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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Markovian diffusions

Markovian diffusions. Consider an Itô process $X:=\left(X_t\right)_{t \geq 0}$ satisfying
$$X_t=X_0+\int_0^t a_s \mathrm{~d} s+\int_0^t b_s \mathrm{~d} W_s, \quad t \geq 0$$

which we usually write in differential form
$$\mathrm{d} X_t=a_t \mathrm{~d} t+b_t \mathrm{~d} W_t,$$
for adapted $a, b$ almost surely satisfying $\int_0\left|a_s\right| \mathrm{d} s<\infty, \int_0^* b_s^2 \mathrm{~d} s<\infty$ (or $\mathbb{E}\left[\int_0^* b_s^2 \mathrm{~d} s\right]<\infty$ if we want the local martingale in (6.6) (or $(6.7)$ ) to be a martingale).
Suppose now that
$$a_t \equiv a\left(X_t\right), \quad b_t \equiv b\left(X_t\right), \quad t \geq 0$$
(or we could also have $a_t \equiv a\left(t, X_t\right), b_t \equiv b\left(t, X_t\right)$ ), for well-behaved functions $a(\cdot), b(\cdot)$ such that the process in (6.6) is well-defined. We call the resulting equation
$$\mathrm{d} X_t=a\left(X_t\right) \mathrm{d} t+b\left(X_t\right) \mathrm{d} W_t,$$
a stochastic differential equation (SDE) for $X$, with initial condition $X_0$. Then $X$ is a Markov process, because its drift and diffuson coefficients depend only on the current value of $X$, and not on its history prior to the current time. (Note that proving this intuitive result rigorously needs quite a bit of preparatory work on the theory of Markov processes, and we will not delve into this theory here.) The process $X$ in (6.8) is also called a (time-homogeneous) diffusion process in the case where the coefficient functions do not have explicit time dependence. (The term diffusion is often also used for the case when the coefficient functions do have explicit time dependence – in this case, the two-dimensional process $\left(t, X_t\right)_{t \geq 0}$ is the (Markov) process, but the “diffusion” $X$ is not time-homogeneous.)
We (of course) interpret the SDE (6.8) as the integral equation
$$X_t=X_0+\int_0^t a\left(X_s\right) \mathrm{d} s+\int_0^t b\left(X_s\right) \mathrm{d} W_s, \quad t \geq 0 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Feynman-Kac theorem

Feynman-Kac theorem. Because of the Markov property, we have, for any $h(\cdot)$ such that $h\left(X_T\right)$ is integrable, for some $T<\infty$, and with $t \in[0, T]$,
$$\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F}t\right]=\mathbb{E}\left[h\left(X_T\right) \mid X_t\right]=: v\left(t, X_t\right), \quad t \in[0, T] .$$ For $X_t=x \in \mathbb{R}(6.9)$, defines a function $v:[0, T] \times \mathbb{R} \mapsto \mathbb{R}$. Lemma 6.4. With $v(\cdot, \cdot)$ defined by $(6.9)$, assume that $Y_t:=v\left(t, X_t\right), t \in[0, T]$ is integrable for all $t \in[0, T]$ (so $\mathbb{E}\left[\left|Y_t\right|\right]<\infty, \forall t \in[0, T]$ ). Then, $Y=\left(Y_t\right){t \in[0, T]}=\left(v\left(t, X_t\right)\right)_{t \in[0, T]}$ is a martingale.

Proof. For $0 \leq s \leq t \leq T$, on exploiting the Markov property and the tower property, we have
\begin{aligned} \mathbb{E}\left[Y_t \mid \mathcal{F}s\right]=\mathbb{E}\left[v\left(t, X_t\right) \mid \mathcal{F}_s\right] & =\mathbb{E}\left[\mathbb{E}\left[h\left(X_T\right) \mid X_t\right] \mid \mathcal{F}_s\right] \ & =\mathbb{E}\left[\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F}_t\right] \mid \mathcal{F}_s\right] \ & =\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F}_s\right] \ & =\mathbb{E}\left[h\left(X_T\right) \mid X_s\right] \ & =v\left(s, X_s\right)=Y_s \end{aligned} This leads to a first statement of the Feynman-Kac theorem. Theorem 6.5 (Feynman-Kac I). With $$\mathrm{d} X_t=a\left(X_t\right) \mathrm{d} t+b\left(X_t\right) \mathrm{d} W_t$$ a continuous diffusion, define the function $v:[0, T] \times \mathbb{R} \mapsto \mathbb{R}$ by $$v(t, x):=\mathbb{E}\left[h\left(X_T\right) \mid X_t=x\right], t \in[0, T],$$ for some Borel function $h(\cdot)$. Assume that $v\left(t, X_t\right)$ is integrable for all $t \in[0, T]$, and that $v \in C^{1,2}([0, T] \times \mathbb{R})$. Then, $v(\cdot, \cdot)$ solves the $P D E$ $$\frac{\partial v}{\partial t}(t, x)+\mathcal{A} v(t, x), \quad v(T, x)=h(x),$$ where $$\mathcal{A} v(t, x):=a(x) v_x(t, x)+\frac{1}{2} b^2(x) v{x x}(t, x)$$
defines the generator of $X$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Markovian diffusions

$$X_t=X_0+\int_0^t a\left(X_s\right) \mathrm{d} s+\int_0^t b\left(X_s\right) \mathrm{d} W_s, \quad t \geq 0 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Feynman-Kac theorem

$$\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F} t\right]=\mathbb{E}\left[h\left(X_T\right) \mid X_t\right]=: v\left(t, X_t\right), \quad t \in[0, T] .$$

$$Y=\left(Y_t\right) t \in[0, T]=\left(v\left(t, X_t\right)\right)_{t \in[0, T]} \text { 是一个轫。 }$$

$$\mathbb{E}\left[Y_t \mid \mathcal{F}_s\right]=\mathbb{E}\left[v\left(t, X_t\right) \mid \mathcal{F}_s\right]=\mathbb{E}\left[\mathbb{E}\left[h\left(X_T\right) \mid X_t\right] \mid \mathcal{F}_s\right] \quad=\mathbb{E}\left[\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F}_t\right] \mid \mathcal{F}_s\right]=\mathbb{E}\left[h\left(X_T\right) \mid \mathcal{F}_s\right] \quad=\mathbb{E}\left[h\left(X_T\right) \mid X_s\right]=v\left(s, X_s\right)=Y_s$$

$$\mathrm{d} X_t=a\left(X_t\right) \mathrm{d} t+b\left(X_t\right) \mathrm{d} W_t$$

$$v(t, x):=\mathbb{E}\left[h\left(X_T\right) \mid X_t=x\right], t \in[0, T],$$

$$\frac{\partial v}{\partial t}(t, x)+\mathcal{A} v(t, x), \quad v(T, x)=h(x),$$

$$\mathcal{A} v(t, x):=a(x) v_x(t, x)+\frac{1}{2} b^2(x) v x x(t, x)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Growth Estimate

