### 统计代写|金融统计代写financial statistics代考| Stochastic Processes in Discrete Time

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## 统计代写|金融统计代写financial statistics代考|Binomial Processes

One of the simplest stochastic processes is an ordinary random walk, a process whose increments $Z_{l}=X_{l}-X_{l-1}$ from time $t-1$ to time $t$ take exclusively the values $+1$ or $-1$. Additionally, we assume the increments to be i.i.d. and independent of the starting value $X_{0}$. Hence, the ordinary random walk can be written as:
$$X_{l}=X_{0}+\sum_{k=1}^{l} Z_{k} \quad, t=1,2, \ldots$$
$X_{0}, Z_{1}, Z_{2}, \ldots$ independent and
$$\mathrm{P}\left(Z_{k}=1\right)=p \quad, \quad \mathrm{P}\left(Z_{k}=-1\right)=1-p \quad \text { for all } k .$$
Letting the process go up by $u$ and go down by $d$, instead, we obtain a more general class of binomial processes:
$$\mathrm{P}\left(Z_{k}=u\right)=p, \quad \mathrm{P}\left(Z_{k}=-d\right)=1-p \quad \text { for all } k,$$

where $u$ and $d$ are constant ( $u=$ up, $d=$ down).
Linear interpolation of the points $\left(t, X_{l}\right)$ reflects the time evolution of the process and is called a path of an ordinary random walk. Starting in $X_{0}=a$, the process moves on the grid of points $\left(t, b_{t}\right), t=0,1,2, \ldots, b_{t}=a-t, a-t+1, \ldots, a+t$. Up to time $t, X_{L}$ can grow at most up to $a+t$ (if $Z_{1}=\ldots=Z_{l}=1$ ) or can fall at least to $a-t$ (if $Z_{1}=\ldots=Z_{l}=-1$ ). Three paths of an ordinary random walk are shown in Figs. $4.1(p=0.5), 4.2(p=0.4)$ and $4.3(p=0.6)$.

For generalized binomial processes the grid of possible paths is more complicated. The values which the process $X_{l}$ starting in $a$ can possibly take up to time $t$ are given by
$$b_{t}=a+n \cdot u-m \cdot d, \text { where } n, m \geq 0, \quad n+m=t .$$
If, from time 0 to time $t$, the process goes up $n$ times and goes down $m$ times then $X_{t}=a+n \cdot u-m \cdot d$. That is, $n$ of $t$ increments $Z_{1}, \ldots, Z_{t}$ takes the value $u$, and $m$ increments take the value $-d$. The grid of possible paths is also called a binomial tree.

## 统计代写|金融统计代写financial statistics代考|Trinomial Processes

In contrast to binomial processes, a trinomial process allows a quantity to stay constant within a given period of time. In the latter case, the increments are described by:
$$\mathrm{P}\left(Z_{k}=u\right)=p, \mathrm{P}\left(Z_{k}=-d\right)=q, \mathrm{P}\left(Z_{k}=0\right)=r=1-p-q,$$
and the process $X_{L}$ is again given by:
$$X_{l}=X_{0}+\sum_{k=1}^{t} Z_{k}$$
where $X_{0}, Z_{1}, Z_{2}, \ldots$ are mutually independent. To solve the Black-Scholes equation, some algorithms use trinomial schemes with time and state dependent probabilities $p, q$ and $r$. Figure $4.5$ shows five simulated paths of a trinomial process with $u=d=1$ and $p=q=0.25$.

Fig. 4.5 Five paths of a trinomial process with $p=q=0.25 .(2 \sigma)$-Intervals around the trend (which is zero) are given as well

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The exact distribution of $X_{t}$ cannot be derived from the binomial distribution but for the trinomial process a similar relations hold:
\begin{aligned} \mathrm{E}\left[X_{l}\right] &=\mathrm{E}\left[X_{0}\right]+t \cdot \mathrm{E}\left[Z_{1}\right]=\mathrm{E}\left[X_{0}\right]+t \cdot(p u-q d) \ \operatorname{Var}\left(X_{l}\right) &=\operatorname{Var}\left(X_{0}\right)+t \cdot \operatorname{Var}\left(Z_{1}\right), \text { where } \ \operatorname{Var}\left(Z_{1}\right) &=p(1-p) u^{2}+q(1-q) d^{2}+2 p q u d \end{aligned}
For large $t, X_{l}$ is approximately $\mathrm{N}\left{\mathrm{E}\left[X_{l}\right], \operatorname{Var}\left(X_{l}\right)\right}$-distributed.

