随机微积分作业代写stochastic calculus代考| REPRESENTATION OF CONTINUOUS LOCAL MARTINGALES

随机微积分作业代写stochastic calculus代考| REPRESENTATION OF CONTINUOUS LOCAL MARTINGALES

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

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  • 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|Representation of positive Brownian martingales

5.e.0. Let $\gamma \in L(W)$ and $M \in \mathcal{S}$. Then the equality $M=\mathcal{E}(\gamma \cdot W)$ is equivalent with $M_{0}=1$ and $d M_{t}=M_{t} \gamma_{t} \cdot d W_{t}$.
Proof. See 4.b.2. $\mathbf{I}$
5.e.1. A positive Brownian martingale $M$ can be represented in the form
$$
M_{t}=\mathcal{E}{t}(\gamma \cdot W)=\exp \left(\int{0}^{t} \gamma_{s} \cdot d W_{s}-\frac{1}{2} \int_{0}^{t}\left|\gamma_{s}\right|^{2} d s\right), \quad t \geq 0,
$$
for some predictable, $R^{d}$-valued process $\gamma \in L(W)$.
Proof. According to $5 . c .4, M$ has a continuous version and we may therefore assume that $M$ is itself continuous. According to $5 . e .0$ it will suffice to find a process $\gamma \in L(W)$ satisfying $d M_{t}=M_{t} \gamma_{t} \cdot d W_{t}$. Using $5 . d .0$ we can write $M_{t}=\int_{0}^{t} H_{s} \cdot d W_{s}$, $t \geq 0$, for some predictable process $H \in L(W)$. Then
$$
\int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, P \text {-as., } \forall t \geq 0,
$$
and $d M_{t}=H_{t} \cdot d W_{t}=M_{t} \gamma_{t} \cdot d W_{t}$, where $\gamma_{t}=M_{t}^{-1} H_{t}$. It remains to be shown only that $\gamma$ is a predictable process in $L(W)$. Since $M$ is continuous it is predictable. Since $H$ is also predictable, it follows that $\gamma$ is predictable. Thus it remains to be shown only that
$$
\int_{0}^{t}\left|\gamma_{s}\right|^{2} d s<\infty, P \text {-as., } \quad \forall t \geq 0 \text {. }
$$
This follows immediately from (1) and the fact that the positive and continuous paths $s \in[0, t] \mapsto M_{s}(\omega)$ are bounded away from zero, $P$-as. on $\Omega$.

微积分作业代写calclulus代考|Kunita-Watanabe decomposition

5.f Kunita-Watanabe decomposition. The representation results above assume that the local martingale to be represented is adapted to the augmented filtration generated by the representing Brownian motion. If this is not the case, we still have a decomposition result. We shall thus no longer assume that our filtration $\left(\mathcal{F}_{t}\right)$ is the augmented filtration generated by some Brownian motion $W$.

Let us call two continuous local martingales $M, N$ orthogonal $(M \perp N)$ iff the covariation process $\langle M, N\rangle$ is identically zero, that is, iff the product $M N$ is a local martingale.

If $M$ and $N$ are square integrable martingales, then $M_{t} N_{t}-\langle M, N\rangle_{t}$ is a martingale (I.11.b.2). Thus $M \perp N$ if and only if the product $M N$ is a martingale.
Assume now that $M_{0}=0$ or $N_{0}=0$, that is, $M_{0} N_{0}=0$. If $\tau$ is a bounded optional time and $T>0$ a number such that $\tau \leq T$, then $M_{\tau}=E_{P}\left[M_{T} \mid \mathcal{F}{\tau}\right]$. Since all conditional expectation operators are contractions on $L^{2}(P)$ (I.2.b.15) it follows that $M{\tau} \in L^{2}(P)$. Likewise $N_{\tau} \in L^{2}(P)$ and consequently $M_{\tau} N_{\tau} \in L^{1}(P)$. According to I.9.c. $4, M_{t} N_{t}$ is a martingale if and only if $E_{P}\left[M_{\tau} N_{\tau}\right]=E_{P}\left[M_{0} N_{0}\right]=$ 0 , that is, $M_{\tau} \perp N_{\tau}$ in $L^{2}(P)$, for all bounded optional times $\tau$ :

5.f.0. Let $M, N$ be continuous, square integrable martingales with $M_{0} N_{0}=0$. Then $M \perp N$ if and only if $M_{\tau} \perp N_{\tau}$ in $L^{2}(P)$, for all bounded optional times $\tau$.