The Metivier-Pellaumail inequality enables us to obtain a growth estimate on $\int f d X$ for any semimartingale $X$. Given a locally bounded predictable process $f$ and a decomposition $X=M+A$ of a semimartingale $X$, where $M$ is a locally square integrable martingale with $M_0=0$ and $A$ is a process with finite variation paths, let $Y=\int f d X, N=\int f d M$ and $B=\int f d A$. Then $Y=N+B, N$ is a locally square integrable martingale and $B \in \mathbb{V}$. Further,
\begin{aligned} {[N, N]t } & =\int_0^t f_s^2 d[M, M]_s, \ \langle N, N\rangle_t & =\int_0^t f_s^2 d\langle M, M\rangle_s \end{aligned} and thus in view of (11.2.1), we have $$\mathrm{E}\left[\sup {t<\tau}\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^{\tau-} f_s^2 d[M, M]s+\int_0^{\tau-} f_s^2 d\langle M, M\rangle_s\right] .$$ Writing $|A|_t=\operatorname{VAR}{[0, t]}(A)$, we have for all $t$,
$$\left|\int_0^t\right| f_s\left|d A_s\right|^2 \leq|A|t \int_0^t\left|f_s^2\right| d|A|_s$$ and hence $$\mathrm{E}\left[\sup {t<\tau}\left|\int_0^t f d A\right|^2\right] \leq \mathrm{E}\left[|A|_{\tau-} \int_0^{\tau-}\left|f_s^2\right| d|A|_s\right]$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Alternate Metric for Emery Topology

We will now introduce another metric on the space of semimartingales in terms of dominating process and then show that this metric is equivalent to the metric introduced earlier for the Emery topology.
Definition 11.16 For semimartingales $X, Y$, let
$\mathbf{d}{s m}(X, Y)=\inf \left{\mathbf{d}{u c p}(V, 0): V\right.$ is a dominating process for $\left.X-Y\right}$.
It is easy to see that if $V$ is a dominating process for $X-Y$ then $V$ is also a dominating process for $Y-X$ and thus $\mathbf{d}{s m}(X, Y)=\mathbf{d}{s m}(Y, X)$. The next two results will show that $\mathbf{d}_{s m}$ is a metric.

Lemma 11.17 Let $X, Y$ be semimartingales such that $\mathbf{d}{s m}(X, Y)=0$. Then $X=Y$. Proof Get $V^k \in \mathbb{V}^{+}$such that $V^k$ dominates $X-Y$ and $\mathbf{d}{u c p}\left(V^k, 0\right) \leq 2^{-k}$.
Then
$$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} 2^{-n} \mathrm{E}\left[V_n^k \wedge 1\right] \leq 1$$
and as a consequence, for every $n$, (using Fubini’s Theorem) we have
$$\mathrm{E}\left[\sum_{k=1}^{\infty} 2^{-n}\left[V_n^k \wedge 1\right]\right] \leq 1$$
Thus (noting $V_t^k \geq 0$ )
$$\sum_{k=1}^{\infty} 2^{-n}\left[V_n^k \wedge 1\right]<\infty \quad \text { a.s. }$$
and hence for every $t<\infty$
$$U_t=\left[\sum_{k=1}^{\infty} V_t^k\right]<\infty \quad \text { a.s. }$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Growth Estimate

Metivier-Pellaumail 不等式使我们能哆获得增长估计 $\int f d X$ 对于任何半鞅 $X$. 给定一个局部有界的可预测过程 $f$ 和分解 $X=M+A$ 半鞅的 $X$ ，在哪里 $M$ 是一个局部平方可积鞅 $M_0=0$ 和 $A$ 是一个具有有限变化路径的过程，让 $Y=\int f d X, N=\int f d M$ 和 $B=\int f d A$. 然后 $Y=N+B, N$ 是局部平方可积鞅， 并且 $B \in \mathbb{V}$. 更远，
$$[N, N] t=\int_0^t f_s^2 d[M, M]s,\langle N, N\rangle_t \quad=\int_0^t f_s^2 d\langle M, M\rangle_s$$ 因此鉴于 (11.2.1)，我们有 $$\mathrm{E}\left[\sup t<\tau\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^{\tau-} f_s^2 d[M, M] s+\int_0^{\tau-} f_s^2 d\langle M, M\rangle_s\right] .$$ 写作 $|A|_t=\operatorname{VAR}0, t$ ，我们有所有 $t$ ， $$\left|\int_0^t\right| f_s\left|d A_s\right|^2 \leq|A| t \int_0^t\left|f_s^2\right| d|A|_s$$ 因此 $$\mathrm{E}\left[\sup t<\tau\left|\int_0^t f d A\right|^2\right] \leq \mathrm{E}\left[|A|{\tau-} \int_0^{\tau-}\left|f_s^2\right| d|A|s\right]$$ Emery Topology

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Alternate Metric for Emery Topology

$$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} 2^{-n} \mathrm{E}\left[V_n^k \wedge 1\right] \leq 1$$

$$\mathrm{E}\left[\sum_{k=1}^{\infty} 2^{-n}\left[V_n^k \wedge 1\right]\right] \leq 1$$

$$\sum_{k=1}^{\infty} 2^{-n}\left[V_n^k \wedge 1\right]<\infty \quad \text { a.s. }$$

$$U_t=\left[\sum_{k=1}^{\infty} V_t^k\right]<\infty \quad \text { a.s. }$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Preliminaries

Throughout this chapter, we will be working with one fixed filtration $(\mathcal{F}$.) such that $\mathcal{F}_0$ contains all null sets. We do not assume that the filtration is right continuous. All notions-martingale, local martingale, stopping time, adapted process, predictable process-are with reference to this fixed filtration. Since in this chapter, we need to deal with martingales which may not have r.c.l.l. paths a priori, we will explicitly assume r.c.l.l. paths when it is needed.

For r.c.l.l.semimartingales $X^1, X^2, \ldots, X^d$, we introduce the class of semimartingales that admit integral representation w.r.t. $X^1, X^2, \ldots, X^d$ :
\begin{aligned} & \mathbb{I}\left(X^1, X^2, \ldots, X^d\right) \ & =\left{Y: \exists g^j \in \mathbb{L}\left(X^j\right), 1 \leq j \leq d \text { with } Y_t=Y_0+\sum_{j=1}^d \int_0^t g^j d X^j \forall t\right} \end{aligned}
Let us note that if $Y \in \mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$ then for any stopping time $\tau, \tilde{Y}$ defined by $\tilde{Y}t=Y{t \wedge \tau}$ also belongs to $\mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$. Also, if $Y \in \mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$ then we can always choose $g^j \in \mathbb{L}\left(X^j\right)$ with $g_0^j=0$ for $1 \leq j \leq d$ such that
$$Y_t=Y_0+\sum_{j=1}^d \int_0^t g^j d X^j \forall t<\infty .$$
If $Y \in \mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$, the semimartingale $Y$ is said to have an integral representation w.r.t. semimartingales $X^1, X^2, \ldots, X^d$. Here is an elementary observation on the class $\mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$.

Lemma 10.1 Let $Y$ be a semimartingale such that for a sequence of stopping times $\tau_n \uparrow \infty, Y^n$ defined by $Y_t^n=Y_{t \wedge \tau_n}$ admits an integral representation w.r.t. r.c.l.l. semimartingales $X^1, X^2, \ldots, X^d$ for each $n \geq 1$. Then $Y$ also admits an integral representation w.r.t. $X^1, X^2, \ldots, X^d$.
Proof Let $f^{n, j} \in \mathbb{L}\left(X^j\right), 1 \leq j \leq d, n \geq 1$ be such that for all $n$,
$$Y_t^n=Y_0^n+\sum_{j=1}^d \int_0^t f^{n, j} d X^j$$
Define $f^j$ by
$$f^j=\sum_{n=1}^{\infty} 1_{\left(\tau_{n-1}, \tau_n\right]} f^{n, j}$$
Then it is easy to check (using Theorem 4.43) that $f^j \in \mathbb{L}\left(X^j\right)$ and
$$Y_t=Y_0+\sum_{i=1}^d \int_0^t f^j d X^j$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|One-Dimensional Case

In this section, we will fix a local martingale $M$ and explore as to when $\mathbb{I}(M)$ contains all martingales. The next lemma gives an important property of $\mathbb{I}(M)$.

