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

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• 随机控制
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• 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|Integration with respect to continuous local martingales

2.a.0. Let $T$ be an optional time. Then $\mu_{M^{T}}(\Delta)=\mu_{M}(\llbracket 0, T \rrbracket \cap \Delta)$, for each set $\Delta \in \mathcal{B} \times \mathcal{F}$ and
and so $H \in L^{2}\left(M^{T}\right) \Longleftrightarrow 1_{[0, T]} H \in L^{2}(M)$, for each measurable process $H$. In particular $L^{2}(M) \subseteq L^{2}\left(M^{T}\right) . \mathbf{I}$
Note that $\mu_{M}(\Pi)=E_{P}\left[\int_{0}^{\infty} 1 d\langle M\rangle_{s}\right]=E_{P}\left[\langle M\rangle_{\infty}\right]$. It follows that the measure $\mu_{M}$ is finite if and only if $M \in \mathbf{H}^{2}$ (I.9.c.0). In general $\mu_{M}$ is $\sigma$-finite.

To see this note that $M$ is indistinguishable from a process every path of which is continuous and we may thus assume that $M$ itself has this property. Then the reducing sequence $\left(T_{n}\right)$ of optional times given in I.8.a.5 satisfies $M^{T_{n}} \in \mathbf{H}^{2}, n \geq 1$, and $T_{n} \uparrow \infty$, at each point of $\Omega$. Consequently $\left[0, T_{n} \rrbracket \uparrow \Pi\right.$, as $n \uparrow \infty$. Moreover
$$\mu_{M}\left(\llbracket 0, T_{n} \rrbracket\right)=\mu_{M T_{n}}(\Pi)<\infty, \quad \text { for each } n \geq 1$$
Recall that $\mathbf{H}^{2}$ denotes the Hilbert space of continuous, $L^{2}$-bounded martingales $N$ with norm $|N|_{2}=\left|N_{\infty}\right|_{L^{2}(P)}$ and inner product $(I, N){\mathbf{H}^{2}}=E{P}\left[I_{\infty} N_{\infty}\right]$, where $N_{\infty}=\lim {t \uparrow \infty} N{t}$ denotes the last element of the martingale $N \in \mathbf{H}^{2}$ and $\langle N\rangle_{\infty}=\lim {t{ }{\infty}}\langle N\rangle_{t}$ is integrable (I.9.a, I.9.c.0).

Recall also from I.9.a.0 that $\mathbf{H}{0}^{2}=\left{N \in \mathbf{H}^{2} \mid N{0}=0\right} \subseteq \mathbf{H}^{2}$ is a closed subspace and hence a Hilbert space itself. $\mathrm{On} \mathrm{H}{0}^{2}$ the norm can also be written as (I.9.c.1) $$|N|{2}=E_{P}\left[\langle N\rangle_{\infty}\right]^{1 / 2}, \quad N \in \mathbf{H}_{0}^{2} .$$
2.a.1. Let $H \in L^{2}(M)$. Then $H \in L^{1}(\langle M, N\rangle)$ and hence the process $H \cdot\langle M, N\rangle$ is defined and is a continuous, bounded variation process, for all $N \in \mathbf{H}^{2}$.

Proof. Let $N \in \mathbf{H}^{2}$. Then $\langle M, N\rangle$ is a continuous bounded variation process and the increasing process $\langle N\rangle$ is integrable, that is $E_{P}\left[\langle N\rangle_{\infty}\right]<\infty$ (I.9.c.0). In particular we have $\langle N\rangle_{\infty}<\infty, P$-as. Similarly, from $H \in L^{2}(M)$ it follows that $\int_{0}^{\infty} H_{s}^{2} d\langle M\rangle_{s}<\infty, P$-as. The Kunita-Watanabe inequality now shows that
\begin{aligned} \int_{0}^{\infty}\left|H_{s}\right|\left|d\langle M, N\rangle_{s}\right| & \leq\left(\int_{0}^{\infty} H_{s}^{2} d\langle M\rangle_{s}\right)^{1 / 2}\left(\int_{0}^{\infty} 1^{2} d\langle N\rangle_{s}\right)^{1 / 2} \ &=\langle N\rangle_{\infty}^{1 / 2}\left(\int_{0}^{\infty} H_{s}^{2} d\langle M\rangle_{s}\right)^{1 / 2}<\infty, \quad P \text {-as. } \end{aligned}
Thus $H \in L^{1}(\langle M, N\rangle)$. The rest now follows from I.10.b.1.(a). I The next theorem introduces the integral process $I=H \bullet M$ for $H \in L^{2}(M)$.

