# 随机微积分作业代写stochastic calculus代考| SEMIMARTINGALES

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## 微积分作业代写calclulus代考|Definition

11.The process $X$ is called a continuous semimartingale if it can be represented as a sum $X_{t}=M_{t}+A_{t}, t \geq 0$, where $M$ is a continuous local martingale and $A$ a continuous (adapted) bounded variation process satisfying $A_{0}=0$.
Equivalently, $X$ is a continuous semimartingale if and only if there exists a continuous bounded variation process $A$ such that $X-A$ is a local martingale. The condition $A_{0}=0$ can always be satisfied by replacing $A$ with $A_{t}-A_{0}$.

The requirement $A_{0}=0$ ensures that the decomposition $X=M+A$ is unique. Indeed, using 9.b.2, $A$ is the unique continuous bounded variation process such that $A_{0}=0$ and $X-A$ is a local martingale, and $M=X-A$. The decomposition $X=M+A$ is referred to as the semimartingale decomposition of $X$, the process $M$ is called the local martingale part of $X$ and the process $A$ the compensator of $X$, denoted
$$A=u_{X} \text {. }$$
Note that each continuous bounded variation process $A$ is a semimartingale with local martingale part $M_{t}=A_{0}$ and compensator $u_{A}(t)=A_{t}-A_{0}$. More precisely, a continuous semimartingale $X$ is a bounded variation process if and only if its compensator satisfies $u_{X}(t)=X_{t}-X_{0}$. If $X_{0}=0$, then $X$ is a bounded variation process if and only if $u_{X}=X$ and $X$ is a local martingale if and only if $u_{X}=0$.
Example. If $X, Y$ are continuous local martingales, then $X^{2}$ and $X Y$ are semimartingales with compensator $u_{X^{2}}=\langle X\rangle$ and $u_{X Y}=\langle X, Y\rangle$ respectively.

Let $\mathcal{S}$ denote the family of all continuous semimartingales with respect to $\left(\Omega, \mathcal{F}, P,\left(\mathcal{F}{t}\right)\right)$. It is easily seen that $\mathcal{S}$ is a real vector space. However, at present it is not even clear that $\mathcal{S}$ is closed under multiplication. Miraculously the Ito formula of the Stochastic Calculus will show that $\mathcal{S}$ is in fact closed under the application of all twice continuously differentiable multivariable functions. That is, if $f \in C^{2}\left(R^{n}\right)$ and if $X{1}, \ldots, X_{n}$ are continuous semimartingales, then $Y=f\left(X_{1}, \ldots, X_{n}\right)$ is also a continuous semimartingale. The Ito formula will also provide a formula for the local martingale part and compensator $u_{Y}$ of $Y$. It thus turns out that the choice of the space $\mathcal{S}$ of continuous semimartingales is extremely fortunate.

11.b.0. Let $X, Y$ be continuous semimartingales, $A$ a continuous bounded variation process and $t \geq 0$. Then
(a) $Q_{\Delta}(X, A) \rightarrow 0, P$-as., as $|\Delta| \rightarrow 0$, and
(b) $Q_{\Delta}(X, Y) \rightarrow\langle X, Y\rangle_{t}$ in probability, as $|\Delta| \rightarrow 0$.
Here the limits are taken over all partitions $\Delta$ of the interval $[0, t]$.
Proof. (a) Let $\omega \in \Omega$ be such that the path $s \rightarrow A_{s}(\omega)$ is of bounded variation and the path $s \rightarrow X_{s}(\omega)$ is continuous and hence uniformly continuous on the interval $[0, t]$. This is the case for $P$-ae. $\omega \in \Omega$. Let $|A|{t}(\omega)<\infty$ denote the total variation of the path $s \rightarrow A{s}(\omega)$ on the interval $[0, t]$ and set, for any partition $\Delta=\left{0=t_{0}<t_{1}<\ldots<t_{n}=t\right}$ of the interval $[0, t]$,
$$C_{\Delta}(\omega)=\sup {1 \leq j \leq n}\left|X{t_{j}}(\omega)-X_{t_{j-1}}(\omega)\right|$$
The uniform continuity of the path $s \rightarrow X_{s}(\omega)$ on the interval $[0, t]$ implies that $\lim {|\Delta| \rightarrow 0} C(\Delta)(\omega)=0$. Thus \begin{aligned} \left|Q{\Delta}(X, A)(\omega)\right| & \leq \sum\left|X_{t_{j}}(\omega)-X_{t_{j-1}}(\omega)\right|\left|A_{t_{j}}(\omega)-A_{t_{j-1}}(\omega)\right| \ & \leq C_{\Delta}(\omega) \sum\left|A_{t_{j}}(\omega)-A_{t_{j-1}}(\omega)\right| \leq\left. C_{\Delta}(\omega)|A|\right|{t}(\omega) \rightarrow 0 \end{aligned} as $|\Delta| \rightarrow 0$. This shows (a). (b) Let $X=M+\tilde{A}$ and $Y=N+B$ be the semimartingale decompositions of $X, Y$. Fix $t \geq 0$ and let $\Delta=\left{0=t{0}<t_{1}<\ldots<t_{n}=t\right}$ be a partition of the interval $[0, t]$. By elementary algebra
\begin{aligned} Q_{\Delta}(X, Y) &=\sum\left[\left(M_{t_{j}}-M_{t_{j-1}}\right)+\left(\tilde{A}{t{j}}-\tilde{A}{t{j-1}}\right)\right]\left[\left(N_{t_{j}}-N_{t_{j-1}}\right)+\left(B_{t_{j}}-B_{t_{j-1}}\right)\right] \ &=Q_{\Delta}(\bar{A}, Y)+Q_{\Delta}(M, N)+Q_{\Delta}(M, B) . \end{aligned}
Now let $|\Delta| \rightarrow 0$. Then we have $Q_{\Delta}(\tilde{A}, Y), Q_{\Delta}(M, B) \rightarrow 0$ according to (a) and, $Q_{\Delta}(M, N) \rightarrow\langle M, N\rangle_{t}$, according to 10.a.eq. (1), in probability, as $n \uparrow \infty$. It follows that $Q_{\Delta}(X, Y) \rightarrow\langle M, N\rangle_{t}=\langle X, Y\rangle_{t}$, in probability.