Lemma 10.3 Let $M$ be an r.c.l.l. local martingale and let $N^n \in \mathbb{I}(M)$ be martingales such that $\mathrm{E}\left[\left|N_t^n-N_t\right|\right] \rightarrow 0$ for all $t$. Then $N \in \mathbb{I}(M)$.

Proof The assumptions imply that $N$ is a martingale (see Theorem 2.23). In view of Theorem $5.39$ and the assumptions on $N^n, N$, it follows that
$N^n$ converges to $N$ in Emery topology.
Thus, invoking Theorem $4.111$, we conclude that
$$\left[N^n-N, N^n-N\right]_T \rightarrow 0 \text { in probability as } n \rightarrow \infty$$
Hence, using $\left[N^n-N^m, N^n-N^m\right]_T \leq 2\left(\left[N^n-N, N^n-N\right]_T+\left[N^m-N, N^m-\right.\right.$ $N]_T$ ) (see $\left.(4.6 .13)\right)$, we have
$$\left[N^n-N^m, N^n-N^m\right]_T \rightarrow 0 \text { in probability as } n, m \rightarrow \infty$$
Since $N^n \in \mathbb{I}(M)$, there exists predictable process $g^n \in \mathbb{L}(M)$ such that

$$N_t^n=N_0^n+\int_0^t g^n d M .$$
As a consequence, for all $T<\infty$,
\begin{aligned} {\left[N^n-N^m, N^n-N^m\right]T } & =\int_0^T\left(g_s^n-g_s^m\right)^2 d[M, M]_s . \ & \rightarrow 0 \text { in probability as } n, m \rightarrow \infty . \end{aligned} By taking a subsequence, if necessary and relabelling, we assume that for $1 \leq k \leq n$, $$\mathrm{P}\left(\left(\int_0^k\left(g_s^n-g_s^k\right)^2 d[M, M]_s\right)^{\frac{1}{2}} \geq \frac{1}{2^k}\right) \leq \frac{1}{2^k} .$$ Then by Borel-Cantelli Lemma, we conclude $$\sum{k=1}^{\infty}\left(\int_0^T\left(g_s^{k+1}-g_s^k\right)^2 d[M, M]_s\right)^{\frac{1}{2}}<\infty \text { a.s. }$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Preliminaries

$\mathbb{I}\left(X^1, X^2, \ldots, X^d\right)$.

$$Y_t^n=Y_0^n+\sum_{j=1}^d \int_0^t f^{n, j} d X^j$$

$$f^j=\sum_{n=1}^{\infty} 1_{\left(\tau_{n-1}, \tau_n\right]} f^{n, j}$$

$$Y_t=Y_0+\sum_{i=1}^d \int_0^t f^j d X^j$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|One-Dimensional Case

$$\left[N^n-N, N^n-N\right]_T \rightarrow 0 \text { in probability as } n \rightarrow \infty$$

$$\left[N^n-N^m, N^n-N^m\right]_T \rightarrow 0 \text { in probability as } n, m \rightarrow \infty$$

$$N_t^n=N_0^n+\int_0^t g^n d M .$$

$$\left[N^n-N^m, N^n-N^m\right] T=\int_0^T\left(g_s^n-g_s^m\right)^2 d[M, M]_s . \quad \rightarrow 0 \text { in probability as } n, m \rightarrow \infty .$$

$$\mathrm{P}\left(\left(\int_0^k\left(g_s^n-g_s^k\right)^2 d[M, M]_s\right)^{\frac{1}{2}} \geq \frac{1}{2^k}\right) \leq \frac{1}{2^k} .$$

$$\sum k=1^{\infty}\left(\int_0^T\left(g_s^{k+1}-g_s^k\right)^2 d[M, M]_s\right)^{\frac{1}{2}}<\infty \text { a.s. }$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Itˆo Integral of a general integrand

3.4. Itô Integral of a general integrand. Let $b$ be a process (not necessarily an elementary process) such that, for any fixed $t \geq 0$ :

$b_s$ is $\mathcal{F}_s$-measurable, $\forall s \in[0, t]$.

$\mathbb{E}\left[\int_0^t b_s^2 \mathrm{~d} s\right]<\infty$.
Then, any such process can be approximated by simple processes, with the approximation becoming suitably precise as we refine the partition used to define the simple processes.
Theorem 3.10. There is a sequence of elementary processes $\left(b^{(n)}\right){n=1}^{\infty}$ such that $$\lim {n \rightarrow \infty} \mathbb{E}\left[\int_0^t\left|b_s^{(n)}-b_s\right|^2 \mathrm{~d} s\right]=0, \quad t \geq 0 .$$
We have shown how to define
$$I_t^{(n)}:=\int_0^t b_s^{(n)} \mathrm{d} W_s, \quad t \geq 0,$$
for every $n \in \mathbb{N}$. We now define the general Itô integral by
$$I_t=\int_0^t b_s \mathrm{~d} W_s:=\lim _{n \rightarrow \infty} \int_0^t b_s^{(n)} \mathrm{d} W_s, \quad t \geq 0 .$$

For each $t \geq 0$, this limit exists, due to the Itô isometry (Theorem 3.5) and the approximation result of Theorem 3.10. This means that the sequence $\left(I_t^{(n)}\right){n \in \mathbb{N}}$ defined in (3.7) is a Cauchy sequence in $L^2(W)$ and so has a limit. Here is the argument. Suppose $m$ and $n$ are large positive integers. Then \begin{aligned} \mathbb{E}\left[\left|I_t^{(n)}-I_t^{(m)}\right|^2\right] & =\operatorname{var}\left(I_t^{(n)}-I_t^{(m)}\right) \ & =\mathbb{E}\left[\left(\int_0^t\left(b_s^{(n)}-b_s^{(m)}\right) \mathrm{d} W_s\right)^2\right] \ \text { (Itô isometry) } & =\mathbb{E}\left[\int_0^t\left(b_s^{(n)}-b_s^{(m)}\right)^2 \mathrm{~d} s\right] \ & \leq \mathbb{E}\left[\int_0^t\left(\left|b_s^{(n)}-b_s\right|+\left|b_s-b_s^{(m)}\right|\right)^2 \mathrm{~d} s\right] \ \left((a+b)^2 \leq 2\left(a^2+b^2\right)\right) & \leq 2 \mathbb{E}\left[\int_0^t\left|b_s^{(n)}-b_s\right|^2 \mathrm{~d} s\right]+2 \mathbb{E}\left[\int_0^t\left|b_s-b_s^{(m)}\right|^2 \mathrm{~d} s\right], \end{aligned} which approaches zero as $m, n \rightarrow \infty$, by Theorem 3.10. This guarantees that the sequence $\left(I_t^{(n)}\right){n=1}^{\infty}$ is a Cauchy sequence in $L^2(\Omega, \mathcal{F}, \mathbb{P}$ ) and so has a limit (which will inherit all the properties of each $\left.I_t^{(n)}\right)$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Properties of the general Itˆo integral

3.5. Properties of the general Itô integral. The Itô integral
$$I_t:=\int_0^t b_s \mathrm{~d} W_s, \quad t \geq 0,$$
as we have defined it, for general integrands satisfying the integrability condition
$$\mathbb{E}\left[\int_0^t b_s^2 \mathrm{~d}[W]_s\right]<\infty, \quad t \geq 0 .$$
inherits all its properties from the properties of Itô integrals of elementary integrands, so we summarise the properties in the following theorem.
Note that we sometimes use the notation
$$I:=\int_0 b_s \mathrm{~d} W_s,$$

to denote the process in (3.8) (so we dispense with writing the time index $t$, with a similar convention for all integrals that might appear).

