### 金融代写|随机偏微分方程代写Stochastic partialResidual

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

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

With Theorem $2.1$ at hand we make the following ansatz
$$u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\mathcal{O}\left(\varepsilon^{2}\right), \quad \text { where } a \in \mathcal{N} .$$
Using a formal calculation completely analogous to the one of Section $1.1 .1$ yields in lowest order of $\varepsilon>0$ the following amplitude equation:
$$d a=A_{c} a+\mathcal{F}{c}(a)+a d \tilde{\beta},$$ where ${\bar{\beta}(T)}{T \geq 0}$ defined by $\bar{\beta}(T)=\varepsilon \beta\left(\varepsilon^{-2} T\right)$ is a rescaled version of the Brownian motion $\beta$. As usual we consider the equation in the Itô sense. Note again, as explained in Section 1.1.1, that a fixed realization of the amplitude equation obviously depends on $\varepsilon$, but in distribution the solutions are independent of $\varepsilon$.
For a solution $a$ of (2.15) we define the residual
\begin{aligned} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)=-\varepsilon a\left(\varepsilon^{2} t\right) &+\varepsilon \mathrm{e}^{t L} a(0)+\varepsilon^{2} \int_{0}^{t} \mathrm{e}^{(t-\tau) L} a\left(\varepsilon^{2} \tau\right) d \beta(\tau) \ &+\varepsilon^{3} \int_{0}^{t} \mathrm{e}^{(t-\tau) L}[A a+\mathcal{F}(a)]\left(\varepsilon^{2} \tau\right) d \tau \end{aligned}
We show:
Theorem 2.2 (Residual) Let Assumptions 2.1, 2.2, and 2.3 be true. Then for all $p>\frac{4}{3}, \delta>0$ and $T_{0}>0$ there is a constant $C>0$ such that
$$\mathrm{P}{\mathrm{c}} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)=0$$ and $$\mathbb{E}\left(\sup {t \in\left[0, T_{0} \varepsilon^{-2}\right]}\left|P_{s} \operatorname{Res}(\varepsilon a)\left(\varepsilon^{2} t\right)\right|^{p}\right) \leq C \varepsilon^{3 p}$$
for all sufficiently small $\varepsilon>0$ and all solutions a of (2.15) with $\mathbb{E}|a(0)|^{3 p} \leq \delta \varepsilon^{3 p}$.

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

Define the remainder $R$, which is the error of our approximation, as
$$\varepsilon^{2} R(t)=u(t)-\varepsilon a\left(\varepsilon^{2} t\right)$$
We split
$$R=R_{c}+R_{s} \quad \text { with } \quad R_{c}=P_{c} R \text { and } R_{s}=P_{s} R .$$
First we treat $R_{s}$ using the a priori estimates on $P_{s} u$. This information on $P_{s} u$ is not necessary for the result, as we can use cut-off techniques to yield local results, but here it helps to simplify the proofs a lot. The a priori estimates on $u$ are only possible because of the very strong stability assumptions on $\mathcal{F}$. Our main result is the following:

Theorem 2.3 (Approximation) Let Assumptions 2.1, 2.2, and $2.3$ be true. For $p>4, T_{0}>0$, and $\delta>0$ there is a constant $C>0$ such that for all strong solutions $u$ of (2.3) in $X$ with
$$\mathbb{E}|u(0)|^{3 p} \leq \delta \varepsilon^{3 p} \quad \text { and } \quad \mathbb{E}\left|P_{s} u(0)\right|^{p} \leq \delta \varepsilon^{3 p}$$
for all $\varepsilon \in(0,1)$, we derive
$$\left.\mathbb{E}\left(\sup {t \in\left[0, T{0} e^{-2}\right]} | P_{s} R(t)\right) |^{p}\right) \leq C \varepsilon^{p}$$
and
$$\left.\mathbb{E}\left(\sup {t \in\left[0, T{\mathrm{b}} e^{-2}\right]} | P_{c} R(t)\right) |^{p}\right) \leq C$$
for all sufficiently small $\varepsilon>0$, where $a$ is a solution of (2.15) such that $a(0)=$ $\varepsilon^{-1} P_{c} u(0)$.

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|A priori Estimates for u

The following section provides standard a priori estimates for solutions of (2.3). Although they are straightforward, they are nevertheless quite technical. We establish bounds for $\mathbb{E}|u(t)|^{p}$ and $\mathbb{E}\left|P_{s} u(t)\right|^{p}$, which are used in the proof of Theorem 2.1. Furthermore, we bound $\mathbb{E} \sup {t \in\left[0, T{0} e^{-2}\right]}|u(t)|^{p}$ and in Lemma $2.1$ $\mathbb{E} \sup {t \in\left[0, T{\mathrm{b}} e^{-2}\right]}\left|P_{s} u(t)\right|^{p}$. The main idea is to apply Itô’s formula to $|u(t)|^{p}$ and to use the strong nonlinear stability condition from (2.7). The main technical obstacle is that a priori we do not know that $\mathbb{E}|u(t)|^{p}$ exists. Therefore we use cut-off techniques.