微积分作业代写calclulus代考|The spaces L2(M) and Λ2(M)

2.f.2 The spaces $L^{2}(M)$ and $\Lambda^{2}(M)$. The spaces $L^{2}(M)$ and $\Lambda^{2}(M)$ of $2 . c .2$ also have vector valued analogues. For an $R^{d}$-valued continuous local martingale $M=$ $\left(M^{1}, M^{2}, \ldots, M^{d}\right)^{\prime}$, we let $L^{2}(M)$ and $\Lambda^{2}(M)$ denote the spaces of all progressively measurable $R^{d}$-valued processes $H=\left(H^{1}, H^{2}, \ldots, H^{d}\right)^{\prime}$ satisfying $H^{j} \in L^{2}\left(M^{j}\right)$ respectively $H^{j} \in \Lambda^{2}\left(M^{j}\right)$, for all $j=1, \ldots, d$. Thus $L^{2}(M)$ is the direct product $L^{2}(M)=L^{2}\left(M^{1}\right) \times L^{2}\left(M^{2}\right) \times \ldots \times L^{2}\left(M^{d}\right)$ and so is a Hilbert space with norm
$$|H|_{L^{2}(M)}^{2}=\sum_{j=1}^{d}\left|H^{j}\right|_{L^{2}\left(M^{j}\right)}^{2}=\sum_{j=1}^{d} E_{P} \int_{0}^{\infty}\left(H_{s}^{j}\right)^{2} d\left\langle M^{j}\right\rangle_{s} .$$
Similarly $\Lambda^{2}(M)$ is the direct product $\Lambda^{2}(M)=\Lambda^{2}\left(M^{1}\right) \times \Lambda^{2}\left(M^{2}\right) \times \ldots \times \Lambda^{2}\left(M^{d}\right)$ and is thus a Fréchet space. As in the one dimensional case $\Lambda^{2}(M)$ consists of all $R^{d}$-valued processes $H$ such that $1_{[0, t]} H \in L^{2}(M)$, for all $t \geq 0$, that is, all $R^{d}$-valued, progressively measurable processes $H$ such that
$$\left|1_{[0, t]} H\right|_{L^{2}(M)}^{2}=\sum_{j=1}^{d}\left|1_{[0, t]} H^{j}\right|_{L^{2}\left(M^{j}\right)}^{2}, \quad \text { for all } t>0 .$$
The subspaces of predictable processes $H$ in $L^{2}(M)$ respectively $\Lambda^{2}(M)$ are closed. We have the inclusion
$$L^{2}(M) \subseteq \Lambda^{2}(M) \subseteq L_{l o c}^{2}(M)=L(M) .$$
Recall that $H \cdot M=\sum_{j=1}^{d} H^{j} \cdot M^{j}$, that is, $\int_{0}^{t} H_{s} \cdot d M_{s}=\sum_{j=1}^{d} \int_{0}^{t} H_{s}^{j} d M_{s}^{j}$, for all $H \in L_{\text {loc }}^{2}(M)$ and $t \geq 0$. If the components $M^{j}$ of the local martingale $M$ are orthogonal in the sense $\left\langle M^{i}, M^{j}\right\rangle=0$, for all $i \neq j$, then we have the analogue of 2.c.3:
2.f.3. Let $M$ be an $R^{d}$-valued continuous local martingale satisfying $\left\langle M^{i}, M^{j}\right\rangle=0$, for all $i \neq j$. Then
(a) For $H \in \Lambda^{2}(M), H \bullet M$ is a square integrable martingale satisfying
$$\left|(H \bullet M){t}\right|{L^{2}(P)}=\left|\int_{0}^{t} H_{s} \cdot d M_{s}\right|_{L^{2}(P)}=\left|1_{[0, t]} H\right|_{L^{2}(M)} .$$
(b) If $H \in L^{2}(M)$ then $H \cdot M \in \mathbf{H}^{2}$ is an $L^{2}$-bounded martingale and the map $H \in L^{2}(M) \mapsto H \cdot M \in \mathbf{H}^{2}$ is an isometry. If $\int_{0}^{\infty} H_{s} \cdot d M_{s}=(H \cdot M){\infty}$ denotes the last element of the martingale $H \cdot M$, then $$\left|\int{0}^{\infty} H_{s} \cdot d M_{s}\right|_{L^{2}(P)}=|H|_{L^{2}(M)^{*}}$$
Proof. (a) For $t \geq 0$ set $U_{j}(t)=\left(H^{j} \bullet M^{j}\right){t}, j=1, \ldots, d$. Then $U{j}$ is a square integrable martingale with $\left|U_{j}(t)\right|_{L^{2}(P)}=\left|1_{[0, t]} H^{j}\right|_{L^{2}\left(M^{j}\right)}$ and $H \cdot M=\sum_{j=1}^{d} U_{j}$. See 2.c.3.(b). Thus $H \cdot M$ is a sum of square integrable martingales and hence itself such a martingale. Fix $t \geq 0$. Using 2.c.5, for $i \neq j$
$$E_{P}\left(U_{i}(t) U_{j}(t)\right)=E_{P}\left[\left(H^{i} \bullet M^{i}\right){t}\left(H^{j} \bullet M^{j}\right){t}\right]=E_{P}\left[\int_{0}^{t} H_{s}^{i} H_{s}^{j} d\left\langle M^{i}, M^{j}\right\rangle_{s}\right]=0 .$$

