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数学代写|随机过程Stochastic Porcess代考|Definitions. General Properties

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数学代写|随机过程代写Stochastic Porcess代考|Definitions. General Properties

Let $\mathscr{X}$ be a linear space with $\sigma$-algebra $\mathfrak{B}$ possessing the following properties: a) for all $x \in \mathscr{X}$ and $A \in \mathfrak{B}$ the set $A_x={y: y-x \in A}$ belongs to $\mathfrak{B}$ (i. e. $\sigma$-algebra $\mathfrak{B}$ is such that all the shifts in $\mathscr{X}$ are measurable with respect to $\mathfrak{B}) ;$ b) for any $A \in \mathfrak{B}$ the set ${(x+y): x+y \in A}$ is $\mathfrak{B} \times \mathfrak{B}$-measurable in $\mathscr{X} \times \mathscr{X}$.

A process $\xi(t)$ defined on a set $T \subset \mathscr{R}$ and taking values in $\mathscr{X}$ is called a process with independent increments if, for all $t_0<t_1<\cdots<t_n$ belonging to $T$, the random variables $\xi\left(t_0\right), \xi\left(t_1\right)-\xi\left(t_0\right), \ldots, \xi\left(t_n\right)-\xi\left(t_{n-1}\right)$ are independent. Conditions imposed on $\mathfrak{B}$ assure that $\xi(t)-\xi\left(t_1\right)$ is a random variable.

Marginal distributions of a process with independent increments are determined by the one-dimensional distributions and the distributions of the increments of the process. Let
$$\mu_t(A)=\mathrm{P}{\xi(t) \in A}, \quad \Phi_{t_1, t_2}(A)=\mathrm{P}\left{\xi\left(t_2\right)-\xi\left(t_1\right) \in A\right}$$
Then
$$\mathrm{P}\left{\xi\left(t_0\right) \in A_0, \xi\left(t_1\right) \in A_1, \ldots, \xi\left(t_n\right) \in A_n\right}=\int \cdots \int \mu_{t_0}\left(d x_0\right) \Phi_{t_0, t_1}\left(d x_1\right), \ldots, \Phi_{t_{n-1}, t_n}\left(d x_n\right),$$
where integration is carried out over the set
$$\left{\left(x_0, \ldots, x_n\right): x_0 \in A_0, x_0+x_1 \in A_1, \ldots, x_0+\cdots+x_n \in A_n\right}$$

数学代写|随机过程代写Stochastic Porcess代考|One-dimensional processes with independent increments

One-dimensional processes with independent increments. Note that for any $\mathfrak{B}$-measurable linear functional $l(x)$ the process $\eta(t)=l(\xi(t))$ will be a onedimensional process with independent increments. Therefore a study of onedimensional processes with independent increments may supply (and, as we shall see in the sequel, indeed does supply) non-trivial information about processes in more complex spaces. On the other hand, the space $\mathscr{R}^1$ is the simplest linear space so that processes in this space are also the simplest in a certain sense.
Thus we shall consider a process $\xi(t)$ taking on real values; the $\sigma$-algebra of all Borel sets on $\mathscr{R}^1$ will serve as the $\sigma$-algebra $\mathfrak{B}$. We shall assume that the process is defined on a set $T$.

Let $\varphi_t(\lambda)$ and $\varphi_{t_1, t_2}(\lambda)$ be characteristic functions of $\xi(t)$ and $\xi\left(t_2\right)-\xi\left(t_1\right)$ respectively. The functions $h_t(\lambda)=\left|\varphi_t(\lambda)\right|^2$ and $h_{t_1, t_2}(\lambda)=\left|\varphi_{t_1, t_2}(\lambda)\right|^2$ are non-negative and, moreover, $h_{t_1, t_2}(\lambda) \leqslant 1$. It follows from the relation
$$h_s(\lambda)=h_t(\lambda) h_{t, s}(\lambda), \quad t<s$$

that $h_s(\lambda)$ is a monotonically non-increasing bounded function of $s$. Consequently the limits $h_{t-0}(\lambda)$ and $h_{t+0}(\lambda)$ exist for $t$ belonging to the closure of $T$ (or only one limit exists if $t$ is a one-sided limit point; these limits are not defined for isolated points).

Consider in addition to $\xi(t)$ a process $\tilde{\xi}(t)$ with the same finite-dimensional distributions as those of $\xi(t)$ but independent of $\xi(t)$. To construct such a process, we take two copies of the same probability space and view the process $\xi(t)$ on the first space and the identical process $\tilde{\xi}(t)$ on the second as processes on the product of these probability spaces. Next set $\xi^(t)=\xi(t)-\tilde{\xi}(t)$. It is easy to see that $$\mathrm{E} e^{i \lambda \xi^(t)}=h_t(\lambda), \quad \mathrm{E} e^{i \lambda\left[\xi^\left(t_2\right)-\xi^\left(t_1\right)\right]}=h_{t_1, t_2}(\lambda)$$

数学代写|随机过程代写Stochastic Porcess代考|Definitions. General Properties

$$\mu_t(A)=\mathrm{P}{\xi(t) \in A}, \quad \Phi_{t_1, t_2}(A)=\mathrm{P}\left{\xi\left(t_2\right)-\xi\left(t_1\right) \in A\right}$$

$$\mathrm{P}\left{\xi\left(t_0\right) \in A_0, \xi\left(t_1\right) \in A_1, \ldots, \xi\left(t_n\right) \in A_n\right}=\int \cdots \int \mu_{t_0}\left(d x_0\right) \Phi_{t_0, t_1}\left(d x_1\right), \ldots, \Phi_{t_{n-1}, t_n}\left(d x_n\right),$$

$$\left{\left(x_0, \ldots, x_n\right): x_0 \in A_0, x_0+x_1 \in A_1, \ldots, x_0+\cdots+x_n \in A_n\right}$$

数学代写|随机过程代写Stochastic Porcess代考|One-dimensional processes with independent increments

$$h_s(\lambda)=h_t(\lambda) h_{t, s}(\lambda), \quad t<s$$

MATLAB代写

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