## 统计代写|金融统计代写financial statistics代考|General Random Walks

Binomial and trinomial processes are simple examples for general random walks, i.e. stochastic processes $\left{X_{l} ; t \geq 0\right}$ satisfying:
$$X_{t}=X_{0}+\sum_{k=1}^{t} Z_{k}, \quad t=1,2, \ldots$$
where $X_{0}$ is independent of $Z_{1}, Z_{2}, \ldots$ which are i.i.d. The increments have a distribution of a real valued random variable. $Z_{k}$ can take a finite or countably infinite number of values; but it is also possible for $Z_{k}$ to take values out of a continuous set.

As an example, consider a Gaussian random walk with $X_{0}=0$, where the finitely many $X_{1}, \ldots, X_{l}$ are jointly normally distributed. Such a random walk can be constructed by assuming identically, independently and normally distributed increments. By the properties of the normal distribution, it follows that $X_{l}$ is $\mathrm{N}\left(\mu t, \sigma^{2} t\right)$-distributed for each $t$. If $X_{0}=0$ and $\operatorname{Var}\left(Z_{1}\right)$ is finite, it holds approximately for all random walks for $t$ large enough:
$$\mathcal{L}\left(X_{t}\right) \approx \mathrm{N}\left(t \cdot \mathrm{E}\left[Z_{1}\right], t \cdot \operatorname{Var}\left(Z_{1}\right)\right)$$
This result follows directly from the central limit theorem for i.i.d. random variables.
Random walks are processes with independent increments. That means, the increment $Z_{t+1}$ of the process from time $t$ to time $t+1$ is independent of the past values $X_{0}, \ldots, X_{t}$ up to time $t$. In general, it holds for any $s>0$ that the increment of the process from time $t$ to time $t+s$
$$X_{t+s}-X_{t}=Z_{t+1}+\ldots+Z_{t+s}$$
is independent of $X_{0}, \ldots, X_{t}$. It follows that the best prediction, in terms of mean squared error, for $X_{t+1}$ given $X_{0}, \ldots, X_{t}$ is just $X_{t}+\mathrm{E}\left[Z_{t+1}\right]$. As long as the price of only one stock is considered, this prediction rule works quite well. As already as 100 years ago, Bachelier postulated (assuming $\mathrm{E}\left[Z_{k}\right]=0$ for all $k$ ): “The best prediction for the stock price of tomorrow is the price of today.”

Processes with independent increments are also Markov-processes. In other words, the future evolution of the process in time $t$ depends exclusively on $X_{L}$, and the value of $X_{t}$ is independent of the past values $X_{0}, \ldots, X_{t-1}$. If the increments $Z_{k}$ and the starting value $X_{0}$, and hence all $X_{t}$, can take a finite or countably infinite number of values, then the Markov-property is formally expressed by:
$$\begin{gathered} \mathrm{P}\left(a_{t+1}<X_{t+1}<b_{t+1} \mid X_{l}=c, a_{t-1}<X_{t-1}<b_{t-1}, \ldots, a_{0}<X_{0}<b_{0}\right) \ =\mathrm{P}\left(a_{t+1}<X_{t+1}<b_{t+1} \mid X_{t}=c\right) \end{gathered}$$
If $X_{t}=c$ is known, additional information about $X_{0}, \ldots, X_{l-1}$ does not influence the opinion about the range in which $X_{I}$ will probably fall.

## 统计代写|金融统计代写financial statistics代考|Binomial Processes

Xl=X0+∑ķ=1l从ķ,吨=1,2,…
X0,从1,从2,…独立和

b吨=一种+n⋅在−米⋅d, 在哪里 n,米≥0,n+米=吨.

Xl=X0+∑ķ=1吨从ķ

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## 统计代写|金融统计代写financial statistics代考|General Random Walks

X吨=X0+∑ķ=1吨从ķ,吨=1,2,…

X吨+s−X吨=从吨+1+…+从吨+s

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