Consequently the random variables $U_{j}(t) \in L^{2}(P)$ are pairwise orthogonal and so
\begin{aligned} \left|(H \bullet M){t}\right|{L^{2}(P)}^{2} &=\left|\sum_{j=1}^{d} U_{j}(t)\right|_{L^{2}(P)}^{2}=\sum_{j=1}^{d}\left|U_{j}(t)\right|_{L^{2}(P)}^{2} \ &=\sum_{j=1}^{d}\left|1_{\llbracket 0, t]} H^{j}\right|_{L^{2}(M j)}^{2}=\left|1_{\llbracket 0, t]} H\right|_{L^{2}(M)^{*}}^{2} \end{aligned}
(b) The last element $Z_{\infty}$ of a martingale $Z \in \mathbf{H}^{2}$ satisfies $\left|Z_{\infty}\right|_{L^{2}}=\sup {t}\left|Z{t}\right|_{L^{2}}$. Thus (b) follows from (a) and the equality $\sup {t \geq 0}\left|1{[0, t]} H\right|_{L^{2}(M)}=|H|_{L^{2}(M)}$. I
2.f.4 Brownian motion. Assume that $W$ is a $d$-dimensional Brownian motion. Then $W$ is a martingale and hence $L(W)=L_{\text {loc }}^{2}(W)$. Moreover each component $W^{j}$ is a one dimensional Brownian motion and so $\left\langle W^{j}\right\rangle_{s}=s$. It follows that the space $L(W)$ of $W$-integrable processes consists exactly of all $R^{d}$-valued, progressively measurable processes $H$ which satisfy
$$\int_{0}^{t}\left|H_{s}\right|^{2} d s=\sum_{j=1}^{d} \int_{0}^{t}\left(H_{s}^{j}\right)^{2} d s<\infty, P \text {-as., } \quad \forall t \geq 0 .$$ Likewise $L^{2}(W)$ and $\Lambda^{2}(W)$ are the spaces of all $R^{d}$ valued, progressively measurable processes $H$ satisfying $$E_{P} \int_{0}^{\infty}\left|H_{s}\right|^{2} d s<\infty, \quad \text { respectively, } \quad E_{P} \int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, \forall t>0 .$$
2.f.5. Let $W$ be an $R^{d}$-valued Brownian motion. For an $R^{d}$-valued, progressively measurable process $H$ the following are equivalent:
(a) $H \in L(W)=L_{l o c}^{2}(W)$.
(b) There exist optional times $T_{n} \uparrow \infty$ such that $H^{T_{n}} \in L^{2}\left(W^{T_{n}}\right)$, for all $n \geq 1$.
(c) There exist optional times $T_{n} \uparrow \infty$ with $E_{P}\left[\int_{0}^{T_{n}}\left|H_{s}\right|^{2} d s\right]<\infty$, for all $n \geq 1$. (d) There exist optional times $T_{n} \uparrow \infty$ such that $1_{\left[0, T_{n}\right]} H \in L^{2}(W)$, for all $n \geq 1$. (e) $\int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, P$-as., for each $t>0$.
Proof. This follows easily from 2.b.0, 2.b.1 if it is observed that $\left\langle W^{j}\right\rangle_{s}=s$, for all $1 \leq j \leq d$.

