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# 数学代写|随机分析代写Stochastic Analysis in Finance代考|IMSE760 Martingale with Continuous Parameter

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## 数学代写|随机分析代写Stochastic Analysis in Finance代考|Several Notions on Stochastic Processes

Let $(\Omega, \mathcal{F}, P)$ be a complete probability space, that is, any subset $B$ of $\Omega$ for which there is an $A \in \mathcal{F}$ such that $B \subset A$ and $P(A)=0$ belongs to $\mathcal{F}$.

Let $\mathcal{N}0$ be the set of $A \in \mathcal{F}$ such that $P(A)=0$ or 1 . Then $\mathcal{N}_0$ is a sub- $\sigma$-algebra. In this section, we consider the case that $\mathbf{T}=[0, \infty)$. Definition 3.1.1 We say that a filtration $\left{\mathcal{F}_t\right}{t \in[0, \infty)}$ satisfies the standard condition, if the following two conditions are satisfied.
(1) $\mathcal{N}0 \subset \mathcal{F}_0$. (2) (Right continuity) $\bigcap{s>t} \mathcal{F}s=\mathcal{F}_t$ for any $t \geqq 0$. We say that $\left(\Omega, \mathcal{F}, P,\left{\mathcal{F}_t\right}{t \in[0, \infty)}\right)$ is a standard filtered probability space, if $(\Omega, \mathcal{F}, P)$ is a complete probability space and a filtration $\left{\mathcal{F}t\right}{t \in[0, \infty)}$ satisfies the standard condition.

We assume that $\left(\Omega, \mathcal{F}, P,\left{\mathcal{F}t\right}{t \in[0, \infty)}\right)$ is a standard filtered probability space from now on.

A stochastic process $X=\left{X_t\right}_{t \in[0, \infty)}$ is a family of random variables. We regard $X$ as a function defined in $[0, \infty) \times \Omega$. We introduce several notions in the following in order to analyze stochastic processes.

Definition $3.1 .2$ (1) We say that a stochastic process $X:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is $\left(\left{\mathcal{F}t\right}{t \in[0, \infty)^{-}}\right.$adapted, if $X(t, \cdot): \Omega \rightarrow \mathbf{R}$ is $\mathcal{F}t$-measurable for any $t \in[0, \infty)$. (2) We say that a stochastic process $X:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is progressively measurable, if $\left.X\right|{[0, T] \times \Omega}:[0, T] \times \Omega \rightarrow \mathbf{R}$ is $\mathcal{B}([0, T]) \otimes \mathcal{F}_T$-measurable for any $T \in$ $[0, \infty)$. Here $\mathcal{B}([0, T])$ is the Borel algebra on $[0, T]$, and $\mathcal{B}([0, T]) \otimes \mathcal{F}_T$ is the product $\sigma$-algebra of $\mathcal{B}([0, T])$ and $\mathcal{F}_T$.

As for a stochastic process $X:[0, \infty) \times \Omega \rightarrow \mathbf{R}$, we denote $X(t, \omega), t \in[0, \infty)$, $\omega \in \Omega$, by $X_t(\omega)$ or $X(t)(\omega)$. We denote by $X_t$ or $X(t)$ a random variable $X(t, \cdot)$ : $\Omega \rightarrow \mathbf{R}$ for $t \in[0, \infty)$.

## 数学代写|随机分析代写Stochastic Analysis in Finance代考|D-Modiﬁﬁcation

In this book, we mainly handle submartingales which are $D$-processes. We give the reason why we may consider that submartingales are $D$-processes.
Proposition 3.2.1 Let $\tilde{w}: \Delta \rightarrow \mathbf{R}$ be a real-valued function defined in $\Delta$. Assume that
$$\sup {n \geqq 1} \sup {K \geqq 1} U_K\left(\left{\tilde{w}\left(2^{-n} k\right)\right}_{k=0}^K ; a, b\right)<\infty$$
for any $a, b \in \mathbf{Q}$ with $a<b$, and that
$$\sup _{t \in \Delta}|\tilde{w}(t)|<\infty .$$
Then we have the following.
(1) For any $t \in[0, \infty)$

$$\lim {s \downarrow t, s \in \Delta} \tilde{w}(s) \in(-\infty, \infty)$$ exists and for any $t \in(0, \infty)$ $$\lim {s \uparrow t, s \in \Delta} \tilde{w}(s) \in(-\infty, \infty)$$
exists.
(2) Let us define $w:[0, \infty) \rightarrow \mathbf{R}$ by
$$w(t)=\lim {s \downarrow t, s \in \Delta} \tilde{w}(s), \quad t \in[0, \infty) .$$ Then $w \in D([0, \infty))$. Proof (1) Suppose that there is a $t \in[0, \infty)$ such that $$\varliminf{s \downarrow t, s \in \Delta} \tilde{w}(s) \neq \varlimsup_{s \downarrow t, s \in \Delta} \tilde{w}(s) .$$
Then there are $a, b \in \mathbf{Q}, as_2>s_3>s_4>\cdots$ and $\tilde{w}\left(s_{2 m-1}\right)>b, \tilde{w}\left(s_{2 m}\right)<a, m=1,2, \ldots$

Let $N=\sup {n \geqq 1} \sup {K \geqq 1} U_K\left(\left{\tilde{w}\left(2^{-n} k\right)\right}_{k=0}^K ; a, b\right)<\infty$. Then there are $n \geqq 1$ and $K \geqq 1$ such that $s_1, s_2, \ldots, s_{2 N+4} \in \Delta_n$ and $s_1<2^{-n} K$. This implies the contradiction. So we obtain the first assertion in (1). The proof of the second assertion of (1) is similar.
Assertion (2) is an easy consequence of Assertion (1).

## 数学代写|随机分析代写Stochastic Analysis in Finance代考|D-Modifification

$\$ \
< \infty forany $\$ a, b \in \mathbf{Q} \$w i t h \$ as_3>s_4>\cdots$和$\tilde{w}\left(s_{2 m-1}\right)>b, \tilde{w}\left(s_{2 m}\right)<a, m=1,2, \ldots$让lleft 的分隔符缺失或无法识别 然后有$n \geqq 1$和$K \geqq 1$这样$s_1, s_2, \ldots, s_{2 N+4} \in \Delta_n$和$s_1<2^{-n} K\$. 这暗

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