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# 数学代写|随机过程Stochastic Porcess代考|The Hille-Yosida theorem and positivity

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## 数学代写|随机过程代写Stochastic Porcess代考|The Hille-Yosida theorem and positivity

We will now discuss the structure of generators of strongly continuous contraction semigroups and Feller semigroups.

7.16 Theorem (Hille 1948; Yosida 1948; Lumer-Phillips 1961). A linear operator $(A, \mathfrak{D}(A))$ on a Banach space $(\mathfrak{B},|\cdot|)$ is the infinitesimal generator of a strongly continuous contraction semigroup if, and only if,
a) $\mathfrak{D}(A)$ is a dense subset of $\mathfrak{B}$;
b) $A$ is dissipative, i.e. $\forall \lambda>0, u \in \mathfrak{D}(A):|\lambda u-A u| \geqslant \lambda|u|$;
c) $\Re(\lambda$ id $-A)=\mathfrak{B}$ for some (or all) $\lambda>0$.
We do not give a proof of this theorem but refer the interested reader to Ethier-Kurtz [61] and Pazy [142]. In fact, Theorem 7.16 is too general for transition semigroups of stochastic processes. Let us now show which role is played by the positivity and subMarkovian property of such semigroups. To keep things simple, we restrict ourselves to the Banach space $\mathfrak{B}=\mathcal{C}_{\infty}$ and Feller semigroups.

7.17 Lemma. Let $\left(P_t\right){t \geqslant 0}$ be a Feller semigroup with generator $(A, \mathfrak{D}(A))$. Then $A$ satisfies the positive maximum principle (PMP) $$u \in \mathfrak{D}(A), u\left(x_0\right)=\sup {x \in \mathbb{R}^d} u(x) \geqslant 0 \Longrightarrow A u\left(x_0\right) \leqslant 0$$
For the Laplace operator (7.18) is quite familiar: At a (global) maximum the second derivative is negative.

Proof. Let $u \in \mathfrak{D}(A)$ and assume that $u\left(x_0\right)=\sup u \geqslant 0$. Since $P_t$ preserves positivity, we get $u^{+}-u \geqslant 0$ and $P_t\left(u^{+}-u\right) \geqslant 0$ or $P_t u^{+} \geqslant P_t u$. Since $u\left(x_0\right)$ is positive, $u\left(x_0\right)=u^{+}\left(x_0\right)$ and so
$$\frac{1}{t}\left(P_t u\left(x_0\right)-u\left(x_0\right)\right) \leqslant \frac{1}{t}\left(P_t u^{+}\left(x_0\right)-u^{+}\left(x_0\right)\right) \leqslant \frac{1}{t}\left(\left|u^{+}\right|_{\infty}-u^{+}\left(x_0\right)\right)=0 .$$

## 数学代写|随机过程代写Stochastic Porcess代考|Dynkin’s characteristic operator

We will now give a probabilistic characterization of the infinitesimal generator. As so often, a simple martingale relation turns out to be extremely helpful. The following theorem should be compared with Theorem 5.6. Recall that a Feller process is a (strong ${ }^2$ ) Markov process with right-continuous trajectories whose transition semigroup $\left(P_t\right){t \geqslant 0}$ is a Feller semigroup. Of course, a Brownian motion is a Feller process. 7.21 Theorem. Let $\left(X_t, \mathcal{F}_t\right){t \geqslant 0}$ be a Feller process on $\mathbb{R}^d$ with transition semigroup $\left(P_t\right)_{t \geqslant 0}$ and generator $(A, \mathfrak{D}(A))$. Then
$$M_t^u:=u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r \text { for all } u \in \mathfrak{D}(A)$$
is an $\mathcal{F}_t$ martingale.

Proof. Let $u \in \mathfrak{D}(A), x \in \mathbb{R}^d$ and $s, t>0$. By the Markov property (6.4c),
\begin{aligned} \mathbb{E}^x & \left(M_{s+t}^u \mid \mathcal{F}t\right) \ & =\mathbb{E}^x\left(u\left(X{s+t}\right)-u(x)-\int_0^{s+t} A u\left(X_r\right) d r \mid \mathcal{F}_t\right) \ & =\mathbb{E}^{X_t} u\left(X_s\right)-u(x)-\int_0^t A u\left(X_r\right) d r-\mathbb{E}^x\left(\int_t^{s+t} A u\left(X_r\right) d r \mid \mathcal{F}_t\right) \ & =P_s u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r-\mathbb{E}^{X_t}\left(\int_0^s A u\left(X_r\right) d r\right) \ & =P_s u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r-\int_0^s \mathbb{E}^{X_t}\left(A u\left(X_r\right)\right) d r \ & =P_s u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r-\int_0^s P_r A u\left(X_t\right) d r \ & =u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r=M_t^u . \end{aligned}

## 数学代写|随机过程代写Stochastic Porcess代考|The Hille-Yosida theorem and positivity

7.16 定理 (Hille 1948；Yosida 1948；Lumer-Phillips 1961)。线性算子 $(A, \mathfrak{D}(A))$ 在巴拿赫空间 $(\mathfrak{B},|\cdot|)$ 是强连续收缩半群 的无穾小生成元当且仅当
a) $\mathfrak{D}(A)$ 是的稠密子集 $\mathfrak{B}$ ；

C) $\mathfrak{R}(\lambda \mid \mathrm{D}-A)=\mathfrak{B}$ 对于一些（或全部) $\lambda>0$.

## 数学代写|随机过程代写Stochastic Porcess代考|Dynkin’s characteristic operator

$$M_t^u:=u\left(X_t\right)-u(x)-\int_0^t A u\left(X_r\right) d r \text { for all } u \in \mathfrak{D}(A)$$

## MATLAB代写

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