$ and some $\psi(0)$, we consider the approximation $\varepsilon w$ given by $(2.47)$. The residual of $\varepsilon w$ is as usual defined as
\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0)+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{0}^{t} \mathrm{e}^{(t-\tau) L}\left\varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{aligned}
Theorem 2.5 (Residual) Let Assumptions 2.1, 2.3, and 2.4 be true.
For $p>4, \delta>0, T_{0}>0$ there is a constant $C>0$ such that for all approximations defined by (2.46) and (2.47), where a is a solution of (2.45), with $\mathbb{E}|a(0)|^{4 p} \leq \delta$ and $\mathbb{E}|\psi(0)|^{2 p} \leq \delta$ we have
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{2 p}$$
and
$$\mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w)(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2}$$
Furthermore $P_{c} \operatorname{Res}(\varepsilon w)$ is differentiable with
$$\partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right]\left(\varepsilon^{2} t\right)$$
The proof below is straightforward using the key Lemma 2.2. This lemma is a purely technical estimate, and we postpone the proof to Section 2.4.4.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Results for Quadratic Nonlinearities

∂吨在=大号在+e2一种在+乙(在,在)+e在b˙,

∂吨一种=一种C一种−2乙C(一种,大号s−1乙s(一种,一种))+一种∂吨b~

$$a(T)=a(0)+\int_{0}^{T}\left A_{c} a-2 B_{c}\left(a, L_{s}^{- 1} B_{s}(a, a)\right)\right d \tau+\int_{0}^{T} a(\tau) d \tilde{\beta}(\tau)$$

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|Residual

\begin{aligned} \operatorname{Res}(\varepsilon w)(t)=&-\varepsilon w(t)+\varepsilon \mathrm{e}^{t L} w(0 )+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} w(\tau) d \beta(\tau) \ &+\int_{ 0}^{t} \mathrm{e}^{(t-\tau) L}\left \varepsilon^{3} A w+\varepsilon^{2} B(w)\right d \tau \end{对齐} 吨H和这r和米2.5(R和s一世d在一种l)大号和吨一种ss在米p吨一世这ns2.1,2.3,一种nd2.4b和吨r在和.F这rp>4,d>0,吨0>0吨H和r和一世s一种C这ns吨一种n吨C>0s在CH吨H一种吨F这r一种ll一种ppr这X一世米一种吨一世这nsd和F一世n和db是(2.46)一种nd(2.47),在H和r和一种一世s一种s这l在吨一世这n这F(2.45),在一世吨H和|一种(0)|4p≤d一种nd和|ψ(0)|2p≤d在和H一种在和 \mathbb{E}\left(\sup {t \in\left[0, T{0} \varepsilon^{-2}\right]}\left|P_{c} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{2 p} 一种nd \mathbb{E}\left(\sup {t \in\left[0, T{0} \epsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon w )(t)\right|^{p}\right) \leq C \varepsilon^{3 p-2} F在r吨H和r米这r和磷C水库⁡(e在)一世sd一世FF和r和n吨一世一种bl和在一世吨H \partial_{t} P_{c} \operatorname{Res}(\varepsilon w)(t)=\varepsilon^{4}\left[A_{c} \psi+B_{c}(\psi)\right] \left(\varepsilon^{2} t\right)

## 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。