Theorem 3.11 (Properties of the Itô integral, martingale version). For $t \geq 0$, let $b$ be an adapted process that satisfies (3.9). Then the Itô integral $I=\left(I_t\right)_{t \geq 0}$ of (3.8) is a continuous adapted process satisfying, in addition:

Linearity: If $I=\int_0^v b_s \mathrm{~d} W_s, J_t=\int_0^v a_s \mathrm{~d} W_s$, then $I \pm J_t=\int_0^s\left(b_s \pm a_s\right) \mathrm{d} W_s$, and for $c \in \mathbb{R}, c I=\int_0^{\prime} c b_s \mathrm{~d} W_s$
Martingale property: $I=\left(I_t\right){t \geq 0}$ is a martingale. Itô isometry: The variance of the Itô integral is $\operatorname{var}(I)=\mathbb{E}\left[I^2\right]$ given by $\mathbb{E}\left[I^2\right]=$ $\mathbb{E}\left[\int_0^c b_s^2 \mathrm{~d} s\right]$ Quadratic variation: The Itô integral has quadratic variation process $[I]=\left([I]_t\right){t \geq 0}$ given by
$$[I]_t \equiv\left[\int_0 b_s \mathrm{~d} W_s\right]_t=\int_0^t b_s^2 \mathrm{~d}[W]_s=\int_0^t b_s^2 \mathrm{~d} s, \quad t \geq 0 .$$
Doob-Meyer decomposition: The process $I^2-[I]$ is a martingale.

## 数学代写随机微积分代写STOCHASTIC CALCULUS代考|l’°° Integral of a general integrand

Itô 一般被积函数的积分。让 $b$ 是一个过程 (不一定是基本过程) 使得对于任何固定的 $t \geq 0$ :
$b_s$ 是 $\mathcal{F}s-$ 可衡量的， $\forall s \in[0, t]$. $\mathbb{E}\left[\int_0^t b_s^2 \mathrm{~d} s\right]<\infty$ 然后，任何这样的过程都可以用简单的过程来近似，随着我们细化用于定义简单过程的分区，近似变得适当皘确。 是理 3.10。存在一系列基本过程 $\left(b^{(n)}\right) n=1^{\infty}$ 这样 $$\lim n \rightarrow \infty \mathbb{E}\left[\int_0^t\left|b_s^{(n)}-b_s\right|^2 \mathrm{~d} s\right]=0, \quad t \geq 0$$ 乎们已弪展示了如何定义 $$I_t^{(n)}:=\int_0^t b_s^{(n)} \mathrm{d} W_s, \quad t \geq 0,$$ 每 个 $n \in \mathbb{N}$. 我们现在定义一般 Itô 积分 $$I_t=\int_0^t b_s \mathrm{~d} W_s:=\lim {n \rightarrow \infty} \int_0^t b_s^{(n)} \mathrm{d} W_s, \quad t \geq 0 .$$

$$\mathbb{E}\left[\left|I_t^{(n)}-I_t^{(m)}\right|^2\right]=\operatorname{var}\left(I_t^{(n)}-I_t^{(m)}\right) \quad=\mathbb{E}\left[\left(\int_0^t\left(b_s^{(n)}-b_s^{(m)}\right) \mathrm{d} W_s\right)^2\right](\text { It 个 isometry })=\mathbb{E}\left[\int_0^t\left(b_s^{(n)}-b_s^{(m)}\right)^2 \mathrm{~d} s\right] \quad \leq \mathbb{E}\left[\int _ { 0 } ^ { t } \left(\left|b_s^{(n)}-b_s\right|\right.\right. \text {. }$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Properties of the general It’o integral

$$I_t:=\int_0^t b_s \mathrm{~d} W_s, \quad t \geq 0,$$

$$\mathbb{E}\left[\int_0^t b_s^2 \mathrm{~d}[W]s\right]<\infty, \quad t \geq 0 .$$ 继承了其本被积函数的 Itô 积分的所有性质，因此我们将这些性质总结为以下定理。 请注意，我们有时使用符昊 $$I:=\int_0 b_s \mathrm{~d} W_s,$$ 来表示 (3.8) 中的过程（所以我们省去了写时间䒺引 $\mid t$ ，对可能出现的所有积分都有类似的约定）。 定理 $3.11$ (Itô 积分的性质，鞅版本) 。为了 $t \geq 0$ ，让 $b$ 是满足 $(3.9)$ 的适应过程。那么伊藤和分 $I=\left(I_t\right){t \geq 0}(3.8)$ 的连续自适 应过程满足，另外:

$$[I]_t \equiv\left[\int_0 b_s \mathrm{~d} W_s\right]_t=\int_0^t b_s^2 \mathrm{~d}[W]_s=\int_0^t b_s^2 \mathrm{~d} s, \quad t \geq 0 .$$
Doob-Meyer 分解: 过程 $I^2-[I]$ 是一个鞅。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Integral with respect to QV

Integral with respect to QV. Let $a$ be an adpated process satisfying $\int_0^t\left|a_s\right|^2 \mathrm{~d} s<$ $\infty, t \geq 0$ almost surely. Let $\mathcal{P}=\left{0=t_0<\cdots<t_n=t\right}$ be our familiar partition of $[0, t]$, for $t \geq 0$.

The fact that $[W]t=t, t \geq 0$, written heuristically in the form (2.27), suggests that the integral of $a$ with respect to time (which, following (2.27), we also write suggestively below as an integral with respect to QV – this is of course well-defined, as QV induces a finite measure, Lebesgue measure, over any finite time interval), $$J_t:=\int_0^t a_s \mathrm{~d} s=: \int_0^t a_s \mathrm{~d}[W]_s, \quad t \geq 0,$$ should coincide, as the mesh of the partition decreases to zero, with the discrete sum $$\mathcal{J}_t:=\sum{k=0}^{n-1} a_{t_k}\left(W_{t_{k+1}}-W_{t_k}\right)^2,$$
of the squared increments of BM, weighted with values of $a$ at the start of each partition interval.

This is true, and we prove this result below. It will come into play when proving the Itô formula in Section 4.

Lemma $2.37$ (Integral with respect to QV). The integral in (2.28) of the adapted process a with respect to $Q V$ and the discrete sum of weighted squared Brownian increments in (2.29), coincide as the mesh of the partition vanishes:
$$\lim {|\mathcal{P}| \rightarrow 0} \mathcal{J}_t:=\lim {|\mathcal{P}| \rightarrow 0} \sum_{k=0}^{n-1} a_{t_k}\left(W_{t_{k+1}}-W_{t_k}\right)^2=J_t:=\int_0^t a_s \mathrm{~d} s=: \int_0^t a_s \mathrm{~d}[W]_s, \quad \text { a.s. }, \quad t \geq 0 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Total variation and path length of BM

Total variation and path length of BM. Given a continuous function $f:[0, t] \rightarrow \mathbb{R}$ its total variation over $[0, t]$, over any partition $\mathcal{P}=\left{0=t_0 \leq t_1 \leq \ldots \leq t_n=t\right}$ of $[0, t]$, is
$$\operatorname{TV}(f)t \equiv[f]_t^{(1)}:=\lim {|\mathcal{P}| \rightarrow 0} \sum_{k=0}^{n-1}\left|f\left(t_{k+1}\right)-f\left(t_k\right)\right|, \quad t \geq 0 .$$
This may be infinite, or some finite number, in which case we say that $f$ has bounded variation. Consider an element of arc length $\Delta s_k$ along $f(\cdot)$ in the interval $\left[t_k, t_{k+1}\right]$. If this interval is small, we have $\left(\Delta s_k\right)^2 \approx\left(\Delta t_k\right)^2+\left(\Delta f_k\right)^2$, where we have written $\Delta t_k=t_{k+1}-t_k$ and $\Delta f_k=f\left(t_{k+1}\right)-f\left(t_k\right)$. By the triangle inequality we have
$$\left|\Delta f_k\right| \leq\left|\Delta s_k\right| \leq\left|\Delta f_k\right|+\left|\Delta t_k\right| .$$

