# 金融代写|风险理论代写Risk theory代考|STAT553

## 金融代写|风险理论代写Risk theory代考|Exponential Families and Change of Measure

Now let $X_1, X_2, \ldots$ be i.i.d. r.v.s with common distribution $F$ and c.g.f. $\kappa(\cdot)$. For each $\theta \in \Theta={\theta \in \mathbb{R}: \kappa(\theta)<\infty}$, we denote by $F_\theta$ the probability distribution with density $\mathrm{e}^{\theta x-\kappa(\theta)}$ w.r.t. $F$. In standard statistical terminology, $\left(F_\theta\right){\theta \in \Theta}$ is the exponential family generated by $F$ (it was encountered already in Sect. II.2). Similarly, $\mathbb{P}\theta$ denotes the probability measure w.r.t. which $X_1, X_2, \ldots$ are i.i.d. with common distribution $F_\theta$.
Proposition 3.4 Let $\kappa_\theta(\alpha)=\log \mathbb{E}\theta \mathrm{e}^{\alpha X_1}$ be the c.g.f. of $F\theta$. Then
$$\kappa_\theta(\alpha)=\kappa(\alpha+\theta)-\kappa(\theta), \mathbb{E}\theta X_1=\kappa^{\prime}(\theta), \operatorname{Var}\theta X_1=\kappa^{\prime \prime}(\theta) .$$
Proof The formula for $\kappa_\theta(\alpha)$ follows from
$$\mathrm{e}^{\kappa_\theta(\alpha)}=\int_{-\infty}^{\infty} \mathrm{e}^{\alpha x} F_\theta(\mathrm{d} x)=\int_{-\infty}^{\infty} \mathrm{e}^{(\alpha+\theta) x-\kappa(\theta)} F(\mathrm{~d} x)=\mathrm{e}^{\kappa(\alpha+\theta)-\kappa(\theta)} .$$
We then get $\mathbb{E}\theta X_1=\kappa\theta^{\prime}(0)=\kappa^{\prime}(\theta), \operatorname{Var}\theta X_1=\kappa\theta^{\prime \prime}(0)=\kappa^{\prime \prime}(\theta)$.
Example $3.5$ Let $F$ be the normal distribution with mean $\mu$ and variance $\sigma^2$. Then $\Theta=\mathbb{R}, \kappa(\alpha)=\mu \alpha+\sigma^2 \alpha^2 / 2$ so that
$$\kappa_\theta(\alpha)=\mu(\alpha+\theta)+\sigma^2(\alpha+\theta)^2 / 2-\mu \theta-\sigma^2 \theta^2 / 2=\left(\mu+\theta \sigma^2\right) \alpha+\sigma^2 \alpha^2 / 2,$$
which shows that $F_\theta$ is the normal distribution with mean $\mu+\theta \sigma^2$ and the same variance $\sigma^2$.

## 金融代写|风险理论代写Risk theory代考|The Chernoff Bound and the Saddlepoint Approximation

Recall the definition of the Legendre-Fenchel transform $\kappa^(\cdot)$ from Sect. 3.2. One has $\kappa^(x)=\theta(x) x-\kappa(\theta(x))$, where $\theta(x)$ is the solution of
$$x=\kappa^{\prime}(\theta(x))=\mathbb{E}{\theta(x)} X_1$$ whenever $x$ is an interior point of the interval where $\kappa^(x)<\infty$. Theorem $3.7$ Let $x>\kappa^{\prime}(0)=\mathbb{E} X_1$ and assume that $(3.8)$ has a solution $\theta=\theta(x)$. Then $$\begin{gathered} \mathbb{P}\left(S_n>n x\right) \leq \mathrm{e}^{-n \kappa^(x)} \ \frac{1}{n} \log \mathbb{P}\left(S_n>n x\right) \rightarrow-\kappa^(x), n \rightarrow \infty \ \mathbb{P}\left(S_n>n x\right) \sim \frac{1}{\theta \sqrt{2 \pi \sigma\theta^2 n}} \mathrm{e}^{-n \kappa^(x)}, n \rightarrow \infty \end{gathered}$$
provided in addition for $(3.10)$ that $\sigma_\theta^2=\kappa^{\prime \prime}(\theta)<\infty$ and for $(3.11)$ that $\left|\kappa^{\prime \prime \prime}(\theta)\right|<$ $\infty$ and that $F$ satisfies Cramér’s condition $(C)$.

Condition $(C)$ is an analytical regularity condition on the characteristic function of $F$ and states that $\lim _{s \rightarrow \pm \infty}\left|\mathbb{E} e^{i s X}\right|=0$. Before giving the proof, we add some remarks.

Remark $3.8$ The inequality (3.9) goes under the name of the Chernoff bound, (3.11) is the saddlepoint approximation, and (3.10) (and some extensions to $\mathbb{P}\left(S_n \in A\right)$ for more general sets than half-lines) is associated with the name of Cramér. See further Sect. XIII.4.1.

## 金融代写|风险理论代写Risk theory代考|Exponential Families and Change of Measure

$$\kappa_\theta(\alpha)=\kappa(\alpha+\theta)-\kappa(\theta), \mathbb{E} \theta X_1=\kappa^{\prime}(\theta), \operatorname{Var} \theta X_1=\kappa^{\prime \prime}(\theta) .$$

$$\mathrm{e}^{\kappa_\theta(\alpha)}=\int_{-\infty}^{\infty} \mathrm{e}^{\alpha x} F_\theta(\mathrm{d} x)=\int_{-\infty}^{\infty} \mathrm{e}^{(\alpha+\theta) x-\kappa(\theta)} F(\mathrm{~d} x)=\mathrm{e}^{\kappa(\alpha+\theta)-\kappa(\theta)} .$$

$$\kappa_\theta(\alpha)=\mu(\alpha+\theta)+\sigma^2(\alpha+\theta)^2 / 2-\mu \theta-\sigma^2 \theta^2 / 2=\left(\mu+\theta \sigma^2\right) \alpha+\sigma^2 \alpha^2 / 2$$

## 金融代写|风险理论代写Risk theory代考|The Chernoff Bound and the Saddlepoint Approximation

$$x=\kappa^{\prime}(\theta(x))=\mathbb{E} \theta(x) X_1$$

$$\left.\mathbb{P}\left(S_n>n x\right) \leq \mathrm{e}^{-n \kappa(x)} \frac{1}{n} \log \mathbb{P}\left(S_n>n x\right) \rightarrow-\kappa^{(} x\right), n \rightarrow \infty \mathbb{P}\left(S_n>n x\right) \sim \frac{1}{\theta \sqrt{2 \pi \sigma \theta^2 n}} \mathrm{e}^{\left.-n \kappa^{(} x\right)}, n \rightarrow \infty$$

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