# 随机微积分作业代写stochastic calculus代考| CONTINUOUS TIME MARTINGALES

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## 微积分作业代写calclulus代考|Filtration, optional times, sampling.gue integrals

7.a.0 Assumption. The probability space $(\Omega, \mathcal{F}, P)$ is complete and the filtration $\left(\mathcal{F}{t}\right){t \geq 0}$ on $(\Omega, \mathcal{F}, P)$ right continuous and augmented.

Augmentation eliminates measurability problems on null sets. Let $X=\left(X_{t}\right), Y=$ $\left(Y_{t}\right), X(n)=\left(X_{t}(n)\right)$ be stochastic processes on $(\Omega, \mathcal{F}, P)$ indexed by $\mathcal{T}=[0, \infty)$ and $t \geq 0$. If $X_{t}$ is $\mathcal{F}{t}$-measurable and $Y{t}=X_{t}, P_{\text {-as. }}$, then $Y_{t}$ is $\mathcal{F}{t}$-measurable. Likewise, if $X{t}(n)$ is $\mathcal{F}{t}$-measurable, for all $n \geq 1$, and $X{t}(n) \rightarrow X_{t}, P$-as., as $n \uparrow \infty$, then $X_{t}$ is $\mathcal{F}{t}$-measurable. If $\omega \in \Omega$, the function $$t \in[0, \infty) \mapsto X{t}(\omega) \in \bar{R}$$
is called the path of $X$ in state $\omega$. The process $X$ is called (right, left) continuous if $X$ is $\left(\mathcal{F}{t}\right)$-adapted and $P$-ae. path of $X$ is finitely valued and (right, left) continuous. Let us call the processes $X, Y$ versions of each other and write $X=Y$ if they satisfy $$X{t}=Y_{t}, \quad P \text {-as., for all } t \geq 0 .$$
Since the filtration $\left(\mathcal{F}{t}\right)$ is augmented, each version of an $\left(\mathcal{F}{t}\right)$-adapted process is again $\left(\mathcal{F}_{t}\right)$-adapted.

The exceptional null set $\left[X_{t} \neq Y_{t}\right]$ in $(0)$ is allowed to depend on $t$. If this null set can be made independent of $t \geq 0$, that is, if there is a $P$-null set $N \subseteq \Omega$ such that $X_{t}(\omega)=Y_{t}(\omega)$, for all $\omega \in \Omega \backslash N$ and all $t \geq 0$, then we call the processes $X$ and $Y$ indistinguishable. Clearly $X$ and $Y$ are indistinguishable if and only if the paths $t \in[0, \infty) \mapsto X_{t}(\omega)$ and $t \in[0, \infty) \mapsto Y_{t}(\omega)$ are identical, for $P$-ae. $\omega \in \Omega$. These notions of equality agree for right continuous processes, to which mostly we shall confine our attention:

## 微积分作业代写calclulus代考|1 Lp-inequalities.

7.e.1 $L^{p}$-inequalities. Let $\left(X_{t}\right){0 \leq t \leq T}$ be a right continuous martingale and $S^{}=$ $\sup {0 \leq t \leq T}\left|X_{t}\right|$. Then
(a) $P\left(S^{ } \geq \lambda\right) \leq \lambda^{-p} E\left(\left|X_{T}\right|^{p}\right)$ for all $\lambda>0, p \geq 1$.
(b) $\left|S^{}\right|_{p} \leq \frac{p}{p-1}\left|X_{T}\right|_{p}$, for all $p>1$. Proof. Let $D \subseteq[0, T]$ be a countable dense subset with $T \in D$. Then $S^{}=$ $\sup {t \in D}\left|X{t}\right|, P$-as. Enumerate the set $D$ as $D=\left{t_{n}\right}_{n \geq 1}$. For $N \geq 1$, set $I_{N}=$ $\left{t_{1}, t_{2}, \ldots, t_{N}, T\right}, S_{N}^{}=\max {t \in I{N}}\left|X_{t}\right|$ and note that
$$S_{N}^{ } \uparrow S^{}, \quad P \text {-as., as } N \uparrow \infty .$$ With the understanding that the index $t$ ranges through the elements of $I_{N}$ in increasing order, the finite sequence $\left(X_{t}\right){t \in I{N}}$ is a martingale. From 6 .b.2 we get
$$P\left(S_{N}^{ } \geq \lambda\right) \leq \lambda^{-p} E\left(\left|X_{T}\right|^{p}\right), \quad \lambda>0, p \geq 1$$
Let $N \uparrow \infty$. From $(0)$ it follows that $\left[S_{N}^{}>\lambda\right] \uparrow\left[S^{}>\lambda\right]$ on the complement of a null set. Here the use of strict inequalities is essential. Thus (1) yields
$$P\left(S^{}>\lambda\right)=\lim {N \uparrow \infty} P\left(S{N}^{}>\lambda\right) \leq \lambda^{-p} E\left(\left|X_{T}\right|^{p}\right)$$
Choose $\lambda_{n}>0$ such that $\lambda_{n}<\lambda$ and $\lambda_{n} \uparrow \lambda$, as $n \uparrow \infty$. (2) applied to $\lambda_{n}$ instead of $\lambda$ yields $$P\left(S^{} \geq \lambda\right) \leq P\left(S^{}>\lambda_{n}\right) \leq\left(\lambda_{n}\right)^{-p} E\left(\left|X_{T}\right|^{p}\right)$$
Letting $n \uparrow \infty$ now yields (a). 6.b.3 applied to the finite martingale sequence $\left(X_{t}\right){t \in I{N}}$ yields $\left|S_{N}^{}\right|_{p} \leq(p /(p-1))\left|X_{T}\right|_{p}$, for all $N \geq 1$. Let $N \uparrow$. Then $0 \leq S_{N}^{} \uparrow S^{}, P$-as., and so $\left|S_{N}^{}\right|_{p} \uparrow\left|S^{*}\right|_{p}$. This establishes (b).