Denoting the total arc length (or path length) of $f$ over $[0, t]$ by $s(f)t$ we therefore have, in the limit $|\mathcal{P}| \rightarrow 0$, Therefore, $$\operatorname{TV}(f)_t \leq s(f)_t \leq \operatorname{TV}(f)_t+t, \quad t \geq 0 . .$$ finite path length $\Longleftrightarrow \operatorname{TV}(f)<\infty$. If, on the other hand, we examine the quadratic variation of $f(\cdot)$ over $[0, t]$, we have \begin{aligned} {[f]_t } & =\lim {|\mathcal{P}| \rightarrow 0} \sum_{k=0}^{n-1}\left|\Delta f_k | \Delta f_k\right| \ & \leq \lim {|\mathcal{P}| \rightarrow 0}\left(\max {j=0, \ldots, n-1}\left|\Delta f_j\right|\right) \lim {|\mathcal{P}| \rightarrow 0} \sum{k=0}^{n-1}\left|\Delta f_k\right| \ & =\lim {|\mathcal{P}| \rightarrow 0}\left(\max {j=0, \ldots, n-1}\left|\Delta f_j\right|\right) \mathrm{TV}(f) . \end{aligned}

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Integral with respect to $\mathrm{Q}$

$$J_t:=\int_0^t a_s \mathrm{~d} s=: \int_0^t a_s \mathrm{~d}[W]s, \quad t \geq 0,$$ 应该重合，因为分区的网格减少到零，与离散总和 $$\mathcal{J}_t:=\sum k=0^{n-1} a{t_k}\left(W_{t_{k+1}}-W_{t_k}\right)^2,$$
$\mathrm{BM}$ 的平方增量，加权值 $a$ 在每个分区间隔的开始。

$$\lim |\mathcal{P}| \rightarrow 0 \mathcal{J}t:=\lim |\mathcal{P}| \rightarrow 0 \sum{k=0}^{n-1} a_{t_k}\left(W_{t_{k+1}}-W_{t_k}\right)^2=J_t:=\int_0^t a_s \mathrm{~d} s=: \int_0^t a_s \mathrm{~d}[W]s, \quad \text { a.s. }, \quad t \geq 0$$ path length of BM $B M$ 的总变化和路径长度。给定一个连紏函数 $f:[0, t] \rightarrow \mathbb{R}$ 它的总变化超过 $[0, t]$, 在任何分区 \left 缺少或无法识别的分隔符 $\quad$ 的 $[0, t]$ ，是 $$\operatorname{TV}(f) t \equiv[f]_t^{(1)}:=\lim |\mathcal{P}| \rightarrow 0 \sum{k=0}^{n-1}\left|f\left(t_{k+1}\right)-f\left(t_k\right)\right|, \quad t \geq 0$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Total variation and path length of BM

$$\left|\Delta f_k\right| \leq\left|\Delta s_k\right| \leq\left|\Delta f_k\right|+\left|\Delta t_k\right| .$$

$$\operatorname{TV}(f)t \leq s(f)_t \leq \operatorname{TV}(f)_t+t, \quad t \geq 0 .$$ 有限路径长度 $\Longleftrightarrow \mathrm{TV}(f)<\infty$. 另一方面，如果我们检萛的二次方差 $f(\cdot)$ 超过 $[0, t]$, 我们有 $$[f]_t=\lim |\mathcal{P}| \rightarrow 0 \sum{k=0}^{n-1}\left|\Delta f_k\right| \Delta f_k|\quad \leq \lim | \mathcal{P}\left|\rightarrow 0\left(\max j=0, \ldots, n-1\left|\Delta f_j\right|\right) \lim \right| \mathcal{P}\left|\rightarrow 0 \sum k=0^{n-1}\right| \Delta f_k|=\lim | \mathcal{P} \mid \rightarrow 0(\max j=0, \ldots, n-1$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Brownian motion

The stochastic process that we shall focus our attention on in this course is Brownian motion (BM) $W=\left(W_t\right){t \geq 0}$ (which is the building block for all stochastic calculus involving processes with continuous sample paths). Summarising it very briefly, BM is a continuous process with stationary, independent, normally distributed (or Gaussian) increments of mean zero and variance equal to the time elapsed during the increment. That is, for $0 \leq s \leq t<\infty$, the increment $W_t-W_s$ is distributed according to $W_t-W_s \sim \mathrm{N}(0, t-s)$ and is independent of $W_s$ (and, indeed, is independent of the entire history $\left(W_u\right){0 \leq u \leq s}$ ). Here, and elsewhere, we shall use the notation whereby $X \sim \mathrm{N}\left(m, s^2\right)$ denotes that a random variable $X$ is normally distributed with mean $m$ and variance $s^2$.

We shall give some rigorous definitions of BM shortly. BM has all the quintessential properties one could hope for: it is a martingale and a Markov process (as well as having continuous paths), and can also be thought of as the limit of a scaled and speeded-up random walk. However, the paths of BM are highly irregular, as we shall see; they are not differentiable, and are of infinite length over any finite time interval.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Remarks on existence of BM

Remarks on existence of $\mathbf{B M}$. The existence of $\mathrm{BM}$ as a well-defined mathematical object is a non-trivial issue, but one that will not concern us – we shall assume the existence of BM, though we now make some brief remarks with references to where constructions of BM can be found.

One approach to showing the existence of $\mathrm{BM}$ is to write down what the finite-dimensional distributions of the process (based on stationarity, independence and normality of its increments) must be, and then construct a probability measure and a process on an appropriate measurable space in such a way that we obtain the prescribed finite-dimensional distributions.

This technique is a standard approach to constructing a Markov process (of which BM is an example), and can be lengthy and technical, see Karatzas and Shreve [13, Section 2.2]. That this procedure works is guaranteed by deep theorems initiated by Kolmogorov (the “consistency” theorem, and the “continuity” theorem). These say, respectively, that given a set of finite-dimensional distributions (FDDs), one can indeed construct a well-defined stochastic process with these FDDs, and under additional conditions on the moments of the increments (so, on $\mathbb{E}\left[\left|X_t-X_s\right|^\alpha\right], s \leq t$, for some positive $\alpha$ ), the process can be assumed to almost surely (so, with probability one) have continuous paths.

There are also a number of more direct constructions of BM. The first rigorous construction was by Norbert Wiener $[26,27]$ using Fourier series methods (his construction is outlined in Bass [1, Section 6.1]). Einstein, in 1905 [8], derived the transition density (the probability density $p(t, x)$ for the BM moving from 0 to $x$ in time $t$ ) for BM by considering the molecular-kinetic theory of heat (as being composed of many random collisions of molecules). An approach similar in spirit to Wiener’s (so also based on Hilbert space theory, namely sets of orthonormal functions), which uses the so-called Haar functions, was carried out by Lévy [18] and later simplified by Ciesielski [5]. Such a construction can be found in Karatzas and Shreve [13, Section 2.3].

