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数学代写|随机偏微分方程代写Stochastic Differential Equation代考|AMATH562 Basic definitions

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数学代写|随机偏微分方程代写Stochastic Differential Equation代考|Basic definitions

Throughout the book $(\Omega, \mathcal{F}, \mathbb{P})$ denotes a complete ${ }^1$ probability space.
Many arguments in the theory of stochastic processes depend on Dynkin’s $\pi-\lambda$ theorem; see Dynkin (1965) and Theorem $3.2$ below. This theorem is concerned with the smallest $\sigma$-field $\sigma(\mathcal{K})$ containing all elements of a family $\mathcal{K}$.

Definition 3.1 A collection $\mathcal{K}$ of subsets of $\Omega$ is said to be a $\pi$-system if it is closed under the formation of finite intersections. A collection $\mathcal{M}$ of subsets of $\Omega$ is a $\lambda$-system if it contains $\Omega$ and is closed under the formation of complements and of countable disjoint unions.

Theorem $3.2$ (Dynkin) If a $\lambda$-system $\mathcal{M}$ contains a $\pi$-system $\mathcal{K}$ then $\mathcal{M} \supset$ $\sigma(\mathcal{K})$

Proof Denote by $\mathcal{K}_0$ the smallest $\lambda$-system containing $\mathcal{K}$. This is equal to the intersection of all $\lambda$-systems containing $\mathcal{K}$. Then $\mathcal{K}_0 \subset \sigma(\mathcal{K})$. To prove the opposite inclusion we first show that $\mathcal{K}_0$ is a $\pi$-system. Let $A \in \mathcal{K}_0$ and define $\mathcal{K}_A=$ $\left{B: B \in \mathcal{K}_0\right.$ and $\left.A \cap B \in \mathcal{K}_0\right}$. It is easy to check that $\mathcal{K}_A$ is closed under the formation of complements and countable disjoint unions and that if $A \in \mathcal{K}$ then $\mathcal{K}_A \supset \mathcal{K}$. Thus, for $A \in \mathcal{K}, \mathcal{K}_A=\mathcal{K}_0$ and we have shown that if $A \in \mathcal{K}$ and $B \in \mathcal{K}_0$ then $A \cap B \in \mathcal{K}_0$. But this implies that $\mathcal{K}_B \supset \mathcal{K}$ and, consequently, that $\mathcal{K}_B=\mathcal{K}_0$ for any $B \in \mathcal{K}_0$. It is now an easy exercise to show that if a $\pi$-system is closed under the formation of complements and countable disjoint unions then it is a $\sigma$-field. This completes the proof.

数学代写|随机偏微分方程代写Stochastic Differential Equation代考|Kolmogorov existence theorem

Let $(E, \mathcal{E})$ be a measurable space. A stochastic process $X$ is usually described in terms of the measures it induces on products of $E$; namely, if $(X(t), t \in I)$ is an $E$-valued process then, for each sequence $\left(t_1, t_2, \ldots, t_k\right)$ of distinct elements of $I$, $\left(X\left(t_1\right), \ldots, X\left(t_k\right)\right)$ is a random vector with values in the product $E \times E \times \cdots \times E$ equipped with the product $\sigma$-field $\mathcal{E} \times \mathcal{E} \times \cdots \times \mathcal{E}$. The probability measures on $E^k$ given by $\lambda_{t_1, \ldots, t_k}:=\mathcal{L}\left(X\left(t_1\right), \ldots, X\left(t_k\right)\right)$ are called finite-dimensional distributions of $X$. Note that the finite-dimensional distributions of a stochastic process $(X(t), t \in I)$ satisfy two consistency conditions.
(i) For arbitrary $A_i \in \mathcal{E}, i=1,2, \ldots, k$, and for any permutation $\pi$ of ${1,2, \ldots, k}, \lambda_{t_1, \ldots, t_k}\left(A_1 \times \cdots \times A_k\right)=\lambda_{t_{\pi 1}, \ldots, t_{\pi k}}\left(A_{\pi 1} \times \cdots \times A_{\pi k}\right)$.
(ii) For arbitrary $A_i \in \mathcal{E}, i=1,2, \ldots, k-1, \lambda_{t_1, \ldots, t_{k-1}}\left(A_1 \times \cdots \times A_{k-1}\right)=$ $\lambda_{t_1, \ldots, t_k}\left(A_1 \times \cdots \times A_{k-1} \times E\right)$.

随机偏微分方程代写

数学代写|随机偏微分方程代写Stochastic Differential Equation代考|基本定义

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MATLAB代写

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