In particular $M \perp N$ implies that $M_{t} \perp N_{t}$ in $L^{2}(P)$, for all $t \geq 0$. If $M, N \in \mathbf{H}^{2}$ are $L^{2}$-bounded martingales, this implies $M_{\infty} \perp N_{\infty}$ in $L^{2}(P)$, that is, $M \perp N$ in $\mathbf{H}^{2}$, since $M_{t} \rightarrow M_{\infty}$ and $N_{t} \rightarrow N_{\infty}$ in $L^{2}(P)$, as $t \uparrow \infty$ (square integrability of the maximal function of an $\mathbf{H}^{2}$-martingale and Dominated Convergence Theorem).
Thus our notion of orthogonality of continuous local martingales $M, N$ is stronger than orthogonality in $\mathbf{H}{0}^{2}$, when specialized to martingales $M, N \in \mathbf{H}{0}^{2}$.

Let $W$ be an $R^{d}$-valued Brownian motion. Recall that $L^{2}(W)$ is a Hilbert space of progressively measurable, $R^{d}$-valued processes $H$, the subspace of all predictable processes $H \in L^{2}(W)$ is closed, the map
$$
\left.H \in L^{2}(W) \mapsto(H \bullet W)\right){\infty} \in L^{2}(P) $$ is an isometry and $\Lambda^{2}(W)$ is the space of all progressively measurable, $R^{d}$-valued processes $H$ such that $1{[0, t]} H \in L^{2}(W)$, for all $t>0$. If $H \in \Lambda^{2}(W)$ and $t>0$, then $(H \bullet W){t}=\left(1{[0, t]} H \cdot W\right){\infty}$, and so, using the isometry $(0)$, $$ \left|(H \cdot W){t}\right|_{L^{2}(P)}^{2}=\left|1_{[0, t]} H\right|_{L^{2}(W)}^{2}=E_{P} \int_{0}^{t}\left|H_{s}\right|^{2} d s
$$
$\operatorname{Fix} T>0$ and set
$$
\mathcal{I}{T}(W)=\left{(H \bullet W){T} \mid H \in \Lambda^{2}(W) \text { is predictable }\right} \subseteq L^{2}(P)
$$
If $H \in \Lambda^{2}(W)$ then $(H \cdot W){T}=(\tilde{H} \cdot W){\infty}$, where $\tilde{H}=1_{[0, T]} H \in L^{2}(W)(2 . f .0 .($ d $))$ satisfies $1_{[0, T]} \tilde{H}=\tilde{H}$. It follows that $\mathcal{I}{T}(W)$ is the image under the isometry $(0)$ of the subspace $L{T}^{2}(W)=\left{H \in L^{2}(W) \mid H=1_{[0, T]} H\right} \subseteq L^{2}(W)$. As $L_{T}^{2}(W) \subseteq$ $L^{2}(W)$ is a closed subspace $\left(H=1_{[0, T]} H\right.$ is equivalent with $\left.1_{(T, \infty)} H=0\right)$ it follows that the subspace $\mathcal{I}{T}(W) \subseteq L^{2}(P)$ is closed also. Letting $$ \mathcal{I}(W)=\left{H \cdot W \mid H \in \Lambda^{2}(W) \text { is predictable }\right} $$ we have $\mathcal{I}{T}(W)=\left{I_{T} \mid I \in \mathcal{I}(W)\right}$. Each integral process $I \in \mathcal{I}(W)$ is a continuous square integrable martingale vanishing at zero. If the underlying filtration $\left(\mathcal{F}_{t}\right)$ is the augmented filtration generated by the Brownian motion $W$, then it follows from 5.c. 3 that conversely every continuous, square integrable martingale vanishing at zero is in $\mathcal{I}(W)$. In general we have the following decomposition result:

