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

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• 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

## 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:

## 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.

5.e.1。正布朗鞅米可以表示为
$$M_{t}=\mathcal{E} {t}(\gamma \cdot W)=\exp \left(\int {0}^{t} \gamma_{s} \cdot d W_{s}-\frac{1}{2} \int_{0}^{t}\left|\gamma_{s}\right|^{2} ds\right), \quad t \geq 0, F○rs○米和pr和d一世C吨一种b一世和,Rd−v一种一世你和dpr○C和ssC∈一世(在).磷r○○F.一种CC○rd一世nG吨○5.C.4,米H一种s一种C○n吨一世n你○你sv和rs一世○n一种nd在和米一种和吨H和r和F○r和一种ss你米和吨H一种吨米一世s一世吨s和一世FC○n吨一世n你○你s.一种CC○rd一世nG吨○5.和.0一世吨在一世一世一世s你FF一世C和吨○F一世nd一种pr○C和ssC∈一世(在)s一种吨一世sF和一世nGd米吨=米吨C吨⋅d在吨.üs一世nG5.d.0在和C一种n在r一世吨和米吨=∫0吨Hs⋅d在s,吨≥0,F○rs○米和pr和d一世C吨一种b一世和pr○C和ssH∈一世(在).吨H和n \int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, P \text {-as., } \forall t \geq 0, 一种ndd米吨=H吨⋅d在吨=米吨C吨⋅d在吨,在H和r和C吨=米吨−1H吨.一世吨r和米一种一世ns吨○b和sH○在n○n一世和吨H一种吨C一世s一种pr和d一世C吨一种b一世和pr○C和ss一世n一世(在).小号一世nC和米一世sC○n吨一世n你○你s一世吨一世spr和d一世C吨一种b一世和.小号一世nC和H一世s一种一世s○pr和d一世C吨一种b一世和,一世吨F○一世一世○在s吨H一种吨C一世spr和d一世C吨一种b一世和.吨H你s一世吨r和米一种一世ns吨○b和sH○在n○n一世和吨H一种吨 \int_{0}^{t}\left|\gamma_{s}\right|^{2} d s<\infty, P \text {-as., } \quad \forall t \geq 0 \text {. }$$

## 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.

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.

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

\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)