微积分作业代写calclulus代考|Integration with respect to continuous local martingales

2.a.0。让吨是一个可选的时间。然后μ米吨(Δ)=μ米(\ll括号0,吨\rr括号∩Δ), 对于每组Δ∈乙×F等等
_H∈一世2(米吨)⟺1[0,吨]H∈一世2(米), 对于每个可测量的过程H. 特别是一世2(米)⊆一世2(米吨).一世

μ米(\ll括号0,吨n\rr括号)=μ米吨n(圆周率)<∞, 对于每个 n≥1

$$2.a.1。让H∈一世2(米). 然后H∈一世1(⟨米,ñ⟩)因此过程H⋅⟨米,ñ⟩被定义并且是一个连续的、有界的变化过程，对于所有ñ∈H2. 证明。让ñ∈H2. 然后⟨米,ñ⟩是一个连续有界变化过程和递增过程⟨ñ⟩是可积的，即和磷[⟨ñ⟩∞]<∞(I.9.c.0)。特别是我们有⟨ñ⟩∞<∞,磷-作为。同样，从H∈一世2(米)它遵循∫0∞Hs2d⟨米⟩s<∞,磷-作为。Kunita-Watanabe 不等式现在表明 ∫0∞|Hs||d⟨米,ñ⟩s|≤(∫0∞Hs2d⟨米⟩s)1/2(∫0∞12d⟨ñ⟩s)1/2 =⟨ñ⟩∞1/2(∫0∞Hs2d⟨米⟩s)1/2<∞,磷-作为。 因此H∈一世1(⟨米,ñ⟩). 其余部分从 I.10.b.1.(a) 开始。I 下一个定理介绍积分过程一世=H∙米为了H∈一世2(米). 微积分作业代写calclulus代考|The spaces L2(M) and Λ2(M) 2.f.2 空间一世2(米)和Λ2(米). 空间一世2(米)和Λ2(米)的2.C.2也有向量值的类似物。为Rd值连续局部鞅米= (米1,米2,…,米d)′，我们让一世2(米)和Λ2(米)表示所有逐渐可测量的空间Rd有价值的过程H=(H1,H2,…,Hd)′令人满意的Hj∈一世2(米j)分别Hj∈Λ2(米j)， 对所有人j=1,…,d. 因此一世2(米)是直接产品一世2(米)=一世2(米1)×一世2(米2)×…×一世2(米d)具有范数的希尔伯特空间也是如此 |H|一世2(米)2=∑j=1d|Hj|一世2(米j)2=∑j=1d和磷∫0∞(Hsj)2d⟨米j⟩s. 相似地Λ2(米)是直接产品Λ2(米)=Λ2(米1)×Λ2(米2)×…×Λ2(米d)因此是一个 Fréchet 空间。与一维情况一样Λ2(米)包括所有Rd有价值的过程H这样1[0,吨]H∈一世2(米)， 对所有人吨≥0， 就这些Rd- 有价值的、可逐步衡量的过程H这样 |1[0,吨]H|一世2(米)2=∑j=1d|1[0,吨]Hj|一世2(米j)2, 对所有人 吨>0. 可预测过程的子空间H在一世2(米)分别Λ2(米)关闭。我们有包容性 一世2(米)⊆Λ2(米)⊆一世一世○C2(米)=一世(米). 回想起那个H⋅米=∑j=1dHj⋅米j， 那是，∫0吨Hs⋅d米s=∑j=1d∫0吨Hsjd米sj， 对所有人H∈一世地方 2(米)和吨≥0. 如果组件米j当地鞅米在这个意义上是正交的⟨米一世,米j⟩=0， 对所有人一世≠j，那么我们就有 2.c.3 的类似物： 2.f.3。让米豆角，扁豆Rd值连续局部鞅满足⟨米一世,米j⟩=0， 对所有人一世≠j. 那么 (a) 对于H∈Λ2(米),H∙米是满足$$
\left|(H \bullet M) {t}\right|的平方可积鞅 {L^{2}(P)}=\left|\int_{0}^{t} H_{s} \cdot d M_{s}\right|_{L^{2}(P)}=\左|1_{[0, t]} H\right|_{L^{2}(M)} 。
$$(b) 如果H∈一世2(米)然后H⋅米∈H2是一个一世2-有界鞅和地图H∈一世2(米)↦H⋅米∈H2是等距。如果 \int_{0}^{\infty} H_{s} \cdot d M_{s}=(H \cdot M) {\infty}d和n○吨和s吨H和一世一种s吨和一世和米和n吨○F吨H和米一种r吨一世nG一种一世和H \cdot M,吨H和n \left|\int {0}^{\infty} H_{s} \cdot d M_{s}\right|_{L^{2}(P)}=|H|_{L^{2} (M)^{*}}$$

