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# 统计代写|鞅论代写Martingale Theory代考|ORIE6640 Stochastic Processes

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## 统计代写|鞅论代写Martingale Theory代考|Arbitrage Pricing Theory

We consider a continuous-time setting with time denoted $t \in[0, \infty)$. We are given a filtered probability space $(\Omega, \mathscr{F}, \mathbb{F}, \mathbb{P})$ where $\Omega$ is the state space with generic element $\omega \in \Omega, \mathscr{F}$ is a $\sigma$-algebra representing the set of events, $\mathbb{F}=\left(\mathscr{F}t\right){0 \leq t \leq \infty}$ is a filtration, and $\mathbb{P}$ is a probability measure defined on $\mathscr{F}$. A filtration is a collection of $\sigma$-algebras, which are increasing, i.e. $\mathscr{F}_s \subseteq \mathscr{F}_t$ for $0 \leq s \leq t \leq \infty$.

A random variable is a mapping $Y: \Omega \rightarrow \mathbb{R}$ such that $Y$ is $\mathscr{F}$-measurable, i.e. $Y^{-1}(A) \in \mathscr{F}$ for all $A \in \mathscr{B}(\mathbb{R})$ where $\mathscr{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$, i.e. the smallest $\sigma$-algebra containing all open intervals $(s, t)$ with $s \leq t$ for $s, t \in \mathbb{R}$ (see Ash [3, p. 8]).

A stochastic process is a collection of random variables indexed by time, i.e. a mapping $X:[0, \infty) \times \Omega \rightarrow \mathbb{R}$, denoted variously depending on the context, $X(t, \omega)=X(t)=X_t$. It is adapted if $X_t$ is $\mathscr{F}_t$-measurable for all $t \in[0, \infty)$.
A sample path of a stochastic process is the graph of $X(t, \omega)$ across time $t$ keeping $\omega$ fixed.

We assume that the filtered probability space satisfies the usual hypotheses. The usual hypotheses are that $\mathscr{F}0$ contains the $\mathbb{P}$ null sets of $\mathscr{F}$ and that the filtration $\mathbb{F}$ is right continuous. Right continuous means that $\mathscr{F}_t=\cap{u>t} \mathscr{F}_u$ for all $0 \leq$ $t<\infty$. Letting $\mathscr{F}_0$ contains the $\mathbb{P}$ null sets of $\mathscr{F}$ facilitates the measurability of various events, random variables, and stochastic processes. Right continuity implies the important result that given a random variable $\tau: \Omega \rightarrow[0, \infty],{\tau(\omega) \leq t} \in \mathscr{F}_t$ for all $t$ if and only if ${\tau(\omega)<t} \in \mathscr{F}_t$ for all $t$, see Protter [158, p. 3]. This fact will be important with respect to the mathematics of stopping times, which are introduced below. One can think of right continuity as implying that the information at time $t^{+}$is known at time $t$, see Medvegyev [143, p. 9].

## 统计代写|鞅论代写Martingale Theory代考|Martingales

A stochastic process $X$ is a martingale with respect to $\mathbb{F}$ if
(i) $X$ is cadlag and adapted,
(ii) $E\left[\left|X_t\right|\right]<\infty$ all $t$, and
(iii) $E\left[X_t \mid \mathscr{F}_s\right]=X_s$ a.s. for all $0 \leq s \leq t<\infty$.
It is a submartingale if (iii) is replaced by $E\left[X_t \mid \mathscr{F}_s\right] \geq X_s$ a.s. It is said to be a strict submartingale if it is a submartingale but not a martingale, i.e. the inequality is strict with positive probability for some $0 \leq s \leq t<\infty$.

It is a supermartingale if (iii) is replaced by $E\left[X_t \mid \mathscr{F}_s\right] \leq X_s$ a.s. It is said to be a strict supermartingale if it is a supermartingale but not a martingale, i.e. the inequality is strict with positive probability for some $0 \leq s \leq t<\infty$.

For the definition of an expectation and a conditional expectation, see Ash [3, Chapter 6]. Within the class of martingales, uniformly integrable martingales play an important role (see Protter [158, Theorem 13, p. 9]).

## 统计代写|鞅论代写Martingale Theory代考|Martingales

(一) $X$ 是 cadlag 和改编的，
(ii) $E\left[\left|X_t\right|\right]<\infty$ 全部 $t$, 和
(iii) $E\left[X_t \mid \mathscr{F}_s\right]=X_s$ 至于所有 $0 \leq s \leq t<\infty$.

## MATLAB代写

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