# 数学代写|随机微积分代写Stochastic calculus代考|MA451A

## 数学代写|随机微积分代写Stochastic calculus代考|The crown jewel: Itˆo’s formula

Unless the integrand has locally bounded variation, stochastic integrals are somewhat inscrutable quantities. They exist, but, like Fourier series, they are not robust and converge only because of intricate cancellations. As a consequence, it is desirable to find more tractable quantities to which stochastic integrals are related, and (3.2.2) provides prime examples of the sort of relationship for which one should be looking.

The key to finding relationships like those in (3.2.2) was discovered by Itô. To describe his result, let $\left(B(t), \mathcal{F}t, \mathbb{P}\right)$ be an $\mathbb{R}^M$-valued Brownian motion, $V:[0, \infty) \times \Omega \longrightarrow \mathbb{R}^{N_1}$ a continuous, progressively measurable function of locally bounded variation, and $\sigma \in P M{\text {loc }}^2\left(\operatorname{Hom}\left(\mathbb{R}^M ; \mathbb{R}^{N_2}\right)\right.$. Then Itô’s formula says that any $\varphi \in C^{1,2}\left(R^{N_1} \times \mathbb{R}^{N_2} ; \mathbb{C}\right)$,
\begin{aligned} \varphi\left(V(t), I_\sigma(t)\right)-\varphi(V(0), 0) \ =\int_0^t\left(\nabla_{(1)} \varphi\left(V(\tau), I_\sigma(\tau)\right), d V(\tau)\right){\mathbb{R}^{N_1}} \ &+\int_0^t\left(\sigma(\tau)^{\top} \nabla{(2)} \varphi\left(I_\sigma(\tau)\right), d B(\tau)\right){\mathbb{R}^M} \ &+\frac{1}{2} \int_0^t \operatorname{Trace}\left(\sigma(\tau) \sigma(\tau)^{\top} \nabla{(2)}^2 \varphi\left(I_\sigma(\tau)\right) d \tau,\right. \end{aligned}

where the subscripts on $\nabla_{(1)}$ and $\nabla_{(2)}$ are used to distinguish between differentiation with respect to variables in $\mathbb{R}^{N_1}$ and those in $\mathbb{R}^{N_2}$. Obviously, (3.3.1) is a version of the fundamental theorem of calculus in which second derivatives appear because, as (2.1.2) makes clear, $d B(t)$ is of order $\sqrt{d t}$, not $d t$, and one therefore has to go out two terms in Taylor’s expansion before getting terms that are truly infinitesimal. For this reason, it is useful to think of the result in (2.1.2) as saying that $d B(t) \otimes d B(t)=\mathbf{I} d t$ and write (3.3.1) in differential form:
\begin{aligned} d \varphi(&\left.V(t), I_\sigma(t)\right) \ =&\left(\nabla_{(1)} \varphi\left(V(t), I_\sigma(t)\right), d V(t)\right){\mathbb{R}^{N_1}}+\left(\sigma(t)^{\top} \nabla{(2)} \varphi\left(V(t), I_\sigma(t)\right), d B(t)\right){\mathbb{R}^{N_2}} \ &+\frac{1}{2} \operatorname{Trace}\left(\nabla{(2)}^2 \varphi\left(V(t), I_\sigma(t)\right)(\sigma(\tau) d B(t)) \otimes(\sigma(\tau) d B(t))\right) . \end{aligned}
To prove (3.3.1), first observe that, by using standard approximation methods and stopping times, one can easily show that it suffices to prove it in the case when $\varphi \in C_{\mathrm{c}}^{\infty}\left(\mathbb{R}^{N_1} \times \mathbb{R}^{N_2} ; \mathbb{R}\right)$ and $\int_0^t|\sigma(\tau)|_{\mathrm{H} . \mathrm{S}}^2 d \tau$ and $V(\mathbf{0})+\operatorname{var}_{[0, t]}(V)$ are bounded. Further, under these conditions, one can reduce to the case when $\tau \rightsquigarrow \sigma(\tau)$ is bounded and continuous. Thus we will proceed under these assumptions.

