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# 数学代写|偏微分方程代考Partial Differential Equations代写|STAT433 Operator-valued angle bracket process

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## 数学代写|偏微分方程代考Partial Differential Equations代写|Operator-valued angle bracket process

Let $M, N \in \mathcal{M}^2(U)$. Denote by $\langle\mathrm{M}, \mathrm{N}\rangle$ the unique predictable process, with trajectories having a locally bounded variation, for which
$$\langle M(t), N(t)\rangle_U-\langle M, N\rangle_t, \quad t \geq 0,$$
is a martingale. By the Doob-Meyer decomposition, the process $\langle M, N\rangle$ always exists (see Remark 3.46) and is called the angle bracket.

In this section we introduce the so-called operator angle bracket $\langle\langle M, N\rangle\rangle$, and in Theorem $8.2$ we will show the absolute continuity of the operator angle bracket with respect to the angle bracket. The relevant density is called the martingale covariance. It plays an important role in the construction of the stochastic integral with respect to $M$. In particular it appears in the fundamental isometric formula.
Denote by $L_1(U)$ the space of all nuclear operators on $U$ equipped with the nuclear norm; see Appendix A. Then $L_1(U)$ is a separable Banach space. Recall that, given $x, y, z \in U, x \otimes y(z)=\langle y, z\rangle_U x$. It is easy to show that $x \otimes y \in$ $L^1(U)$ and $|x \otimes y|_{L_1(U)}=|x|_U|y|_U$. We denote by $L_1^{+}(U)$ the subspace of $L_1(U)$ consisting of all self-adjoint non-negative nuclear operators. If $M \in \mathcal{M}^2(U)$ then the process $(M(t) \otimes M(t), t \geq 0)$ is an $L_1(U)$-valued right-continuous process such that
$$\mathbb{E}|M(t) \otimes M(t)|_{L_1(U)}=\mathbb{E}|M(t)|_U^2 \leq \mathbb{E}|M(T)|_U^2<\infty, \quad t \geq 0 .$$
We will need the following result.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Construction of the stochastic integral

To deal with stochastic equations one needs the concept of the stochastic integral, $I_t^M(\Psi):=\int_0^t \Psi(s) \mathrm{d} M(s)$, where $M \in \mathcal{M}^2(U)$ and $\Psi(s, \omega)$ are operators from $U$ to another Hilbert space $H$. As for real-valued martingales, first we define the stochastic integral for simple processes $\Psi$. Then, in the next section, we extend the class of integrands using the isometric formula (8.3) below. The isometric formula in the general case appeared for the first time in Métivier and Pistone (1975). We will denote by $Q$ the martingale covariance of $M$ introduced in Definition 8.3.
Definition 8.5 Let $L(U, H)$ be the Banach space of continuous linear operators from $U$ into $H$. An $L(U, H)$-valued stochastic process $\Psi$ is said to be simple if there exist a sequence of non-negative numbers $t_0=0<t_1<\cdots<t_m$, a sequence of operators $\Psi_j \in L(U, H), j=1, \ldots, m$, and a sequence of events $A_j \in \mathcal{F}{t_j}, j=$ $0, \ldots, m-1$, such that $$\Psi(s)=\sum{j=0}^{m-1} \chi_{A_j} \chi_{\left(t_j, t_{j+1}\right]}(s) \Psi_j, \quad s \geq 0 .$$
We shall denote by $\mathcal{S}:=\mathcal{S}(U, H)$ the class of all simple processes with values in $L(U, H)$. For a simple process $\Psi$, we set
$$I_t^M(\Psi):=\sum_{j=0}^{m-1} \chi_{A_j} \Psi_j\left(M\left(t_{j+1} \wedge t\right)-M\left(t_j \wedge t\right)\right), \quad t \geq 0$$
Let $L_{(H S)}(U, H)$ be the space of all Hilbert-Schmidt operators from $U$ into $H$ equipped with the Hilbert-Schmidt norm $|\cdot|_{L_{(H S)}(U, H)}$. We prove the isometric formula first for simple processes.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|Operator-valued angle bracket process

$$\langle M(t), N(t)\rangle_U-\langle M, N\rangle_t, \quad t \geq 0,$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Construction of the stochastic integral

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