Yet another another proof for the existence of BM is based on the idea of a weak limit (so a convergence in distribution) of a sequence of random walks. This construction can be found in Karatzas and Shreve [13, Section 2.4]. It involves constructing a sequence of probability spaces such that the corresponding sequence of probability measures satisfies a property known as tightness. One has a sequence of processes $\left(X^n\right){n \in \mathbb{N}}$ which induce a sequence of tight probability measures on $(C([0, \infty)), \mathcal{B}(C([0, \infty))))$ (the space of continuous functions on $[0, \infty)$ equipped with the Borel $\sigma$-field on this space), such that there is a limiting measure $\mathbb{P}*$ (called Wiener measure) under which the coordinate mapping process $W_t(\omega):=\omega(t), t \geq 0$ on $C([0, \infty))$ is a standard, one-dimensional Brownian motion. This is a lengthy and technical construction, to say the least. It is also possible to construct a probability space on which all the random walks are defined and converge to BM almost surely, rather than merely in distribution, as shown by Knight .

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Remarks on existence of BM

BM 也有许多更直接的结构。第一个严格的构造是由 Norbert Wiener $[26,27]$ 使用傅立叶级数方法（他的构造在 Bass [1，第 $6.1$ 节] 中进行了概述) 。咾因斯坦在1905年 [8]推导了跃迁密度 (概率密度 $p(t, x)$ 对于 BM 从 0 移动到 $x$ 及时 $t$ ) 对于 BM，通过考虑 热的分子动力学理论 (由许多分子的随机碰噇组成)。Lévy [18] 实施了一种与 Wiener 的方法 (因此也甚于 Hilbert 空间理论， 即正交函数集) 在本质上类似的方法，该方法使用所调的 Haar 函数，由 Lévy [18] 进行，后来由 Ciesielski [5] 进行了简化。这 种结构可以在 Karatzas 和 Shreve [13，第 $2.3$ 节] 中找到。

BM 存在的另一个证明是基于随机游走序列的弱极限（因此分布收敛）的想法。这种结构可以在 Karatzas 和 Shreve [13，第 $2.4$ 节] 中找到。它涉及构建概率空间序列，使得相应的概率测度序列满足称为苭度的属性。一个有一系列的过程 $\left(X^n\right) n \in \mathbb{N}$ 这引起 了一手列严格的概率测量 $(C([0, \infty)), \mathcal{B}(C([0, \infty)))$ ) (连紏函数的空间 $[0, \infty)$ 配备了 Borel $\sigma$-这个空间上的场)，这样就有一 个限制措施 $\mathbb{P} *$ (称为维妠测度) 下的坐标映射过程 $W_t(\omega):=\omega(t), t \geq 0$ 上 $C([0, \infty))$ 是标准的一维布朗运动。至少可以说， 这是一个几长的技术结构。还可以构建一个概率空间，在该空间上定义了所有随机游走并几夹肯定地收敛到 BM，而不是仅仅在分 布中，如 Knight 所示。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Natural FV Processes Are Predictable

The main result of this section is to show that a process $A \in \mathbb{V}$ is natural if and only if it is predictable. To achieve this, we need to consider the right continuous filtration $\left(\mathcal{F}_{.}^{+}\right)$along with the given filtration.

Recall that we had observed in Corollary $4.5$ that a $\left(\mathcal{F}^{+}\right)$predictable process $f$ such that $f_0$ is $\mathcal{F}0$ measurable is $\left(\mathcal{F}{.}\right)$predictable.

In his work on decomposition of submartingales, P. A. Meyer had introduced a notion of natural increasing process. It was an ad hoc definition, given with the aim of showing uniqueness in the Doob-Meyer decomposition.

Definition 8.31 Let $A \in \mathbb{V}0$; i.e. $A$ is an adapted process with finite variation paths and $A_0=0$. Suppose $|A|$ is locally integrable where $|A|_t=\operatorname{Var}{[0, t]}(A) . A$ is said to be natural if for all bounded r.c.l.I. martingales $M$
$[M, A]$ is a local martingale.
Let $\mathbb{W}=\left{V \in \mathbb{V}0:|V|\right.$ is locally integrable where $\left.|V|_t=\operatorname{VaR}{[0, t]}(V)\right}$.
Remark 8.32 Let $A \in \mathbb{V}_0$ be such that $[A, A]$ is locally integrable. Since $\Delta[A, A]$ $=(\Delta A)^2$, it follows that $(\triangle A)$ is locally integrable and as a consequence $A$ is locally integrable and thus $A \in \mathbb{W}$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Decomposition of Semimartingales Revisited

In view of this identification of natural $\mathrm{FV}$ processes as predictable, we can recast Theorem $5.50$ as follows.
Theorem $8.38$ Let $X$ be a stochastic integrator such that
(i) $X_t=X_{t \wedge T}$ for all $t$.
(ii) $\mathrm{E}\left[\sup {s \leq T}\left|X_s\right|\right]<\infty$. (iii) $\mathrm{E}\left[[X, X]_T\right]<\infty$. Then $\mathrm{X}$ admits a decomposition $$X=M+A, M \in \mathbb{M}^2, A \in \mathbb{V}, A_0=0 \text { and } A \text { is predictable } .$$ Further, the decomposition (8.4.1) is unique. Proof Let $X=M+A$ be the decomposition in Theorem 5.50. As seen in Corollary 5.50, the process $A$ satisfies $A_t=A{t \wedge T}$. Since $\mathrm{E}\left[[M, A]_T\right]=0$, we have
$$\mathrm{E}\left[[X, X]_T\right]=\mathrm{E}\left[[M, M]_T\right]+\mathrm{E}\left[[A, A]_T\right]$$
and hence $\mathrm{E}\left[[A, A]_T\right]<\infty$ and so $A \in \mathbb{W}$. Thus $A$ satisfies conditions of Theorem $8.37$ and hence $A$ is predictable. For uniqueness, if $X=N+B$ is another decomposition with $N \in \mathbb{M}^2$ and $B \in \mathbb{V}$ and $B$ being predictable, then $M-N=B-A$ is a predictable process with finite variation paths which is also a martingale and hence by Theorem $8.29, M=N$ and $B=A$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Doob-Meyer Decomposition

$$0 \leq C_t \leq A_t \quad \forall t$$

$$U=D+B$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales

$$M_t=\xi 1_{[\tau, \infty)}(t)$$

$$A_t=\mathrm{E}\left[\xi^2 \mid \mathcal{F} \tau-\right] 1[\tau, \infty)(t) .$$

$$M_t^2-A_t=\left(\xi^2-\mathrm{E}\left[\xi^2 \mid \mathcal{F}_\tau-\right]\right) 1[\tau, \infty)(t)$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

Let us consider the stochastic differential equation (3.5.1) where instead of a Brownian motion as in Chap. 3, here $W=\left(W^1, W^2, \ldots, W^d\right)$ is a amenable semimartingale. The growth estimate (7.2.5) enables one to conclude that in this case too, Theorem $3.30$ is true and the same proof works essentially-using (7.2.5) instead of (3.4.4). Moreover, using random time change, one can conclude that the same is true even when $W$ is any continuous semimartingale. We will prove this along with some results on approximations to the solution of an SDE.

We are going to consider the following general framework for the SDE driven by continuous semimartingales, where the evolution from a time $t_0$ onwards could depend upon the entire past history of the solution rather than only on its current value as was the case in Eq. (3.5.1) driven by a Brownian motion.