随机微积分作业代写stochastic calculus代考| REPRESENTATION OF CONTINUOUS LOCAL MARTINGALES

微积分作业代写calclulus代考|Representation of positive Brownian martingales

5.e.0。让C∈一世(在)和米∈小号. 那么等式米=和(C⋅在)相当于米0=1和d米吨=米吨C吨⋅d在吨.
证明。见 4.b.2。一世
5.e.1。正布朗鞅米可以表示为
$$
M_{t}=\mathcal{E} {t}(\gamma \cdot W)=\exp \left(\int {0}^{t} \gamma_{s} \cdot d W_{s}-\frac{1}{2} \int_{0}^{t}\left|\gamma_{s}\right|^{2} ds\right), \quad t \geq 0,
F○rs○米和pr和d一世C吨一种b一世和,$Rd$−v一种一世你和dpr○C和ss$C∈一世(在)$.磷r○○F.一种CC○rd一世nG吨○$5.C.4,米$H一种s一种C○n吨一世n你○你sv和rs一世○n一种nd在和米一种和吨H和r和F○r和一种ss你米和吨H一种吨$米$一世s一世吨s和一世FC○n吨一世n你○你s.一种CC○rd一世nG吨○$5.和.0$一世吨在一世一世一世s你FF一世C和吨○F一世nd一种pr○C和ss$C∈一世(在)$s一种吨一世sF和一世nG$d米吨=米吨C吨⋅d在吨$.üs一世nG$5.d.0$在和C一种n在r一世吨和$米吨=∫0吨Hs⋅d在s$,$吨≥0$,F○rs○米和pr和d一世C吨一种b一世和pr○C和ss$H∈一世(在)$.吨H和n
\int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, P \text {-as., } \forall t \geq 0,
一种nd$d米吨=H吨⋅d在吨=米吨C吨⋅d在吨$,在H和r和$C吨=米吨−1H吨$.一世吨r和米一种一世ns吨○b和sH○在n○n一世和吨H一种吨$C$一世s一种pr和d一世C吨一种b一世和pr○C和ss一世n$一世(在)$.小号一世nC和$米$一世sC○n吨一世n你○你s一世吨一世spr和d一世C吨一种b一世和.小号一世nC和$H$一世s一种一世s○pr和d一世C吨一种b一世和,一世吨F○一世一世○在s吨H一种吨$C$一世spr和d一世C吨一种b一世和.吨H你s一世吨r和米一种一世ns吨○b和sH○在n○n一世和吨H一种吨
\int_{0}^{t}\left|\gamma_{s}\right|^{2} d s<\infty, P \text {-as., } \quad \forall t \geq 0 \text {. }
$$
这紧随 (1) 以及正向和连续路径的事实s∈[0,吨]↦米s(ω)有界远离零,磷-作为。在Ω.

微积分作业代写calclulus代考|Kunita-Watanabe decomposition

5.f Kunita-Watanabe 分解。上面的表示结果假设要表示的局部鞅适应由表示布朗运动产生的增强过滤。如果不是这样,我们仍然有分解结果。因此,我们将不再假设我们的过滤(F吨)是一些布朗运动产生的增强过滤在.

让我们称两个连续的局部鞅米,ñ正交(米⊥ñ)iff协变过程⟨米,ñ⟩相同为零,也就是说,如果产品米ñ是局部鞅。

如果米和ñ是平方可积鞅,则米吨ñ吨−⟨米,ñ⟩吨是鞅 (I.11.b.2)。因此米⊥ñ当且仅当产品米ñ是鞅。
现在假设米0=0要么ñ0=0, 那是,米0ñ0=0. 如果τ是一个有界的可选时间并且吨>0一个这样的数字τ≤吨, 然后米τ=和磷[米吨∣Fτ]. 由于所有条件期望运算符都是收缩的一世2(磷)(I.2.b.15)它遵循米τ∈一世2(磷). 同样地ñτ∈一世2(磷)因此米τñτ∈一世1(磷). 根据 I.9.c.4,米吨ñ吨是鞅当且仅当和磷[米τñτ]=和磷[米0ñ0]=0 ,即米τ⊥ñτ在一世2(磷), 对于所有有界可选时间τ:

5.f.0。让米,ñ是连续的平方可积鞅米0ñ0=0. 然后米⊥ñ当且仅当米τ⊥ñτ在一世2(磷), 对于所有有界可选时间τ.

