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# 数学代写|随机分析代写Stochastic Calculus代考|Iterations of the Integration by Parts Formula

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## 数学代写|随机分析代写Stochastic Calculus代考|Iterations of the Integration by Parts Formula

We will iterate the integration by parts formula given in Proposition 2. We recall that if we iterate $l$ times the integration by parts formula, we will integrate by parts successively with respect to the variables $\left(V_i\right){i \in I_k}$ for $1 \leq k \leq l$. In order to give some estimates of the weights appearing in these formulas, we introduce the following norm on $\mathcal{S}^l\left(\cup{k=1}^l I_k\right)$, for $1 \leq l \leq L$.
$$|F|l=|F|{\infty}+\sum_{k=1}^l \sum_{1 \leq l_1<\ldots<l_k \leq l}\left|D_{l_1} \ldots D_{l_k} F\right|{\infty}$$ where $|.|{\infty}$ is defined on $\mathcal{S}^0$ by
$$|F|{\infty}=\sup {v \in O_J}\left|f^J(\omega, v)\right|$$
For $l=0$, we set $|F|0=|F|{\infty}$. We remark that we have for $1 \leq l_1<\ldots<l_k \leq l$
$$\left|D_{l_1} \ldots D_{l_k} F\right|{\infty}=\sum{i_1 \in I_{l_1}, \ldots, i_k \in I_{l_k}}\left(\prod_{j=1}^k 1_{\Lambda_{l_j, i_j}}\right)\left|\partial_{v_{i_1}} \ldots \partial_{v_{i_k}} F\right|{\infty}$$ and since for each $l\left(\Lambda{l, i}\right){i \in I_l}$ is a partition of $\Omega$, for $\omega$ fixed, the preceding sum has only one term not equal to zero. This family of norms satisfies for $F \in \mathcal{S}^{l+1}\left(\cup{k=1}^{l+1} I_k\right)$
$$|F|{l+1}=\left|D{l+1} F\right|l+|F|_l \quad \text { so } \quad\left|D{l+1} F\right|l \leq|F|{l+1}$$

## 数学代写|随机分析代写Stochastic Calculus代考|Notations and Hypotheses

We consider a Poisson point process $p$ with measurable state space $(E, \mathcal{B}(E))$. We refer to Ikeda and Watanabe [9] for the notation. We denote by $N$ the counting measure associated to $p$ so $N_t(A):=N((0, t) \times A)=#\left{s0$.
We consider the one-dimensional stochastic equation
$$X_t=x+\int_0^t \int_E c\left(s, a, X_{s^{-}}\right) \mathrm{d} N(s, a)+\int_0^t g\left(s, X_s\right) \mathrm{d} s .$$
Our aim is to give sufficient conditions on the coefficients $c$ and $g$ in order to prove that the law of $X_t$ is absolutely continuous with respect to the Lebesgue measure and has a smooth density. We make the following assumptions on the coefficients $c$ and $g$.

H1. We assume that the functions $c$ and $g$ are infinitely differentiable with respect to the variables $(t, x)$ and that there exist a bounded function $\bar{c}$ and a constant $\bar{g}$, such that
\begin{aligned} \forall(t, a, x) \quad|c(t, a, x)| \leq \bar{c}(a)(1+|x|), \quad \sup {l+l^{\prime} \geq 1}\left|\partial_t^{l^{\prime}} \partial_x^l c(t, a, x)\right| \leq \bar{c}(a) \ \forall(t, x) \quad|g(t, x)| \leq \bar{g}(1+|x|), \sup {l+l^{\prime} \geq 1}\left|\partial_t^{l^{\prime}} \partial_x^l g(t, x)\right| \leq \bar{g} \end{aligned}
We assume moreover that $\int_E \bar{c}(a) \mathrm{d} \mu(a)<\infty$.

## 数学代写|随机分析代写Stochastic Calculus代考|Iterations of the Integration by Parts Formula

$$|F|l=|F|{\infty}+\sum_{k=1}^l \sum_{1 \leq l_1<\ldots<l_k \leq l}\left|D_{l_1} \ldots D_{l_k} F\right|{\infty}$$ 在哪里 $|.|{\infty}$ 定义为 $\mathcal{S}^0$ 通过
$$|F|{\infty}=\sup {v \in O_J}\left|f^J(\omega, v)\right|$$

$$\left|D_{l_1} \ldots D_{l_k} F\right|{\infty}=\sum{i_1 \in I_{l_1}, \ldots, i_k \in I_{l_k}}\left(\prod_{j=1}^k 1_{\Lambda_{l_j, i_j}}\right)\left|\partial_{v_{i_1}} \ldots \partial_{v_{i_k}} F\right|{\infty}$$ 因为对于每个人 $l\left(\Lambda{l, i}\right){i \in I_l}$ 是的分割 $\Omega$，为 $\omega$ 固定的，前面的和只有一项不等于零。这个范数族满足 $F \in \mathcal{S}^{l+1}\left(\cup{k=1}^{l+1} I_k\right)$
$$|F|{l+1}=\left|D{l+1} F\right|l+|F|_l \quad \text { so } \quad\left|D{l+1} F\right|l \leq|F|{l+1}$$

## 数学代写|随机分析代写Stochastic Calculus代考|Notations and Hypotheses

$$X_t=x+\int_0^t \int_E c\left(s, a, X_{s^{-}}\right) \mathrm{d} N(s, a)+\int_0^t g\left(s, X_s\right) \mathrm{d} s .$$

h1。我们假设函数$c$和$g$对变量$(t, x)$是无限可微的，并且存在有界函数$\bar{c}$和常数$\bar{g}$，使得
\begin{aligned} \forall(t, a, x) \quad|c(t, a, x)| \leq \bar{c}(a)(1+|x|), \quad \sup {l+l^{\prime} \geq 1}\left|\partial_t^{l^{\prime}} \partial_x^l c(t, a, x)\right| \leq \bar{c}(a) \ \forall(t, x) \quad|g(t, x)| \leq \bar{g}(1+|x|), \sup {l+l^{\prime} \geq 1}\left|\partial_t^{l^{\prime}} \partial_x^l g(t, x)\right| \leq \bar{g} \end{aligned}

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