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# 数学代写|随机分析作业代写stochastic analysis代考|Continuous Local Martingale

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## 数学代写|随机分析作业代写stochastic analysis代考|non-decreasing stopping times

(1) We say that $\left{\tau_{n}\right}_{n=1}^{\infty}$ is a sequence of non-decreasing stopping times, if $\tau_{n}, n=1,2, \ldots$, are stopping times, $\tau_{k}(\omega) \leqq \tau_{k+1}(\omega), k=1,2, \ldots$, and $\tau_{n}(\omega) \rightarrow \infty, n \rightarrow \infty$, for all $\omega \in \Omega$.
(2) We say that a stochastic process $M:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is a continuous local martingale, if $M$ is an adapted continuous process, and if there is a sequence of non-decreasing stopping times $\left{\tau_{n}\right}_{n=1}^{\infty}$ such that $M^{\tau_{n}}-M_{0} \in \mathcal{M}{b}^{c}$ for all $n \geqq 1$. (3) We say that a stochastic process $A:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is an adapted nondecreasing continuous process, if $A$ is an adapted continuous process, and if $A(\cdot, \omega):[0, \infty) \rightarrow \mathbf{R}$ is an non-decreasing function for all $\omega \in \Omega$. (4) We say that a stochastic process $A:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is an adapted continuous process with finite variation, if there are adapted continuous non-decreasing processes $A{i}, i=0,1$, such that $A=A_{1}-A_{0}$.

In this book, we denote by $\mathcal{M}{\text {loc }}^{c}$ the set of continuous local martingales $M$ with $M{0}=0$, denote by $\mathcal{A}{+, c}$ the set of adapted non-decreasing processes $A$ with $A{0}=0$, and denote by $\mathcal{A}{c}$ the set of adapted continuous processes with finite variation with $A{0}=0$.

If $M:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is a continuous process and is a martingale with $M_{0}=0$, then $M$ is a continuous local martingale.

Proof Let $\tau_{n}=\inf \left{t \geqq 0 ;\left|M_{t}\right| \geqq n\right} \wedge n, n \geqq 1$. Then $\left{\tau_{n}\right}_{n=1}^{\infty}$ is a sequence of non-decreasing stopping times. By Proposition 3.3.12 we see that $M^{\tau_{n}} \in \mathcal{M}_{b}^{c}$, and so we have our assertion.

Proposition 3.6.2 (1) If $\left{\tau_{n}\right}_{n=1}^{\infty}$ and $\left{\sigma_{n}\right}_{n=1}^{\infty}$ are sequences of non-decreasing stopping times, then $\left{\tau_{n} \wedge \sigma_{n}\right}_{n=1}^{\infty}$ is also a sequence of non-decreasing stopping times.
(2) Let $\left{\tau_{n}\right}_{n=1}^{\infty}$ be a sequence of non-decreasing stopping times and let $X_{n}$ : $[0, \infty) \times \Omega \rightarrow \mathbf{R}, n=1,2, \ldots$, be adapted continuous process such that $X_{n}^{\tau_{n}}=$ $X_{n+1}^{\tau_{n}}, n=1,2, \ldots$, with probability 1. Then there exists an adapted continuous process $X:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ such that $X^{\tau_{n}}=X_{n}^{\tau_{n}}$ a.s., $n=1,2, \ldots$ Moreover, if $\tilde{X}:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is an adapted continuous process such that $\tilde{X}^{\tau_{n}}=X_{n}^{\tau_{n}}$ a.s., $n=1,2, \ldots$, then $X=\tilde{X}$.

## 数学代写|随机分析作业代写STOCHASTIC ANALYSIS代考|NON-DECREASING STOPPING TIMES

1我们说\left{\tau_{n}\right}_{n=1}^{\infty}\left{\tau_{n}\right}_{n=1}^{\infty}是非递减停止时间的序列，如果τn,n=1,2,…, 是停止时间,τ到(ω)≦τ到+1(ω),到=1,2,…， 和τn(ω)→∞,n→∞， 对全部ω∈Ω.
2我们说一个随机过程米:[0,∞)×Ω→R是一个连续局部鞅，如果米是一个适应的连续过程，如果存在一系列非递减的停止时间$M:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is a continuous local martingale, if $M$ is an adapted continuous process, and if there is a sequence of non-decreasing stopping times $\left{\tau_{n}\right}_{n=1}^{\infty}$ such that $M^{\tau_{n}}-M_{0} \in \mathcal{M}{b}^{c}$ for all $n \geqq 1$. (3) We say that a stochastic process $A:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is an adapted nondecreasing continuous process, if $A$ is an adapted continuous process, and if $A(\cdot, \omega):[0, \infty) \rightarrow \mathbf{R}$ is an non-decreasing function for all $\omega \in \Omega$. (4) We say that a stochastic process $A:[0, \infty) \times \Omega \rightarrow \mathbf{R}$ is an adapted continuous process with finite variation, if there are adapted continuous non-decreasing processes $A{i}, i=0,1$, such that $A=A_{1}-A_{0}$.

2让\left{\tau_{n}\right}_{n=1}^{\infty}\left{\tau_{n}\right}_{n=1}^{\infty}是一个非递减停止时间的序列并且让Xn:[0,∞)×Ω→R,n=1,2,…, 适应连续过程，使得Xnτn= Xn+1τn,n=1,2,…, 概率为 1. 那么存在一个适应的连续过程X:[0,∞)×Ω→R这样Xτn=Xnτn作为，n=1,2,…此外，如果X~:[0,∞)×Ω→R是一个适应的连续过程，使得X~τn=Xnτn作为，n=1,2,…， 然后X=X~.