## 微积分作业代写calclulus代考|Filtration, optional times, sampling.gue integrals

7.a.0 假设。概率空间(Ω,F,磷)完成过滤 $\left(\mathcal{F} {t}\right) {t \geq 0}○n(\Omega, \mathcal{F}, P)$ 右连续和增广。

$$称为路径X处于状态ω. 过程X被称为（右，左）连续如果X是 \left(\mathcal{F} {t}\right)−一种d一种p吨和d一种nd磷−一种和.p一种吨H○FX一世sF一世n一世吨和一世和v一种一世你和d一种nd(r一世GH吨,一世和F吨)C○n吨一世n你○你s.一世和吨你sC一种一世一世吨H和pr○C和ss和sx-yv和rs一世○ns○F和一种CH○吨H和r一种nd在r一世吨和X=Y一世F吨H和和s一种吨一世sF和 X {t}=Y_{t}, \quad P \text {-as., 对于所有 } t \geq 0 。$$

## 微积分作业代写calclulus代考|1 Lp-inequalities.

7.e.1一世p-不平等。设 $\left(X_{t}\right) {0 \leq t \leq T}b和一种r一世GH吨C○n吨一世n你○你s米一种r吨一世nG一种一世和一种ndS^{}=\sup {0 \leq t \leq T}\left|X_{t}\right|.吨H和n(一种)P\left(S^{ } \geq \lambda\right) \leq \lambda^{-p} E\left(\left|X_{T}\right|^{p}\right)F○r一种一世一世\ λ > 0, p \ geq 1.(b)\left|S^{ }\right|_{p} \leq \frac{p}{p-1}\left|X_{T}\right|_{p},F○r一种一世一世p>1.磷r○○F.一世和吨D \subseteq[0, T]b和一种C○你n吨一种b一世和d和ns和s你bs和吨在一世吨HT \in D.吨H和nS^{ }=\sup {t \in D}\left|X {t}\right|, P−一种s.和n你米和r一种吨和吨H和s和吨D一种sD=\left{t_{n}\right}_{n \geq 1}.F○rN \ geq 1,s和吨我_{N}=\left{t_{1}, t_{2}, \ldots, t_{N}, T\right}, S_{N}^{ }=\max {t \in I{N}}\left|X_{ t}\对|一种ndn○吨和吨H一种吨$
S_{N}^{
} \uparrow S^{ }, \quad P \text {-as., as } N \uparrow \infty 。在一世吨H吨H和你nd和rs吨一种nd一世nG吨H一种吨吨H和一世nd和X$吨$r一种nG和s吨Hr○你GH吨H和和一世和米和n吨s○F$一世ñ$一世n一世nCr和一种s一世nG○rd和r,吨H和F一世n一世吨和s和q你和nC和$(X吨)吨∈一世ñ$一世s一种米一种r吨一世nG一种一世和.Fr○米6.b.2在和G和吨
P\left(S_{N}^{
} \geq \lambda\right) \leq \lambda^{-p} E\left(\left|X_{T}\right|^{p}\right), \四元 \lambda>0, p \geq 1
$$让ñ↑∞. 从(0)它遵循 \left[S_{N}^{ }>\lambda\right] \uparrow\left[S^{ }>\lambda\right]○n吨H和C○米p一世和米和n吨○F一种n你一世一世s和吨.H和r和吨H和你s和○Fs吨r一世C吨一世n和q你一种一世一世吨一世和s一世s和ss和n吨一世一种一世.吨H你s(1)和一世和一世ds P\left(S^{ }>\lambda\right)=\lim {N \uparrow \infty} P\left(S{N}^{ }>\lambda\right) \leq \lambda^{-p } E\left(\left|X_{T}\right|^{p}\right) CH○○s和λn>0s你CH吨H一种吨λn<λ一种ndλn↑λ,一种sn↑∞.(2)一种pp一世一世和d吨○λn一世ns吨和一种d○Fλ和一世和一世dsP\left(S^{ } \geq \lambda\right) \leq P\left(S^{ }>\lambda_{n}\right) \leq\left(\lambda_{n}\right)^{- p} E\left(\left|X_{T}\right|^{p}\right)$$