Let $X$ be a continuous semimartingale and $X=M+A$ the semimartingale decomposition of $X$. If $T$ is any optional time $X^{T}$ is again a continuous semimartingale

with semimartingale decomposition $X^{T}=M^{T}+A^{T}$. Most properties of the covariation process $\langle M, N\rangle$ of continuous local martingales $M, N$ extend easily to the case of continuous semimartingales $X, Y$.

It is of course no longer true that $X^{2}-\langle X\rangle$ is a local martingale (it is a continuous semimartingale, but this is no longer interesting, since much stronger statements are true for the space $\mathcal{S}$ ). Thus, it is also no longer true that $u_{X^{2}}=\langle X\rangle$. Likewise $\langle X\rangle=0$ no longer implies that $X$ is constant in time, merely that $X$ is a bounded variation process. We collect these results below:

## 微积分作业代写calclulus代考|Definition

11.过程X如果它可以表示为和，则称为连续半鞅X吨=米吨+一种吨,吨≥0， 在哪里米是一个连续的局部鞅，并且一种一个连续的（适应的）有界变化过程，满足一种0=0.

11.b.0。让X,和是连续的半鞅，一种一个连续的有界变化过程和吨≥0. 那么
（一）问Δ(X,一种)→0,磷-as.，作为|Δ|→0, 和
(b)问Δ(X,和)→⟨X,和⟩吨在概率上，如|Δ|→0.

C_{\Delta}(\omega)=\sup {1 \leq j \leq n}\left|X {t_{j}}(\omega)-X_{t_{j-1}}(\omega) \对|
路径的均匀连续性s→Xs(ω)在区间[0,吨]意味着 \lim {|\Delta| \rightarrow 0} C(\Delta)(\omega)=0.吨H你s \begin{对齐} \left|Q {\Delta}(X, A)(\omega)\right| & \leq \sum\left|X_{t_{j}}(\omega)-X_{t_{j-1}}(\omega)\right|\left|A_{t_{j}}(\omega) -A_{t_{j-1}}(\omega)\right| \ & \leq C_{\Delta}(\omega) \sum\left|A_{t_{j}}(\omega)-A_{t_{j-1}}(\omega)\right| \leq\左。C_{\Delta}(\omega)|A|\right| {t}(\omega) \rightarrow 0 \end{aligned} as|Δ|→0. 这表明（a）。(b) 让X=米+一种~和和=ñ+乙是的半鞅分解X,和. 使固定吨≥0并令 $\Delta=\left{0=t {0}<t_{1}<\ldots<t_{n}=t\right}b和一种p一种r吨一世吨一世○n○F吨H和一世n吨和rv一种一世[0, 吨].乙和和一世和米和n吨一种r和一种一世G和br一种$
\begin{aligned}
Q_{\Delta}(X, Y) &=\sum\left[\left(M_{t_{j}}-M_{t_{j-1}}\right)+\left( \tilde{A} {t {j}}-\tilde{A} {t {j-1}}\right)\right]\left[\left(N_{t_{j}}-N_{t_{j -1}}\right)+\left(B_{t_{j}}-B_{t_{j-1}}\right)\right] \
&=Q_{\Delta}(\bar{A}, Y )+Q_{\Delta}(M, N)+Q_{\Delta}(M, B) 。
\end{aligned}