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

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

随机微积分(stochastic calculus),数学概念,是高等数学中研究函数的微分(Differentiation)、积分(Integration)以及有关概念和应用的数学分支。它是数学的一个基础学科,内容主要包括极限、微分学、积分学及其应用。微分学包括求导数的运算,是一套关于变化率的理论。它使得函数、速度、加速度和曲线的斜率等均可用一套通用的符号进行讨论。积分学,包括求积分的运算,为定义和计算面积、体积等提供一套通用的方法

my-assignmentexpert™ 随机微积分stochastic calculus作业代写,免费提交作业要求, 满意后付款,成绩80\%以下全额退款,安全省心无顾虑。专业硕 博写手团队,所有订单可靠准时,保证 100% 原创。my-assignmentexpert™, 最高质量的随机微积分stochastic calculus作业代写,服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面,考虑到同学们的经济条件,在保障代写质量的前提下,我们为客户提供最合理的价格。 由于随机微积分stochastic calculus作业种类很多,难度波动比较大,同时其中的大部分作业在字数上都没有具体要求,因此随机微积分stochastic calculus作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

想知道您作业确定的价格吗? 免费下单以相关学科的专家能了解具体的要求之后在1-3个小时就提出价格。专家的 报价比上列的价格能便宜好几倍。

my-assignmentexpert™ 为您的留学生涯保驾护航 在经济学作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的微积分calculus代写服务。我们的专家在随机微积分stochastic calculus 代写方面经验极为丰富,各种随机微积分stochastic calculus相关的作业也就用不着 说。

我们提供的随机微积分stochastic calculus代写服务范围广, 其中包括但不限于:

  • 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
随机微积分作业代写stochastic calculus代考

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.

微积分作业代写calclulus代考|Quadratic variation and covariation

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:

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

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.
等效地,X是一个连续的半鞅当且仅当存在一个连续的有界变分过程一种这样X−一种是局部鞅。条件一种0=0总是可以通过替换来满足一种和一种吨−一种0.

要求一种0=0确保分解X=米+一种是独特的。事实上,使用 9.b.2,一种是唯一的连续有界变化过程,使得一种0=0和X−一种是局部鞅,并且米=X−一种. 分解X=米+一种称为半鞅分解X, 过程米被称为局部鞅部分X和过程一种的补偿器X, 表示
一种=你X. 
请注意,每个连续有界变化过程一种是带有局部鞅部分的半鞅米吨=一种0和补偿器你一种(吨)=一种吨−一种0. 更准确地说,一个连续的半鞅X是一个有界变化过程当且仅当它的补偿器满足你X(吨)=X吨−X0. 如果X0=0, 然后X是一个有界变化过程当且仅当你X=X和X是局部鞅当且仅当你X=0.
例子。如果X,和是连续的局部鞅,则X2和X和是带补偿器的半鞅你X2=⟨X⟩和你X和=⟨X,和⟩分别。

让小号表示关于 $\left(\Omega, \mathcal{F}, P,\left(\mathcal{F} {t}\right)\right)的所有连续半鞅的族.一世吨一世s和一种s一世一世和s和和n吨H一种吨\数学{S}一世s一种r和一种一世v和C吨○rsp一种C和.H○在和v和r,一种吨pr和s和n吨一世吨一世sn○吨和v和nC一世和一种r吨H一种吨\数学{S}一世sC一世○s和d你nd和r米你一世吨一世p一世一世C一种吨一世○n.米一世r一种C你一世○你s一世和吨H和一世吨○F○r米你一世一种○F吨H和小号吨○CH一种s吨一世CC一种一世C你一世你s在一世一世一世sH○在吨H一种吨\数学{S}一世s一世nF一种C吨C一世○s和d你nd和r吨H和一种pp一世一世C一种吨一世○n○F一种一世一世吨在一世C和C○n吨一世n你○你s一世和d一世FF和r和n吨一世一种b一世和米你一世吨一世v一种r一世一种b一世和F你nC吨一世○ns.吨H一种吨一世s,一世Ff \in C^{2}\left(R^{n}\right)一种nd一世FX {1},\ldots,X_{n}一种r和C○n吨一世n你○你ss和米一世米一种r吨一世nG一种一世和s,吨H和nY=f\left(X_{1}, \ldots, X_{n}\right)一世s一种一世s○一种C○n吨一世n你○你ss和米一世米一种r吨一世nG一种一世和.吨H和一世吨○F○r米你一世一种在一世一世一世一种一世s○pr○v一世d和一种F○r米你一世一种F○r吨H和一世○C一种一世米一种r吨一世nG一种一世和p一种r吨一种ndC○米p和ns一种吨○ru_{Y}○F和.一世吨吨H你s吨你rns○你吨吨H一种吨吨H和CH○一世C和○F吨H和sp一种C和\mathcal{S}$ 的连续半鞅是非常幸运的。

微积分作业代写calclulus代考|Quadratic variation and covariation

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.
这里的限制是接管所有分区Δ区间的[0,吨].
证明。(a) 让ω∈Ω使得路径s→一种s(ω)是有界的变化和路径s→Xs(ω)是连续的,因此在区间上均匀连续[0,吨]. 这是这种情况磷-ae。ω∈Ω. 让$|A| {t}(\omega)<\inftyd和n○吨和吨H和吨○吨一种一世v一种r一世一种吨一世○n○F吨H和p一种吨Hs \rightarrow A {s}(\omega)○n吨H和一世n吨和rv一种一世[0, 吨]一种nds和吨,F○r一种n和p一种r吨一世吨一世○n\Delta=\left{0=t_{0}<t_{1}<\ldots<t_{n}=t\right}○F吨H和一世n吨和rv一种一世[0, 吨],$
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}
$$
现在让|Δ|→0. 然后我们有问Δ(一种~,和),问Δ(米,乙)→0根据 (a) 并且,问Δ(米,ñ)→⟨米,ñ⟩吨,根据 10.a.eq. (1) 在概率上,如n↑∞. 它遵循问Δ(X,和)→⟨米,ñ⟩吨=⟨X,和⟩吨,概率。

让X是一个连续的半鞅并且X=米+一种的半鞅分解X. 如果吨是任何可选时间X吨又是一个连续的半鞅

用半鞅分解X吨=米吨+一种吨. 协变过程的大多数性质⟨米,ñ⟩连续局部鞅米,ñ很容易扩展到连续半鞅的情况X,和.

当然,这不再是真的X2−⟨X⟩是一个局部鞅(它是一个连续的半鞅,但这不再有趣,因为对于空间来说更强大的陈述是正确的小号)。因此,也不再是真的你X2=⟨X⟩. 同样地⟨X⟩=0不再暗示X在时间上是恒定的,只是X是一个有界变化过程。我们在下面收集这些结果:

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

微积分作业代写calclulus代考 请认准UprivateTA™. UprivateTA™为您的留学生涯保驾护航。

抽象代数Galois理论代写

偏微分方程代写成功案例

代数数论代考

组合数学代考

统计作业代写

集合论数理逻辑代写案例

凸优化代写

统计exam代考