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

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## 数学代写|随机分析代写Stochastic Calculus代考|Notations-Derivative Operators

The abstract framework is quite similar to the one developed in Bally and Clément [2], but we introduce here some modifications in order to take into account the border terms appearing in the integration by parts formula. We consider a sequence of random variables $\left(V_i\right)_{i \in \mathbb{N} *}$ on a probability space $(\Omega, \mathcal{F}, P)$, a sub $\sigma$-algebra $\mathcal{G} \subseteq \mathcal{F}$ and a random variable $J, \mathcal{G}$ measurable, with values in $\mathbb{N}$. Our aim is to establish a differential calculus based on the variables $\left(V_i\right)$, conditionally on $\mathcal{G}$. In order to derive an integration by parts formula, we need some assumptions on the random variables $\left(V_i\right)$. The main hypothesis is that conditionally on $\mathcal{G}$, the law of $V_i$ admits a locally smooth density with respect to the Lebesgue measure.

H0. (a) Conditionally on $\mathcal{G}$, the random variables $\left(V_i\right)_{1 \leq i \leq J}$ are independent and for each $i \in{1, \ldots, J}$ the law of $V_i$ is absolutely continuous with respect to the Lebesgue measure. We note $p_i$ the conditional density.
(b) For all $i \in{1, \ldots, J}$, there exist some $\mathcal{G}$ measurable random variables $a_i$ and $b_i$ such that $-\infty0\right}$. We also assume that $p_i$ admits a continuous bounded derivative on $\left(a_i, b_i\right)$ and that $\ln p_i$ is bounded on $\left(a_i, b_i\right)$

We define now the class of functions on which this differential calculus will apply. We consider in this paper functions $f: \Omega \times \mathbb{R}^{\mathbb{N}^*} \rightarrow \mathbb{R}$ which can be written as
$$f(\omega, v)=\sum_{m=1}^{\infty} f^m\left(\omega, v_1, \ldots, v_m\right) 1_{{J(\omega)=m}}$$
where $f^m: \Omega \times \mathbb{R}^m \rightarrow \mathbb{R}$ are $\mathcal{G} \times \mathcal{B}\left(\mathbb{R}^m\right)$-measurable functions.

## 数学代写|随机分析代写Stochastic Calculus代考|Duality and Basic Integration by Parts Formula

In our framework, the duality between $\delta_l$ and $D_l$ is given by the following proposition. In the sequel, we denote by $E_{\mathcal{G}}$ the conditional expectation with respect to the sigma-algebra $\mathcal{G}$.
Proposition 1. Assuming HO then $\forall F, U \in \mathcal{S}^1\left(I_l\right)$ we have
$$E_{\mathcal{G}}\left(U D_l F\right)=-E_{\mathcal{G}}\left(F \delta_l(U)\right)+E_{\mathcal{G}}[F U]_l$$
For simplicity, we assume in this proposition that the random variables $F$ and $U$ take values in $\mathbb{R}$ but such a result can easily be extended to $\mathbb{R}^d$ value random variables.

Proof. We have $E_{\mathcal{G}}\left(U D_l F\right)=\sum_{i \in I_l} 1_{\Lambda_{l, i}} E_{\mathcal{G}} 1_{O_J}(V)\left(\partial_{v_i} f^J(\omega, V) u^J(\omega, V)\right)$. From $\mathrm{H} 0$ we obtain
$$E_{\mathcal{G}} 1_{O_J}(V)\left(\partial_{v_i} f^J(\omega, V) u^J(\omega, V)\right)=E_{\mathcal{G}} 1_{O_{J, i}}\left(V_{(i)}\right) \int_{a_i}^{b_i} \partial_{v_i}\left(f^J\right) u^J p_i\left(v_i\right) \mathrm{d} v_i$$
By using the classical integration by parts formula, we have
$$\int_{a_i}^{b_i} \partial_{v_i}\left(f^J\right) u^J p_i\left(v_i\right) \mathrm{d} v_i=\left[f^J u^J p_i\right]{a_i}^{b_i}-\int{a_i}^{b_i} f^J \partial_{v_i}\left(u^J p_i\right) \mathrm{d} v_i$$
Observing that $\partial_{v_i}\left(u^J p_i\right)=\left(\partial_{v_i}\left(u^J\right)+u^J \partial_{v_i}\left(\ln p_i\right)\right) p_i$, we have
\begin{aligned} E_{\mathcal{G}}\left(1_{O_J}(V) \partial_{v_i} f^J u^J\right)= & E_{\mathcal{G}} 1_{O_{J, i}}\left[\left(V_{(i)}\right) f^J u^J p_i\right]{a_i}^{b_i} \ & -E{\mathcal{G}} 1_{O_J}(V) F\left(\partial_{v_i}(U)+U \partial_{v_i}\left(\ln p_i\right)\right) \end{aligned}
and the proposition is proved.
We can now state a first integration by parts formula.

## 数学代写|随机分析代写Stochastic Calculus代考|Notations-Derivative Operators

(b)对于所有$i \in{1, \ldots, J}$，存在一些$\mathcal{G}$可测量的随机变量$a_i$和$b_i$，使得$-\infty0\right}$。我们还假设$p_i$允许在$\left(a_i, b_i\right)$上有连续的有界导数，并且$\ln p_i$在$\left(a_i, b_i\right)$上有界

$$f(\omega, v)=\sum_{m=1}^{\infty} f^m\left(\omega, v_1, \ldots, v_m\right) 1_{{J(\omega)=m}}$$
，其中$f^m: \Omega \times \mathbb{R}^m \rightarrow \mathbb{R}$为$\mathcal{G} \times \mathcal{B}\left(\mathbb{R}^m\right)$ -可测函数

## 数学代写|随机分析代写Stochastic Calculus代考|Duality and Basic Integration by Parts Formula

$$E_{\mathcal{G}}\left(U D_l F\right)=-E_{\mathcal{G}}\left(F \delta_l(U)\right)+E_{\mathcal{G}}[F U]l$$

$$E_{\mathcal{G}} 1_{O_J}(V)\left(\partial_{v_i} f^J(\omega, V) u^J(\omega, V)\right)=E_{\mathcal{G}} 1_{O_{J, i}}\left(V_{(i)}\right) \int_{a_i}^{b_i} \partial_{v_i}\left(f^J\right) u^J p_i\left(v_i\right) \mathrm{d} v_i$$

$$\int_{a_i}^{b_i} \partial_{v_i}\left(f^J\right) u^J p_i\left(v_i\right) \mathrm{d} v_i=\left[f^J u^J p_i\right]{a_i}^{b_i}-\int{a_i}^{b_i} f^J \partial_{v_i}\left(u^J p_i\right) \mathrm{d} v_i$$

\begin{aligned} E_{\mathcal{G}}\left(1_{O_J}(V) \partial_{v_i} f^J u^J\right)= & E_{\mathcal{G}} 1_{O_{J, i}}\left[\left(V_{(i)}\right) f^J u^J p_i\right]{a_i}^{b_i} \ & -E{\mathcal{G}} 1_{O_J}(V) F\left(\partial_{v_i}(U)+U \partial_{v_i}\left(\ln p_i\right)\right) \end{aligned}

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