特别是米⊥ñ暗示米吨⊥ñ吨在一世2(磷), 对所有人吨≥0. 如果米,ñ∈H2是一世2-有界鞅,这意味着米∞⊥ñ∞在一世2(磷), 那是,米⊥ñ在H2, 自从米吨→米∞和ñ吨→ñ∞在一世2(磷), 作为吨↑∞(最大函数的平方可积性H2-马丁格尔和支配收敛定理)。
因此,我们的连续局部鞅正交性的概念米,ñ强于 $\mathbf{H} {0}^{2}中的正交性,在H和nsp和C一世一种一世一世和和d吨○米一种r吨一世nG一种一世和sM, N \in \mathbf{H} {0}^{2}$。

让在豆角,扁豆Rd值布朗运动。回想起那个一世2(在)是一个渐进可测的希尔伯特空间,Rd有价值的过程H, 所有可预测过程的子空间H∈一世2(在)关门了,地图
H∈一世2(在)↦(H∙在))∞∈一世2(磷)是等距并且Λ2(在)是所有逐渐可测量的空间,Rd有价值的过程H这样1[0,吨]H∈一世2(在), 对所有人吨>0. 如果H∈Λ2(在)和吨>0, 然后(H∙在)吨=(1[0,吨]H⋅在)∞,因此,使用等距(0),|(H⋅在)吨|一世2(磷)2=|1[0,吨]H|一世2(在)2=和磷∫0吨|Hs|2ds
使固定⁡吨>0并设置
\mathcal{I}{T}(W)=\left{(H \bullet W){T} \mid H \in \Lambda^{2}(W) \text { 是可预测的 }\right} \subseteq L ^{2}(P)\mathcal{I}{T}(W)=\left{(H \bullet W){T} \mid H \in \Lambda^{2}(W) \text { 是可预测的 }\right} \subseteq L ^{2}(P)
如果H∈Λ2(在)然后(H⋅在)吨=(H~⋅在)∞, 在哪里H~=1[0,吨]H∈一世2(在)(2.F.0.(d))满足1[0,吨]H~=H~. 它遵循一世吨(在)是等距下的图像(0)子空间的L{T}^{2}(W)=\left{H \in L^{2}(W) \mid H=1_{[0, T]} H\right} \subseteq L^{2}( W)L{T}^{2}(W)=\left{H \in L^{2}(W) \mid H=1_{[0, T]} H\right} \subseteq L^{2}( W). 作为一世吨2(在)⊆ 一世2(在)是一个封闭的子空间(H=1[0,吨]H相当于1(吨,∞)H=0)由此得出子空间一世吨(在)⊆一世2(磷)也关闭了。让\mathcal{I}(W)=\left{H \cdot W \mid H \in \Lambda^{2}(W) \text { 是可预测的 }\right}\mathcal{I}(W)=\left{H \cdot W \mid H \in \Lambda^{2}(W) \text { 是可预测的 }\right}我们有\mathcal{I}{T}(W)=\left{I_{T} \mid I \in \mathcal{I}(W)\right}\mathcal{I}{T}(W)=\left{I_{T} \mid I \in \mathcal{I}(W)\right}. 每个积分过程一世∈一世(在)是一个在零处消失的连续平方可积鞅。如果底层过滤(F吨)是布朗运动产生的增强过滤在, 然后从 5.c 开始。3 相反,每个在零处消失的连续平方可积鞅在一世(在). 一般来说,我们有以下分解结果:

随机微积分作业代写stochastic calculus代考| REPRESENTATION OF CONTINUOUS LOCAL MARTINGALES
微积分作业代写calclulus代考

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