Let $Y^1, Y^2, \ldots Y^m$ be continuous semimartingales w.r.t. the filtration $(\mathcal{F}$.). Let $Y=\left(Y^1, Y^2, \ldots Y^m\right)$. Here we will consider an SDE
$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$
where the functional $b$ is given as follows. Recall that $\mathbb{C}d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. Let $$a:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$$ be such that for all $\zeta \in \mathbb{C}_d$, $(t, \omega) \mapsto a(t, \omega, \zeta)$ is an r.c.l.l. $(\mathcal{F}$.$) adapted process$ and there is an increasing r.c.l.l. adapted process $K$ such that for all $\zeta_1, \zeta_2 \in \mathbb{C}_d$, $$\sup {0 \leq s \leq t}\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup _{0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$
Finally, $b:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$ be given by
$$b(s, \omega, \zeta)=a(s-, \omega, \zeta)$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Pathwise Formula for Solution of SDE

In this section, we will consider the SDE
$$d V_t=f(t-, H, V) d X_t$$
for an $\mathbb{R}^d$-valued process $V$ where $f:[0, \infty) \times \mathbb{D}r \times \mathbb{C}_d \mapsto \mathrm{L}(d, m), H$ is an $\mathbb{R}^r$-valued r.c.l.l. adapted process, $X$ is a $\mathbb{R}^m$-valued continuous semimartingale. Here $\mathbb{D}_r=\mathbb{D}\left([0, \infty), \mathbb{R}^r\right), \mathbb{C}_d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. For $t<\infty, \zeta \in \mathbb{C}_d$ and $\gamma \in \mathbb{D}_r$, let $\gamma^t(s)=\gamma(t \wedge s)$ and $\zeta^t(s)=\zeta(t \wedge s)$. We assume that $f$ satisfies \begin{aligned} f(t, \gamma, \zeta) & =f\left(t, \gamma^t, \zeta^t\right), \quad \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d, 0 \leq t<\infty \ t & \mapsto f(t, \gamma, \zeta) \text { is an r.c.l.l. function } \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d \end{aligned} We also assume that there exists a constant $C_T<\infty$ for each $T<\infty$ such that $\forall \gamma \in \mathbb{D}_r, \zeta_1, \zeta_2 \in \mathbb{C}_d, 0 \leq t \leq T$ $$\left|f\left(t, \gamma, \zeta_1\right)-f\left(t, \gamma, \zeta_2\right)\right| \leq C_T\left(1+\sup {0 \leq s \leq t}|\gamma(s)|\right)\left(\sup _{0 \leq s \leq t}\left|\zeta_1(s)-\zeta_2(s)\right|\right)$$
As in Sect. 6.2, we will now obtain a mapping $\Psi$ that yields a pathwise solution to the SDE (7.4.1).

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$

$$a:[0, \infty) \times \Omega \times \mathbb{C}d \rightarrow \mathrm{L}(d, m)$$ 对所有人来说 $\zeta \in \mathbb{C}_d,(t, \omega) \mapsto a(t, \omega, \zeta)$ 是一个 $r c |(\mathcal{F}$. adaptedprocess并且有一个越来越多的 rcll 适应过程K这样对于所 有人 $\zeta_1, \zeta_2 \in \mathbb{C}_d$ $$\sup 0 \leq s \leq t\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup {0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## avatest™帮您通过考试

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Quadratic Variation of a Square Integrable Martingale

The next lemma connects the quadratic variation map $\Psi$ and r.c.l.l. martingales.
Lemma 5.16 Let $\left(N_t, \mathcal{F}t\right)$ be an r.c.l.l. martingale such that $\mathrm{E}\left(N_t^2\right)<\infty$ for all $t>0$. Suppose there is a constant $C<\infty$ such that with $$\tau=\inf \left{t>0:\left|N_t\right| \geq C \text { or }\left|N{t-}\right| \geq C\right}$$
one has
$$N_t=N_{t \wedge \tau} .$$
Let
$$A_t(\omega)=\Psi(N .(\omega))(t) .$$
Then $\left(A_t\right)$ is an $\left(\mathcal{F}t\right)$ adapted r.c.l.l. increasing process such that $X_t:=N_t^2-A_t$ is also a martingale. Proof Let $\Psi_n(\gamma)$ and $t_i^n(\gamma)$ be as in the previous section. \begin{aligned} A_t^n(\omega) & =\Psi_n(N .(\omega))(t) \ \sigma_i^n(\omega) & =t_i^n(N .(\omega)) \ Y_t^n(\omega) & =N_t^2(\omega)-N_0^2(\omega)-A_t^n(\omega) \end{aligned} It is easy to see that for each $n,\left{\sigma_i^n: i \geq 1\right}$ are stopping times (see Theorem 2.46) and that $$A_t^n=\sum{i=0}^{\infty}\left(N_{\sigma_{i+1}^n \wedge t}-N_{\sigma_i^n \wedge t}\right)^2 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales Are Stochastic Integrators

The main aim of this section is to show that square integrable martingales are stochastic integrators.

The treatment is essentially classical, as in Kunita-Watanabe [46], but with an exception. The role of $\langle M, M\rangle$-the predictable quadratic variation in the KunitaWatanabe treatment-is here played by the quadratic variation $[M, M]$.

Recall that $\mathbb{M}^2$ denotes the class of r.c.l.l. martingales $M$ such that $\mathrm{E}\left[M_t^2\right]<\infty$ for all $t<\infty$ with $M_0=0$.

Lemma 5.27 Let $M, N \in \mathbb{M}^2$ and $f, g \in \mathbb{S}$. Let $X=J_M(f)$ and $Y=J_N(g)$. Let $Z_t=X_t Y_t-\int_0^t f_s g_s d[M, N]s^\psi$. Then $X, Y, Z$ are martingales. Proof The proof is almost the same as proof of Lemma 3.10, and it uses $M_t N_t-$ $[M, N]_t^\psi$ is a martingale along with Theorem $2.59$, Corollary $2.60$ and Theorem $2.61$. Corollary 5.28 Let $M \in \mathbb{M}^2$ and $f \in \mathbb{S}$. Then $Y_t=\int_0^t f d M$ and $Z_t=\left(Y_t\right)^2-$ $\int_0^t f_s^2 d[M, M]_s^\psi$ are martingales and $$\mathrm{E}\left[\sup {0 \leq t \leq T}\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\nu\right] .$$
Proof Lemma $5.27$ gives $Y, Z$ are martingales. The estimate (5.4.1) now follows from Doob’s inequality.

Theorem 5.29 Let $M \in \mathbb{M}^2$. Then $M$ is a stochastic integrator. Further, for $f \in$ $\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$, the processes $Y_t=\int_0^t f d M$ and $Z_t=Y_t^2-\int_0^t f_s^2 d[M, M]s^*$ are martingales, $[Y, Y]_t^\psi=\int_0^t f_s^2 d[M, M]_s^\psi$ and $$\mathrm{E}\left[\sup {0 \leq t \leq T}\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^T f_s^2 d[M, M]s^\psi\right], \quad \forall T<\infty .$$ Proof Fix $T<\infty$. Suffices to prove the result for the case when $M_t=M{t \wedge T}$. The rest follows by localization. See Theorem $4.49$. Recall that $\widetilde{\Omega}=[0, \infty) \times \Omega$ and $\mathcal{P}$ is the predictable $\sigma$-field on $\widetilde{\Omega}$. Let $\mu$ be the measure on $(\widetilde{\Omega}, \mathcal{P})$ defined for $A \in \mathcal{P}$
$$\mu(A)=\int\left[\int_0^T 1_A(\omega, s) d[M, M]s^\psi(\omega)\right] d \mathrm{P}(\omega) .$$ Note that $$\mu(\widetilde{\Omega})=\mathrm{E}\left[[M, M]_T^{\nsim}\right]=\mathrm{E}\left[\left|M_T\right|^2\right]<\infty$$ and for $f \in \mathbb{B}(\tilde{\Omega}, \mathcal{P})$ the norm on $\mathbb{L}^2(\tilde{\Omega}, \mathcal{P}, \mu)$ is given by $$|f|{2, \mu}=\sqrt{\mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\psi\right]}$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Quadratic Variation of a Square Integrable Martingale

