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# 数学代写|矩阵方法代写Applied Matrix Theory代考|MATH40550 Continuous-Time Martingales

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## 数学代写|矩阵方法代写Applied Matrix Theory代考|Continuous-Time Martingales

Consider a stochastic process with index set $[0, \infty)$ taking values in $\mathbb{R}$ (or $\mathbb{R}^d$ ), i.e., a collection of random variables (or vectors) $\left{X_t\right}_{t \geq 0}$. The process is said to be adapted to a $\sigma$-algebra $\left{\mathscr{F}t\right}{t \geq 0}$ if $X_t$ is $\mathscr{F}t$-measurable for all $t \geq 0$. Let $(\Omega, \mathscr{F}, \mathbb{P})$ denote a basic probability space on which the process $\left{X_t\right}{t \geq 0}$ is defined. Then $\left(\Omega, \mathscr{F},\left{\mathscr{F}t\right}{t \geq 0}, \mathbb{P}\right)$ is called a filtered probability space, where $\mathscr{F}t \subseteq \mathscr{F}$ for all $t \geq 0$. We think of $\mathscr{F}$ as the collection of all possible events, while $\mathscr{F}_t$ contains only all possible events up to time $t$. The filtration $\left{\mathscr{F}_t\right}{t \geq 0}$ is called complete if $\mathscr{F}_0$, and hence all the $\sigma$-algebras including $\mathscr{F}$, contains all the sets $A$ for which $\mathbb{P}(A)=0$. Completion of filtrations is often important in proving that certain properties are valid almost surely.

Definition 2.2.30. Let $\left{\mathscr{F}t\right}{t \geq 0}$ be a filtration. Let $\left{X_t\right}_{t \geq 0}$ be an adapted stochastic process with values in $\mathbb{R}$ such that $\mathbb{E}\left|X_t\right|<\infty$ for all $t \geq 0$. Then we say that $\left{X_t\right}_{t \geq 0}$ is an $\left{\mathscr{F}s\right}{s \geq 0}$
(a) submartingale if
$$s \leq t \Rightarrow X_s \leq \mathbb{E}\left(X_t \mid \mathscr{F}_s\right) \text { a.s. }$$
(b) supermartingale if
$$s \leq t \Rightarrow X_s \geq \mathbb{E}\left(X_t \mid \mathscr{F}_s\right) a . s .$$
(c) martingale if it is both a sub- and supermartingale, i.e., if
$$s \leq t \Rightarrow X_s=\mathbb{E}\left(X_t \mid \mathscr{F}_s\right) \text { a.s. }$$
The condition $\mathbb{E}\left|X_t\right|<\infty$ ensures the existence of the conditional expectations to be used in (b), and it is referred to as the integrability condition.

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Regularization of submartingales

Assume that the filtration $\left{\mathscr{F}t\right}_t \geq 0$ is right continuous and complete, i.e., contains all sets of measure zero. Then the submartingale $\left{X_t\right}{t \geq 0}$ has a càdlàg modification, i.e., with probability one the process is right continuous and has limits from the left.
Proof. From the proof of Theorem 2.2.31 we recall that the total number of upcrossings in $[0, t]$ for fixed $t>0$ satisfies $U<\infty$ a.s. Now let $\Omega_0={\omega \in \Omega \mid U(\omega)<\infty} \subseteq$ $\Omega$. Then both right and left limits exist on this subset. Since $\Omega_0^C$ is a set of measure zero, by completeness it is contained in all $\sigma$-algebras $\mathscr{F}t$ and so is $\Omega_0$. Now define $$\tilde{X}_t(\omega)=\left{\begin{array}{cc} X{t+}(\omega) & \omega \in \Omega_0, \ 0 & \omega \notin \Omega_0 \end{array}\right.$$
Then $\tilde{X}t$ is measurable, since $\Omega_0$ is contained in the $\sigma$-algebras. By definition, $X{t+}$ is right continuous, and since $\Omega_0$ has probability one, $\tilde{X}t$ is a right-continuous modification of $X ;\left{X{t+}\right}$ is an $\left{\mathscr{F}{t+}\right}$-submartingale, and by right continuity of the filtration, $\left{X{t+}\right}$ is hence also an $\left{\mathscr{F}_t\right}$-submartingale. By completeness of the filtration, then $\left{\tilde{X}_t\right}$ is an $\mathscr{F}_t$-submartingale. That the process also possesses left limits at all points is clear from the construction using the upcrossing inequality.

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Continuous-Time Martingales

\left 的分隔符汻失或无法识别 . 据兑该过程适用于 $\sigma$-代数lleft 的分隔符秝失或无法识别

\left 的分隔符缺失或无法识别 称为过澞概

## 数学代写|矩阵方法代写Applied Matrix Theory代考|Regularization of submartingales

\left 的分隔符秝失或无法识别 有一个 càdlàg 修改，即概率为 1 的过程是右连续的并且从左开始有限制。 证明。从定理 2.2.31 的证明中，我们回杝起在 $[0, t]$ 对于固定 $t>0$ 满足 $U<\infty$ 现在让 $\Omega_0=\omega \in \Omega \mid U(\omega)<\infty \subseteq \Omega$. 那 $\angle$ 这 个子集上同时存在左右限制。自从 $\Omega_0^C$ 是一组零测度，通过完整生它包含在所有 $\sigma-$ 代数 $\mathscr{F} t$ 也是如此 $\Omega_0$. 现在定义 $\$ \$$\mid tilde {\mathrm{x}} t(\mid lomega )=\mid left {$$
$$【正确的。 \ \$$

\left 的分隔符缺失或无法识别 是一个〈left 的分隔符午失或无法识别

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