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# 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|STAT4021 Estimation

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## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Estimation

The problem now is: suppose the counting process $Y$, or equivalently the jump times $\tau_{1}, \tau_{2}, \ldots$, are observed. We wish to estimate the state of $Z$ and the parameters in $\alpha=\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{N}\right)^{\prime}$
A Filter
We shall use a ‘reference probability’ $\bar{P}$. Suppose that under $\bar{P}$
1) $Z$ is a Markov chain with rate matrix $A$
2) $Y$ is a counting process with compensation $\lambda t$.
Then as in Section $1, \bar{Q}{t}=Y{t}-\lambda t$ is a $\bar{P}$ martingale.
Definition 1 Write
$$\Lambda_{t}=1+\int_{0}^{t} \Lambda_{s-}\left(\frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda}-1\right)\left(d Y_{s}-\lambda d s\right)$$

SO:
$$\Lambda_{t}=\exp \left(-\int_{0}^{t}\left(\frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda}-1\right) \lambda d s+\int_{0}^{t} \log \frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda} d Y_{s}\right) .$$
Note that $\Lambda$ is a $(\bar{P}, \mathcal{G})$ martingale. Define a probability measure $P$ by
$$\left.\frac{d P}{d \bar{P}}\right|{G{t}}=\Lambda_{t} .$$

## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Parameter Estimation

We have seen that to change the rate from $\lambda$ to $\left\langle\alpha, Z_{t}\right\rangle$ the Girsanov density given by (1) is used.
Write this density as $\Lambda_{t}^{\alpha}$ and the related probability $P^{\alpha}$.
Suppose there is a second possible set of parameter values
$$\alpha^{\prime}=\left(\alpha_{1}^{\prime}, \alpha_{2}^{0}, \ldots, \alpha_{N}^{\prime}\right)^{\prime} \in R^{N}$$
giving a related probability $P^{\alpha^{\prime}}$.
Then the Girsanov density $\frac{\Lambda_{t}^{\alpha}}{\Lambda_{t}^{\alpha^{\prime}}}$ will change the probability $P^{\alpha^{\prime}}$ to $P^{\alpha}$ and the compensator of $Y$ from $\left\langle\alpha^{\prime}, Z_{t}\right\rangle$ to $\left\langle\alpha, Z_{t}\right\rangle$.
Suppose the model has been implemented with a parameter set
$$\left{A=\left(a_{j i}\right), \alpha^{\prime}=\left(\alpha_{1}^{\prime}, \ldots, \alpha_{N}^{\prime}\right)^{\prime}\right} .$$
Given the observations of $Y$ we wish to re-estimate the parameters in $\alpha^{\prime}$. The conditional expectation of the log-likelihood to change parameters $\alpha^{\prime}$ to $\alpha$ is
$$E\left[\log \frac{\Lambda_{t}^{\alpha}}{\Lambda_{t}^{\alpha^{\prime}}} \mid y_{t}\right]=E\left[-\int_{0}^{t}\left(\frac{\left\langle\alpha, Z_{s}\right\rangle}{\lambda}-1\right) \lambda d s+\int_{0}^{t} \log \frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda} d Y_{s} \mid y_{t}\right]+R$$
where $R$ represents terms which do not depend on $\alpha$.
In turn, this is

$$\begin{gathered} =E\left[-\int_{0}^{t}\left\langle\alpha, Z_{s}\right\rangle+\int_{0}^{t} \log \left\langle\alpha, Z_{s-}\right\rangle, d Y_{s} \mid y_{t}\right] \ +\text { terms which do not depend on } \alpha \end{gathered}$$
Write $J_{t}^{i}=\int_{0}^{t}\left\langle e_{i}, Z_{s}\right\rangle d s$ for the amount of time $Z$ has spent in state $e_{i}$ upto time $t$.
Also,note
$$\int_{0}^{t} \log \left\langle\alpha, Z_{s-}\right\rangle d Y_{s}=\sum_{i=1}^{N} \log \alpha_{i} \int_{0}^{t}\left\langle e_{i}, Z_{s-}\right\rangle d Y_{s}$$
so
$$E\left[\log \frac{\Lambda_{t}^{\alpha}}{\lambda_{t}^{\alpha^{\prime}}} \mid y_{t}\right]=E\left[-\sum_{i=1}^{N} \alpha_{i} J_{t}^{i}+\sum_{i=1}^{N} \log \alpha_{i} \int_{0}^{t}\left\langle e_{i}, Z_{s-}\right\rangle d Y_{s} \mid y_{t}\right]$$
$+$ terms which do not depend on $\alpha$.

## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Estimation

1) $Z$ 是具有速率矩阵的马尔可夫链 $A$
2) $Y$ 是一个有补偿的计数过程 $\lambda t$.

$$\Lambda_{t}=1+\int_{0}^{t} \Lambda_{s-}\left(\frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda}-1\right)\left(d Y_{s}-\lambda d s\right)$$
A所L:
$$\Lambda_{t}=\exp \left(-\int_{0}^{t}\left(\frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda}-1\right) \lambda d s+\int_{0}^{t} \log \frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda} d Y_{s}\right) .$$

\$\$
$\backslash$ left. $\backslash$ frac ${d P}{d \backslash$ bar ${P}} \backslash$ right $\mid{G{t}}=\backslash$ Lambda_ ${t}$ 。
$\$ \$$## 金融代写|随机分析代写STOCHASTIC ANALYSIS代考|Parameter Estimation 我们已经看到，从 \lambda 至 \left\langle\alpha, Z_{t}\right\rangle 使用由 (1) 给出的 Girsanov 密度。 将此密度写为 \Lambda_{t}^{\alpha} 和相关的概率 P^{\alpha}. 假设有第二组可能的参数值$$
\alpha^{\prime}=\left(\alpha_{1}^{\prime}, \alpha_{2}^{0}, \ldots, \alpha_{N}^{\prime}\right)^{\prime} \in R^{N}
$$给出相关嘅率 P^{\alpha^{\prime}}. 假设模型已经使用参数集实现 \left 的分隔符缺失或无法识别 鉴于观覍到 Y 我们希望重新估计参数 \alpha^{\prime}. 改变参数的对数似然的条件期望 \alpha^{\prime} 至 \alpha 是$$
E\left[\log \frac{\Lambda_{t}^{\alpha}}{\Lambda_{t}^{\alpha^{\alpha}}} \mid y_{t}\right]=E\left[-\int_{0}^{t}\left(\frac{\left\langle\alpha, Z_{s}\right\rangle}{\lambda}-1\right) \lambda d s+\int_{0}^{t} \log \frac{\left\langle\alpha, Z_{s-}\right\rangle}{\lambda} d Y_{s} \mid y_{t}\right]+R
$$在䂙里 R 表示不依赖于的项 \alpha. 反过来，这是$$
=E\left[-\int_{0}^{t}\left\langle\alpha, Z_{s}\right\rangle+\int_{0}^{t} \log \left\langle\alpha, Z_{s-}\right\rangle, d Y_{s} \mid y_{t}\right]+\text { terms which do not depend on } \alpha
$$写 J_{t}^{i}=\int_{0}^{t}\left\langle e_{i}, Z_{s}\right\rangle d s 在一段时间内 Z 已经在州度过 e_{i} 及时 t. 另外，请注意$$
\int_{0}^{t} \log \left\langle\alpha, Z_{s-}\right\rangle d Y_{s}=\sum_{i=1}^{N} \log \alpha_{i} \int_{0}^{t}\left\langle e_{i}, Z_{s_{-}}\right\rangle d Y_{s}
$$所以$$
E\left[\log \frac{\Lambda_{t}^{\alpha}}{\lambda_{t}^{\alpha^{\prime}}} \mid y_{t}\right]=E\left[-\sum_{i=1}^{N} \alpha_{i} J_{t}^{i}+\sum_{i=1}^{N} \log \alpha_{i} \int_{0}^{t}\left\langle e_{i}, Z_{s-}\right\rangle d Y_{s} \mid y_{t}\right]

$+$ 不依赖于的术语 $\alpha$.

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