E_{P}\left(U_{i}(t) U_{j}(t)\right)=E_{P}\left[\left(H^{i} \bullet M^{i}\right ) {t}\left(H^{j} \bullet M^{j}\right) {t}\right]=E_{P}\left[\int_{0}^{t} H_{s}^ {i} H_{s}^{j} d\left\langle M^{i}, M^{j}\right\rangle_{s}\right]=0 。
$$因此随机变量üj(吨)∈一世2(磷)是成对正交的，所以$$
\begin{aligned}
\left|(H \bullet M) {t}\right| {L^{2}(P)}^{2} &=\left|\sum_{j=1}^{d} U_{j}(t)\right|_{L^{2}(P) }^{2}=\sum_{j=1}^{d}\left|U_{j}(t)\right|_{L^{2}(P)}^{2} \
&=\sum_ {j=1}^{d}\left|1_{\llbracket 0, t]} H^{j}\right|_{L^{2}(M j)}^{2}=\left|1_ {\llbracket 0, t]} H\right|_{L^{2}(M)^{*}}^{2}
\end{aligned}

(b) 最后一个元素和∞鞅和∈H2满足 $\left|Z_{\infty}\right|_{L^{2}}=\sup {t}\left|Z {t}\right|_{L^{2}}.吨H你s(b)F○一世一世○在sFr○米(一种)一种nd吨H和和q你一种一世一世吨和\sup {t \geq 0}\left|1 {[0, t]} H\right|_{L^{2}(M)}=|H|_{L^{2}(M)}.一世2.F.4乙r○在n一世一种n米○吨一世○n.一种ss你米和吨H一种吨在一世s一种d−d一世米和ns一世○n一种一世乙r○在n一世一种n米○吨一世○n.吨H和n在一世s一种米一种r吨一世nG一种一世和一种ndH和nC和L(W)=L_{\text {loc }}^{2}(W).米○r和○v和r和一种CHC○米p○n和n吨W^{j}一世s一种○n和d一世米和ns一世○n一种一世乙r○在n一世一种n米○吨一世○n一种nds○\left\langle W^{j}\right\rangle_{s}=s.一世吨F○一世一世○在s吨H一种吨吨H和sp一种C和长(宽)○F在−一世n吨和Gr一种b一世和pr○C和ss和sC○ns一世s吨s和X一种C吨一世和○F一种一世一世R^{d}−v一种一世你和d,pr○Gr和ss一世v和一世和米和一种s你r一种b一世和pr○C和ss和sH在H一世CHs一种吨一世sF和∫0吨|Hs|2ds=∑j=1d∫0吨(Hsj)2ds<∞,磷-作为。， ∀吨≥0.一世一世到和在一世s和L^{2}(W)一种nd\ λ ^ {2} (W)一种r和吨H和sp一种C和s○F一种一世一世R^{d}v一种一世你和d,pr○Gr和ss一世v和一世和米和一种s你r一种b一世和pr○C和ss和sHs一种吨一世sF和一世nG和磷∫0∞|Hs|2ds<∞, 分别， 和磷∫0吨|Hs|2ds<∞,∀吨>0.2.F.5.一世和吨在b和一种nR^{d}−v一种一世你和d乙r○在n一世一种n米○吨一世○n.F○r一种nR^{d}−v一种一世你和d,pr○Gr和ss一世v和一世和米和一种s你r一种b一世和pr○C和ssH吨H和F○一世一世○在一世nG一种r和和q你一世v一种一世和n吨:(一种)H \in L(W)=L_{loc}^{2}(W).(b)吨H和r和和X一世s吨○p吨一世○n一种一世吨一世米和sT_{n} \uparrow \inftys你CH吨H一种吨H^{T_{n}} \in L^{2}\left(W^{T_{n}}\right),F○r一种一世一世n \ geq 1.(C)吨H和r和和X一世s吨○p吨一世○n一种一世吨一世米和sT_{n} \uparrow \infty在一世吨HE_{P}\left[\int_{0}^{T_{n}}\left|H_{s}\right|^{2} ds\right]<\infty,F○r一种一世一世n \ geq 1.(d)吨H和r和和X一世s吨○p吨一世○n一种一世吨一世米和sT_{n} \uparrow \inftys你CH吨H一种吨1_{\left[0, T_{n}\right]} H \in L^{2}(W),F○r一种一世一世n \ geq 1.(和)\int_{0}^{t}\left|H_{s}\right|^{2} d s<\infty, P−一种s.,F○r和一种CHt>0.磷r○○F.吨H一世sF○一世一世○在s和一种s一世一世和Fr○米2.b.0,2.b.1一世F一世吨一世s○bs和rv和d吨H一种吨\left\langle W^{j}\right\rangle_{s}=s,F○r一种一世一世1 \ leq j \ leq d$。