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# 数学代写|随机过程Stochastic Porcesses代考|ENAS496 Integrated Brownian motion

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## 数学代写|随机过程Stochastic Porcesses代考|Integrated Brownian motion

Definition 4.2.4. Let ${Y(t), t \geq 0}$ be a Brownian motion with drift coefficient $\mu$ and diffusion coefficient $\sigma^2$, and let
$$Z(t):=Z(0)+\int_0^t Y(s) d s$$
The stochastic process ${Z(t), t \geq 0}$ is called an integrated Brownian motion.

Proposition 4.2.2. The integrated Brownian motion is a Gaussian process.
Proof. First, we use the definition of an integral as the limit of a sum:
$$Z(t)=Z(0)+\lim {n \rightarrow \infty} \frac{t}{n} \sum{k=1}^n Y\left(\frac{t k}{n}\right)$$
Since the Wiener process is a Gaussian process, the $Y(t k / n)$ ‘s are Gaussian random variables. From this, it can be shown that the variable $Z(t)$ has a Gaussian distribution and also that the random vector $\left(Z\left(t_1\right), Z\left(t_2\right), \ldots\right.$, $\left.Z\left(t_n\right)\right)$ has a multinormal distribution, for all $t_1, t_2, \ldots, t_n$ and for any $n$, so that the process ${Z(t), t \geq 0}$ is Gaussian.

## 数学代写|随机过程Stochastic Porcesses代考|Brownian bridge

The processes that we studied so far in this chapter were all defined for values of the variable $t$ in the interval $[0, \infty)$. However, an interesting process that is based on a standard Brownian motion, ${B(t), t \geq 0}$, and that has been the subject of many research papers in the last 20 years or so, is defined for $t \in[0,1]$ only. Moreover, it is a conditional diffusion process, because we suppose that $B(1)=0$. Since it is as if the process thus obtained were tied at both ends, it is sometimes called the tied Wiener process, but most often the expression Brownian bridge is used.

Definition 4.2.5. Let ${B(t), t \geq 0}$ be a standard Brownian motion. The conditional stochastic process ${Z(t), 0 \leq t \leq 1}$, where
$$Z(t):=B(t) \mid{B(1)=0}$$
is called a Brownian bridge.
Remark. We deduce from the properties of the Brownian motion that the Brownian bridge is a Gaussian diffusion process (thus, it is also Markovian).

## 数学代写|随机过程Stochastic processes代考|Integrated brown motion

.

4.2.4.

$$Z(t):=Z(0)+\int_0^t Y(s) d s$$

$$Z(t)=Z(0)+\lim {n \rightarrow \infty} \frac{t}{n} \sum{k=1}^n Y\left(\frac{t k}{n}\right)$$

## 数学代写|随机过程Stochastic processes代考| brown bridge

. brown bridge

$$Z(t):=B(t) \mid{B(1)=0}$$

## MATLAB代写

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