Proof. (of Theorem 2.1) For $p \geq 2$ and $\gamma>0$ consider smooth bounded $\varphi_{\gamma, p}:[0, \infty) \rightarrow \mathbb{R}$ such that $0 \leq \varphi_{\gamma, p}(z) \nearrow \varphi_{p}(z)=z^{p / 2}$. To be more precise, define
$$\varphi_{\gamma, p}(z):=\left(\frac{z}{1+\gamma z}\right)^{p / 2} \text { for } z \geq 0 .$$
It is now easy to check that there are constants $C_{p}$ and $c_{p}$ independent of $\gamma$ such that for $z \geq 0$
$$\begin{gathered} 0 \leq \varphi_{\gamma, p}^{\prime}(z) z \leq C_{p} \varphi_{\gamma, p}(z), \quad-p \varphi_{\gamma, p}(z) \leq \varphi_{\gamma, p}^{\prime \prime}(z) z^{2} \leq C_{p} \varphi_{\gamma, p}(z), \ \varphi_{\gamma, p}^{\prime}(z) z^{2}=\frac{p}{2} \varphi_{\gamma, p}(z)^{(p+2) / p}, \quad \varphi_{\gamma, p}^{\prime}(z) z^{2} \leq \frac{p}{2} \varphi_{\gamma, p-2}(z)=\frac{p}{2} \varphi_{\gamma, p}(z)^{(p-2) / p} . \end{gathered}$$
Apply Itô’s formula to $\varphi_{\gamma, p}\left(|u(t)|^{2}\right)$ for $t<\tau_{e}$ to derive
$$\begin{gathered} d \varphi_{\gamma, p}\left(|u(t)|^{2}\right)=\varphi_{\gamma, p}^{\prime}\left(|u(t)|^{2}\right)\left\langle u(t), L u(t)+\varepsilon^{2} A u(t)+\mathcal{F}(u(t))\right\rangle d t \ +\varphi_{\gamma, p}^{\prime}\left(|u(t)|^{2}\right)|u(t)|^{2}\left[\varepsilon d \beta(t)+\frac{1}{2} \varepsilon^{2} d t\right] \ +\varphi_{\gamma, p}^{\prime \prime}\left(|u(t)|^{2}\right)|u(t)|^{4} \varepsilon^{2} d t . \end{gathered}$$
Hence, for $t<\tau_{0}$ as we are dealing with strong solutions in the sense of Definition $2.4$.
\begin{aligned} &\mathbb{E} \varphi_{\gamma, p}\left(|u(t)|^{2}\right)-\mathbb{E} \varphi_{\gamma, p}\left(|u(0)|^{2}\right) \ &=\int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime}\left(|u(\tau)|^{2}\right)\left(u(\tau), L u(\tau)+\varepsilon^{2} A u(\tau)+\mathcal{F}(u(\tau))\right\rangle d \tau \ &\quad+\frac{1}{2} \varepsilon^{2} \int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime}\left(|u(\tau)|^{2}\right)|u(\tau)|^{2} d \tau \ &+\varepsilon^{2} \int_{0}^{t} \mathbb{E} \varphi_{\gamma, p}^{\prime \prime}\left(|u(\tau)|^{2}\right)|u(\tau)|^{4} d \tau \end{aligned}

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

u(t)=\varepsilon a\left(\varepsilon^{2} t\right)+\mathcal{O}\left(\varepsilon^{2}\right)， \quad \text { 其中 } 一个 \in \mathcal{N} 。使用与第1.1 节 .1u(t)=εa(ε2t)+O(ε2), where a∈N.

Res⁡(εa)(ε2t)=−εa(ε2t)+εetLa(0)+ε2∫0te(t−τ)La(ε2τ)dβ(τ) +ε3∫0te(t−τ)L[Aa+F(a)](ε2τ)dτ

PcRes⁡(εa)(ε2t)=0和E(supt∈[0,T0ε−2]|PsRes⁡(εa)(ε2t)|p)≤Cε3p

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

ε2R(t)=u(t)−εa(ε2t)

R=Rc+Rs with Rc=PcR and Rs=PsR.

E|u(0)|3p≤δε3p and E|Psu(0)|p≤δε3p
ε∈(0,1)
E(supt∈[0,T0e−2]|PsR(t))|p)≤Cεp

E(supt∈[0,Tbe−2]|PcR(t))|p)≤C

## 金融代写|随机偏微分方程代写Stochastic partial differential equations代考|A priori Estimates for u

φγ,p(z):=(z1+γz)p/2 for z≥0.

0≤φγ,p′(z)z≤Cpφγ,p(z),−pφγ,p(z)≤φγ,p′′(z)z2≤Cpφγ,p(z), φγ,p′(z)z2=p2φγ,p(z)(p+2)/p,φγ,p′(z)z2≤p2φγ,p−2(z)=p2φγ,p(z)(p−2)/p.

dφγ,p(|u(t)|2)=φγ,p′(|u(t)|2)⟨u(t),Lu(t)+ε2Au(t)+F(u(t))⟩dt +φγ,p′(|u(t)|2)|u(t)|2[εdβ(t)+12ε2dt] +φγ,p′′(|u(t)|2)|u(t)|4ε2dt.

Eφγ,p(|u(t)|2)−Eφγ,p(|u(0)|2) =∫0tEφγ,p′(|u(τ)|2)(u(τ),Lu(τ)+ε2Au(τ)+F(u(τ))⟩dτ +12ε2∫0tEφγ,p′(|u(τ)|2)|u(τ)|2dτ +ε2∫0tEφγ,p′′(|u(τ)|2)|u(τ)|4dτ

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