数学代写|随机过程统计代写Stochastic process statistics代考|STAT3921

数学代写|随机过程统计代写Stochastic process statistics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 $\left(X_t, \mathcal{F}_t\right){t \gtrless 0}$ be a Feller process on $\mathbb{R}^d$ with transition semigroup $\left(P_i\right)_{i \geqslant 0}$ 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 \ &=P_s u\left(X_t\right)-u\left(X_t\right) \text { by }(7.11 c) \ &=u\left(X_t\right)-u(x)-\int_0^u A u\left(X_r\right) d r=M_t^u . \end{aligned}
The following result, due to Dynkin, could be obtained from Theorem $7.21$ by optional stopping, but we prefer to give an independent proof.

数学代写|随机过程统计代写Stochastic process statistics代考|The PDE connection

We want to discuss some relations between partial differential equations (PDEs) and Brownian motion. For many classical PDE problems probability theory yields concrete representation formulae for the solutions in the form of expected values of a Brownian functional. These formulae can be used to get generalized solutions of PDEs (which require less smoothness of the initial/boundary data or the boundary itself) and they are amenable to Monte-Carlo simulations. Purely probabilistic existence proofs for classical PDE problems are, however, rare: Classical solutions require smoothness, which does usually not follow from martingale methods. ${ }^1$ This explains the role of Proposition $7.3 \mathrm{~g})$ and Proposition 8.10. Let us point out that $\frac{1}{2} \Delta$ has two meanings: On the domain $\mathfrak{D}(\Delta)$, it is the generator of a Brownian motion, but it can also be seen as partial differential operator $L=\sum_{j, k=1}^d \partial_j$ which acts on all $\mathrm{C}^2$ functions. Of course, on $\mathcal{C}{\infty}^2$ both meanings coincide, and it is this observation which makes the method work. The origin of the probabilistic approach are the pioneering papers by Kakutani $[96,97]$, Kac $[90,91]$ and Doob [41]. As an illustration of the method we begin with the elementary (inhomogeneous) heat equation. In this context, classical PDE methods are certainly superior to the clumsy-looking Brownian motion machinery. Its elegance and effectiveness become obvious in the Feynman-Kac formula and the Dirichlet problem which we discuss in Sections $8.3$ and 8.4. Moreover, since many second-order differential operators generate diffusion processes, see Chapter 19, only minor changes in our proofs yield similar representation formulae for the corresponding PDE problems. A martingale relation is the key ingredient. Let $\left(B_t, \mathcal{F}_t\right){t \geqslant 0}$ be a $\mathrm{BM}^d$. Recall from Theorem $5.6$ that for $u \in \mathfrak{C}^{1,2}\left((0, \infty) \times \mathbb{R}^d\right) \cap \mathcal{C}\left([0, \infty) \times \mathbb{R}^d\right)$ satisfying
$$|u(t, x)|+\left|\frac{\partial u(t, x)}{\partial t}\right|+\sum_{j=1}^d\left|\frac{\partial u(t, x)}{\partial x_j}\right|+\sum_{j, k=1}^d\left|\frac{\partial^2 u(t, x)}{\partial x_j \partial x_k}\right| \leqslant c(t) e^{C|x|}$$
for all $t>0$ and $x \in \mathbb{R}^d$ with some constant $C>0$ and a locally bounded function $c:(0, \infty) \rightarrow[0, \infty)$, the process $M_s^u=u\left(t-s, B_s\right)-u\left(t, B_0\right)+\int_0^s\left(\frac{\partial}{\partial t}-\frac{1}{2} \Delta_x\right) u\left(t-r, B_r\right) d r, s \in[0, t)$, (8.2) is an $\mathcal{F}_s$ martingale for every measure $\mathbb{P}^x$, i. e. for every starting point $B_0=x$ of a Brownian motion.

数学代写|随机过程统计代写Stochastic process statistics代考|Dynkin’s characteristic operator

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

数学代写|随机过程统计代写Stochastic process statistics代考|The PDE connection

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