## 数学代写|随机微积分代写Stochastic calculus代考|Burkholder’s inequality

Let $\sigma \in P M_{\text {loc }}^2\left(\operatorname{Hom}\left(\mathbb{R}^M ; \mathbb{R}^N\right)\right)$, and set $A(t)=\int_0^t \sigma(\tau) \sigma(\tau)^{\top} d \tau$. When $\sigma$ is deterministic, $I_\sigma(t)$ is a centered Gaussian random variable with covariance $A(t)$, and therefore, for each $p \in[1, \infty)$
$$\mathbb{E}^{\mathbb{P}}\left[\left|\left(\xi, I_\sigma(t)\right){\mathbb{R}^N}\right|^p\right]=C_p(\xi, A(t) \xi){\mathbb{R}^N}^{\frac{p}{2}} \quad \text { for all } \xi \in \mathbb{R}^N,$$
where $C_p=\int|y|^p \gamma_{0,1}(d y)$. Our first application of (3.3.1) shows that moments of $I_\sigma(\cdot)$ can be estimated in terms of $A(\cdot)$ even when $\sigma$ is random. Namely, given $p \in[2, \infty)$, observe that
$$\nabla^2|\mathbf{y}|^p=p(p-2)|\mathbf{y}|^{p-2} \frac{\mathbf{y} \otimes \mathbf{y}}{|\mathbf{y}|^2}+p|\mathbf{y}|^{p-2} \mathbf{I} .$$
Therefore, by (3.3.1),
$$\left|I_\sigma(t)\right|^p-\frac{1}{2} \int_0^t\left(p(p-2)\left|I_\sigma(\tau)\right|^{p-2} \frac{\left|\sigma^{\top}(\tau) I_\sigma(\tau)\right|^2}{\left|I_\sigma(\tau)\right|^2}+p|\sigma(\tau)|_{\mathrm{H} . S .}^2\left|I_\sigma(\tau)\right|^{p-2}\right) d \tau$$
is a local $\mathbb{P}$-martingale relative to $\left{\mathcal{F}t: t \geq 0\right}$. Thus, if $$\zeta_R=\inf \left{t \geq 0:\left|I\sigma(\tau)\right| \vee \operatorname{Trace}(A(\tau)) \geq R\right},$$
then, by Hölder’s inequality, one sees that
\begin{aligned} &\mathbb{E}^{\mathbb{P}}\left[\left|I_\sigma\left(t \wedge \zeta_R\right)\right|^p\right] \leq \frac{p(p-1)}{2} \mathbb{E}^{\mathbb{P}}\left[\int_0^{t \wedge \zeta_R}|\sigma(\tau)|_{\mathrm{H} . \mathrm{S} .}^2\left|I_\sigma(\tau)\right|^{p-2} d \tau\right] \ &\quad \leq \frac{p(p-1)}{2} \mathbb{E}^{\mathbb{P}}\left[\operatorname{Trace}\left(A\left(t \wedge \zeta_R\right)\right)\left|I_\sigma(\cdot)\right|_{\left[0, t \wedge \zeta_R\right]}^{p-2}\right] \ &\quad \leq \frac{p(p-1)}{2} \mathbb{E}^{\mathbb{P}}\left[\operatorname{Trace}(A(t))^{\frac{p}{2}}\right]^{\frac{2}{p}} \mathbb{E}^{\mathbb{P}}\left[\left|I_\sigma(\cdot)\right|_{\left[0, t \wedge \zeta_R\right]}^p\right]^{1-\frac{2}{p}}, \end{aligned}
and then, using Doob’s inequality, one concludes that
$$\mathbb{E}^{\mathbb{P}}\left[\left|I_\sigma(\cdot)\right|_{\left[0, t \wedge \zeta_R\right]}^p\right]^{\frac{2}{p}} \leq\left(\frac{p^{p+1}}{2(p-1)^{p-1}}\right) \mathbb{E}^{\mathbb{P}}\left[\operatorname{Trace}(A(t))^{\frac{p}{2}}\right]^{\frac{2}{p}} .$$

# 随机微积分代考

## 数学代写|随机微积分代写Stochastic calculus代考|The crown jewel: Itˆo’s formula

$$\varphi\left(V(t), I_\sigma(t)\right)-\varphi(V(0), 0)=\int_0^t\left(\nabla_{(1)} \varphi\left(V(\tau), I_\sigma(\tau)\right), d V(\tau)\right) \mathbb{R}^{N_1}+\int_0^t\left(\sigma(\tau)^{\top} \nabla(2) \varphi\left(I_\sigma(\tau)\right), d B(\tau\right.$$

$$d \varphi\left(V(t), I_\sigma(t)\right)=\quad\left(\nabla_{(1)} \varphi\left(V(t), I_\sigma(t)\right), d V(t)\right) \mathbb{R}^{N_1}+\left(\sigma(t)^{\top} \nabla(2) \varphi\left(V(t), I_\sigma(t)\right), d B(t)\right) \mathbb{R}^{N_2}+\frac{1}{2}$$

## 数学代写|随机微积分代写Stochastic calculus代考|Burkholder’s inequality

$$\mathbb{E}^{\mathbb{P}}\left[\left|\left(\xi, I_\sigma(t)\right) \mathbb{R}^N\right|^p\right]=C_p(\xi, A(t) \xi) \mathbb{R}^{N \frac{p}{2}} \quad \text { for all } \xi \in \mathbb{R}^N,$$

$$\nabla^2|\mathbf{y}|^p=p(p-2)|\mathbf{y}|^{p-2} \frac{\mathbf{y} \otimes \mathbf{y}}{|\mathbf{y}|^2}+p|\mathbf{y}|^{p-2} \mathbf{I} .$$

$$\left|I_\sigma(t)\right|^p-\frac{1}{2} \int_0^t\left(p(p-2)\left|I_\sigma(\tau)\right|^{p-2} \frac{\left|\sigma^{\top}(\tau) I_\sigma(\tau)\right|^2}{\left|I_\sigma(\tau)\right|^2}+p|\sigma(\tau)|{\mathrm{H} . S .}^2\left|I\sigma(\tau)\right|^{p-2}\right) d \tau$$

$$\mathbb{E}^{\mathbb{P}}\left[\left|I_\sigma\left(t \wedge \zeta_R\right)\right|^p\right] \leq \frac{p(p-1)}{2} \mathbb{E}^{\mathbb{P}}\left[\int_0^{t \wedge \zeta_R}|\sigma(\tau)|{\mathrm{H} . S .}^2\left|I\sigma(\tau)\right|^{p-2} d \tau\right] \quad \leq \frac{p(p-1)}{2} \mathbb{E}^{\mathbb{P}}\left[\operatorname{Trace}\left(A\left(t \wedge \zeta_R\right)\right)\right.$$

$$\mathbb{E}^{\mathbb{P}}\left[\left|I_\sigma(\cdot)\right|_{\left[0, t \wedge \zeta_R\right]}^p\right]^{\frac{2}{p}} \leq\left(\frac{p^{p+1}}{2(p-1)^{p-1}}\right) \mathbb{E}^{\mathbb{P}}\left[\operatorname{Trace}(A(t))^{\frac{p}{2}}\right]^{\frac{2}{p}} .$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: