# 随机微积分作业代写stochastic calculus代考| MEASURABILITY PROPERTIES OF STOCHASTIC PROCESSES

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• 随机偏微分方程
• 随机控制
• Ito积分
• black-Scholes-Merton option pricing formula
• Fokker–Planck equation
• 布朗运动 Brownian motion

## 微积分作业代写calclulus代考|The progressive and predictable σ-fields on

1.a.0. Let $X$ be a progressively measurable process and $T$ an $\left(\mathcal{F}{t}\right)$-optional time. Then the process $X^{T}$ is progressively measurable and the random variable $X{T}$ is $\mathcal{F}_{T}$-measurable.

Remark. Here $X_{T}=X_{\infty}$ on the set $[T=\infty]$, where $X_{\infty}$ is any $\mathcal{F}_{\infty}$-measurable random variable.

Proof. Fix $t \geq 0$. The maps $(s, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right) \rightarrow u=T(\omega) \wedge s \in\left([0, t], \mathcal{B}{t}\right)$ and $(s, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right) \rightarrow \omega \in\left(\Omega, \mathcal{F}{t}\right)$ are measurable and hence so is
$$(s, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right) \rightarrow(u, \omega)=(T(\omega) \wedge s, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right)$$
Likewise the map $(u, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right) \rightarrow X(u, \omega) \in(\bar{R}, \overline{\mathcal{B}})$ is measurable by progressive measurability of $X$ and hence so is the composition of the last two maps, that is the map
$$(s, \omega) \in\left([0, t] \times \Omega, \mathcal{B}{t} \times \mathcal{F}{t}\right) \rightarrow X(T(\omega) \wedge s, \omega)=X_{s}^{T}(\omega) \in(\bar{R}, \overline{\mathcal{B}}) .$$
This shows that the process $X^{T}$ is progressively measurable and hence in particular adapted. To see that the random variable $X_{T}$ is $\mathcal{F}{T}$-measurable let $B \subseteq \bar{R}$ be a Borel set. We must show that $\left[X{T} \in B\right] \in \mathcal{F}{T}$, equivalently $\left[X{T} \in B\right] \cap[T \leq t] \in \mathcal{F}_{t}$, for all $0 \leq t \leq \infty$.

If $t<\infty$ then $\left[X_{T} \in B\right] \cap[T \leq t]=\left[X_{t \wedge T} \in B\right] \cap[T \leq t] \in \mathcal{F}{t}$, as $[T \leq t] \in \mathcal{F}{t}$ and the process $X^{T}$ is adapted. This implies that $\left[X_{T} \in B\right] \cap[T<\infty] \in \mathcal{F}{\infty}$ and since $\left[X{T} \in B\right] \cap[T=\infty] \in \mathcal{F}{\infty}$, by $\mathcal{F}{\infty}$-measurability of $X_{\infty}$, it follows that $\left[X_{T} \in B\right] \cap[T \leq t] \in \mathcal{F}_{t}$ for $t=\infty$ also.

A set $R$ of the form $R={0} \times F$, where $F \in \mathcal{F}{0}$, or $R=(s, t] \times F$, where $0 \leq s{s}$, is called a predictable rectangle. Note that the predictable rectangles form a $\pi$-system. The predictable $\sigma$-field is the $\sigma$-field $\mathcal{P}$ generated by the predictable rectangles on the set $\Pi$. The sets in $\mathcal{P}$ are called the predictable sets. The process $X: \Pi \rightarrow \bar{R}$ is called predictable, if it is measurable relative to the predictable $\sigma$-field $\mathcal{P}$. The process $X$ is called simple predictable, if it is a finite sum of processes of the form
$$Z_{0}(\omega) 1_{{0}}(t), Z(\omega) 1_{(a, b]}(t)$$
where $0 \leq a<b, Z_{0}$ is $\mathcal{F}{0}$-measurable and $Z$ is an $\mathcal{F}{a}$-measurable random variable. If for example $R$ is a predictable rectangle, then $X=1_{R}$ is a simple predictable process.

## 微积分作业代写calclulus代考|Stochastic intervals and the optional σ-field

1.b Stochastic intervals and the optional $\sigma$-field. For optional times $S, T: \Omega \rightarrow$ $[0, \infty]$ define the stochastic interval $[S, T]$ to be the set
$$\llbracket S, T \rrbracket={(t, \omega) \in \Pi \mid S(\omega) \leq t \leq T(\omega)} .$$
The stochastic intervals $] S, T \rrbracket, \llbracket S, T[$ and $] S, T[$ are defined similarly and are easily seen to be $\mathcal{B} \times \mathcal{F}$-measurable. A stochastic interval is a subset of $\Pi$ and hence does not contain a point of the form $(\infty, \omega)$, even if $T(\omega)=\infty$. It is not assumed that $S \leq T$. If $S(\omega)>T(\omega)$ then the $\omega$-section of any of the above stochastic intervals is empty. Note that
$$1_{[S, T]}(t, \omega)=1_{[S(\omega), T(\omega)]}(t),$$
and similar relations hold for all stochastic intervals. Real numbers $0 \leq s<t$ can be interpreted as constant optional times. Then the stochastic interval $\llbracket s, t \rrbracket$ is the set $[s, t] \times \Omega \subseteq \Pi$.

Every predictable rectangle $R$ is a stochastic interval. To see this, assume first that $R$ is of the form $R=(s, t] \times F$, with $F \in \mathcal{F}{s}, 0 \leq s{F e}+t 1_{F}$. For the optionality of $T$ see I.7.a.6. Note that the simpler representation $R=\rrbracket S, T \rrbracket$, where $S=s 1_{F}$ and $T=t 1_{F}$, does not work since in general neither of these is an optional time. In a similar way a predictable rectangle $R$ of the form $R={0} \times F$, with $F \in \mathcal{F}{0}$, can be written as $R=[S, T]$ with $T=0$ and $S=1{F^{c}} . S$ is optional since $F \in \mathcal{F}_{0}$.

The optional $\sigma$-field $\mathcal{O}$ on $\Pi$ is the $\sigma$-field generated by the family of all stochastic intervals. The sets in $\mathcal{O}$ are called the optional sets. From the above it follows that $\mathcal{P} \subseteq \mathcal{O} \subseteq \mathcal{B} \times \mathcal{F}$.

A process $X: \Pi \rightarrow \bar{R}$ is called optional if it is measurable relative to the optional $\sigma$-field $\mathcal{O}$ on $\Pi$. Thus every predictable process is optional.

A more thorough investigation of the measurability properties of processes [CW] yields the following result (which we do not need):
1.b.0. (a) Every optional process is progressively measurable.
(b) Every right continuous process is optional.
Certain stochastic intervals are predictable:
1.b.1. Let $S, T$ be optional times. Then the stochastic intervals $[0, T]$ and $\rrbracket S, T]$ are predictable sets.

Proof. The processes $X=1_{[0, T]}$ and $Y=1_{] S, T]}$ are left continuous and hence

## 微积分作业代写calclulus代考|The progressive and predictable σ-fields on

1.a.0。让X是一个逐步可衡量的过程，并且吨一个(F吨)- 可选时间。然后流程X吨是渐进可测量的，随机变量X吨是F吨- 可测量的。

(s,ω)∈([0,吨]×Ω,乙吨×F吨)→(你,ω)=(吨(ω)∧s,ω)∈([0,吨]×Ω,乙吨×F吨)

(s,ω)∈([0,吨]×Ω,乙吨×F吨)→X(吨(ω)∧s,ω)=Xs吨(ω)∈(R¯,乙¯).

## 微积分作业代写calclulus代考|Stochastic intervals and the optional σ-field

1.b 随机区间和可选σ-场地。对于可选时间小号,吨:Ω→ [0,∞]定义随机区间[小号,吨]成为集合
\ll括号小号,吨\rr括号=(吨,ω)∈圆周率∣小号(ω)≤吨≤吨(ω).

1[小号,吨](吨,ω)=1[小号(ω),吨(ω)](吨),

1.b.0。(a) 每个可选过程都是可逐步衡量的。
(b) 每个正确的连续过程都是可选的。

1.b.1。让小号,吨是可选的时间。然后是随机区间[0,吨]和\rr括号小号,吨]是可预测的集合。