\left 缺少或无法识别的分隔符

$$N_t=N_{t \wedge \tau} .$$

$$A_t(\omega)=\Psi(N .(\omega))(t) .$$

$$A_t^n(\omega)=\Psi_n(N .(\omega))(t) \sigma_i^n(\omega) \quad=t_i^n(N .(\omega)) Y_t^n(\omega)=N_t^2(\omega)-N_0^2(\omega)-A_t^n(\omega)$$

$$A_t^n=\sum i=0^{\infty}\left(N_{\sigma_{i+1}^n \wedge t}-N_{\sigma_2^n \wedge t}\right)^2 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales Are Stochastic Integrators

$$\mu(A)=\int\left[\int_0^T 1_A(\omega, s) d[M, M] s^\psi(\omega)\right] d \mathrm{P}(\omega) .$$

$$\mu(\widetilde{\Omega})=\mathrm{E}\left[[M, M]_T^{\infty}\right]=\mathrm{E}\left[\left|M_T\right|^2\right]<\infty$$

$$|f| 2, \mu=\sqrt{\mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\psi\right]}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 金融代写|随机微积分代写STOCHASTIC CALCULUS代考|Locally Bounded Processes

We will introduce an important class of integrands, namely that of locally bounded predictable processes, that is contained in $\mathbb{L}(X)$ for every stochastic integrator $X$. For a stopping time $\tau,[0, \tau]$ will denote the set ${(t, \omega) \in \widetilde{\Omega}: 0 \leq t \leq \tau(\omega)}$ and thus $g=f 1_{[0, \tau]}$ means the following: $g_{t}(\omega)=f_{t}(\omega)$ if $t \leq \tau(\omega)$ and $g_{t}(\omega)$ is zero if $t>\tau(\omega)$.

The next result gives interplay between stopping times and stochastic integration.
Lemma 4.36 Let $X$ be a stochastic integrator and $f \in \mathbb{L}(X)$. Let $\tau$ be a stopping time. Let $g=f 1_{[0, \tau]}$. Let
$$Y_{t}=\int_{0}^{t} f d X$$
and $V=\int g d X .$ Then $V_{t}=Y_{t \wedge \tau}$, i.e.
$$Y_{t \wedge \tau}=\int_{0}^{t} f 1_{[0, \tau]} d X .$$
Proof When $f \in \mathbb{S}$ is a simple predictable process and $\tau$ is a stopping time taking only finitely many values, then $g \in \mathbb{S}$ and (4.4.1)-(4.4.2) can be checked as in that case, the integrals $\int f d X$ and $\int g d X$ are both defined directly by (4.2.2). Thus fix $f \in \mathbb{S}$. Approximating a bounded stopping time $\tau$ from above by stopping time taking finitely many values (as seen in the proof of Theorem 2.54), it follows that (4.4.2) is true for any bounded stopping time, then any stopping time $\tau$ can be approximated by $\tilde{\tau}^{n}=\tau \wedge n$ and one can check that (4.4.2) continues to be true. Thus we have proven the result for simple integrands.
Now fix a stopping time $\tau$ and let
$$\mathbb{K}={f \in \mathbb{B}(\widetilde{\Omega}, \mathcal{P}):(4.4 .1)-(4.4 .2) \text { is true for all } t \geq 0 .} \text {. }$$

## 金融代写|随机微积分代写STOCHASTIC CALCULUS代考|Approximation by Riemann Sums

The next result shows that for an r.c.l.l. process $Y$ and a stochastic integrator $X$, the stochastic integral $\int Y^{-} d X$ can be approximated by Riemann-like sums. The difference is that the integrand must be evaluated at the lower end point of the interval as opposed to any point in the interval in the Riemann-Stieltjes integral.
Theorem 4.55 Let $Y$ be an r.c.l.l. adapted process and $X$ be a stochastic integrator. Let
$$0=t_{0}^{m}<t_{1}^{m}<\ldots<t_{n}^{m}<\ldots ; \quad t_{n}^{m} \uparrow \infty \text { as } n \uparrow \infty$$
be a sequence of partitions of $[0, \infty)$ such that for all $T<\infty$,
$$\delta_{m}(T)=\left(\sup {\left{n::{n}^{m} \leq T\right}}\left(t_{n+1}^{m}-t_{n}^{m}\right)\right) \rightarrow 0 \quad \text { as } m \uparrow \infty$$
Let
$$Z_{t}^{m}=\sum_{n=0}^{\infty} Y_{t_{n}^{m} \wedge t}\left(X_{t_{n+1}^{m} \wedge t}-X_{t_{n}^{m} \wedge t}\right)$$
and $Z=\int Y^{-} d X$. Note that for each $t, m$, the sum in (4.5.3) is a finite sum since $t_{n}^{m} \wedge t=t$ from some n onwards. Then
$$Z^{m} \stackrel{u c p}{\longrightarrow} Z$$
or in other words
$$\sum_{n=0}^{\infty} Y_{t_{n}^{m} \wedge t}\left(X_{t_{n+1}^{\mathrm{m}} \wedge t}-X_{t_{n}^{\mathrm{m}} \wedge t}\right) \stackrel{u c p}{\longrightarrow} \int_{0}^{t} Y^{-} d X$$
Proof Let $Y^{m}$ be defined by
$$Y_{t}^{m}=\sum_{n=0}^{\infty} Y_{t_{n}^{m} \wedge t} 1_{\left(t_{n}^{m}, t_{n+1}^{m}\right]}(t)$$
We will first prove
$$\int Y^{m} d X=Z^{m}$$

## 金融代写|随机微积分代写STOCHASTIC CALCULUS代考|Locally Bounded Processes

$$Y_{t}=\int_{0}^{t} f d X$$

$$Y_{t \wedge \tau}=\int_{0}^{t} f 1_{[0, \tau]} d X .$$

$$\mathbb{K}=f \in \mathbb{B}(\widetilde{\Omega}, \mathcal{P}):(4.4 .1)-(4.4 .2) \text { is true for all } t \geq 0 .$$

## 金融代写|随机微积分代写STOCHASTIC CALCULUS代考|Approximation by Riemann Sums

$$0=t_{0}^{m}<t_{1}^{m}<\ldots<t_{n}^{m}<\ldots ; \quad t_{n}^{m} \uparrow \infty \text { as } n \uparrow \infty$$

\left 的分隔符缺失或无法识别

$$Z_{t}^{m}=\sum_{n=0}^{\infty} Y_{t_{n} \cap t}\left(X_{t_{m 11}^{m} \wedge t}-X_{t_{m} \cap t}\right)$$

$$Z^{m} \stackrel{u c p}{\longrightarrow} Z$$

$$\sum_{n=0}^{\infty} Y_{t_{n}^{m} \wedge t}\left(X_{t_{n+1}^{\mathrm{m}} \wedge t}-X_{t_{\Pi}^{\mathrm{m}} \wedge t}\right) \stackrel{u c p}{\longrightarrow} \int_{0}^{t} Y^{-} d X$$

$$Y_{t}^{m}=\sum_{n=0}^{\infty} Y_{t_{m} \wedge t} 1_{\left(t_{m}, t_{n+1}^{m}\right]}(t)$$

$$\int Y^{m} d X=Z^{m}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。