## 金融代写|金融数值计算代写Market Risk, Numerical Analysis for Finance代考|Parlial derivalives

The most natural way to define derivatives of functions of several variables is to allow only one variable at a time to move, while freezing the others. Thus, if $f: V \rightarrow \mathbb{R}$ is a function of $n$ variables, whose domain is the open set $V$, we define the set $\left{x_1\right} \times \cdots \times\left{x_{j-1}\right} \times[a, b] \times\left{x_{j+1}\right} \times \cdots \times\left{x_n\right}$, where $[a, b]$ is chosen so to have $\left{x_1\right} \times \cdots \times\left{x_{j-1}\right} \times{t} \times\left{x_{j+1}\right} \times \cdots \times\left{x_n\right} \subset V$ for any $t \in[a, b]$. We shall denote the function:
$$g(t):=f\left(x_1, \ldots, x_{j-1}, t, x_{j+1}, \ldots, x_n\right)$$
by
$$f\left(x_1, \ldots, x_{j-1}, \cdots, x_{j+1}, \ldots, x_n\right) \text {. }$$
If $g$ is differentiable (see Definition 3.7) at some $t_0 \in(a, b)$, then the firstorder partial derivative of $f$ at $\left(x_l, \ldots, x_{j-1}, t_0, x_{j+1}, \ldots, x_n\right)$, with respect to $x_j$, is defined by:
\begin{aligned} f x_j\left(x_1, \ldots, x_{j-1}, t_0, x_{j+1}, \ldots, x_n\right) &:=\frac{\partial f}{\partial x_j}\left(x_1, \ldots, x_{j-1}, t_0, x_{j+1}, \ldots, x_n\right) \ &:=g^{\prime}\left(t_0\right) . \end{aligned}
Therefore, the partial derivative $f_{x_j}$ exists at a point $\boldsymbol{a}$ if and only if the following limit exists:
$$\frac{\partial f}{\partial x_j}(\boldsymbol{a}):=\lim _{h \rightarrow 0} \frac{f\left(\boldsymbol{a}+h \boldsymbol{e}_j\right)-f(\boldsymbol{a})}{h} .$$ Partial derivatives of order higher than one are defined by iteration.

## 金融代写|金融数值计算代写Market Risk, Numerical Analysis for Finance代考|Differentiability

In this section, we define what it means for a vector function $f$ to be differentiable at a point $\boldsymbol{a}$. Whatever our definition, if $f$ is differentiable at $\boldsymbol{a}$, then we expect two things:
(1) $f$ will be continuous at $\boldsymbol{a}$;
(2) all first-order partial derivatives of $f$ will exist at $\boldsymbol{a}$.
To appreciate the following Definition $3.7$ of total derivative of a function of $n$ variables, we consider one peculiar aspect of differentiable functions of one variable. Recall that $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable at $x \in \mathbb{R}$ if the following limit is finite, i.e., it is a real number:
$$\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h}:=f^{\prime}(x) .$$ The definition above is equivalent to the following: $f$ is differentiable at $x \in \mathbb{R}$ if there exist $\alpha \in \mathbb{R}$ and a function $\omega:(-\delta, \delta) \rightarrow \mathbb{R}$, with $\omega(0)=0$ and $\lim {h \rightarrow 0} \frac{\omega(h)}{h}=0$, such that:
$$f(x+h)=f(x)+\alpha h+\omega(h) h .$$

The definition of differentiability for functions of several variables extends Property (3.1).

Definition 3.7. Let $f$ be a real function of $n$ variables. $f$ is said to be differentiable, at a point $a \in \mathbb{R}^n$, if and only if there exists an open set $V \subset \mathbb{R}^n$, such that $\boldsymbol{a} \in V$ and $f: V \rightarrow \mathbb{R}$, and there exists $\boldsymbol{d} \in \mathbb{R}^n$ such that:
$$\lim _{\boldsymbol{h} \rightarrow 0} \frac{f(\boldsymbol{a}+\boldsymbol{h})-f(\boldsymbol{a})-\boldsymbol{d} \cdot \boldsymbol{h}}{|\boldsymbol{h}|}=0$$
$\boldsymbol{d}$ is called total derivative of $f$ at $\boldsymbol{a}$
Theorem 3.8. If $f$ is differentiable at $a$, then:
(i) $f$ is continuous at $\boldsymbol{a}$;
(ii) all first-order partial derivatives of $f$ exist at $\boldsymbol{a}$;
(iii) $\boldsymbol{d}=\nabla f(\boldsymbol{a}):=\left(\frac{\partial f}{\partial x_1}(\boldsymbol{a}), \ldots, \frac{\partial f}{\partial x_n}(\boldsymbol{a})\right)$.
$\nabla f(\boldsymbol{a})$ is called the gradient (or nabla) of $f$ at $\boldsymbol{a}$.
A reverse implication to Theorem $3.8$ also holds true.

# 金融数值计算代考

## 金融代写|金融数值计算代写市场风险，金融数值分析代考|议会衍生品

$$g(t):=f\left(x_1, \ldots, x_{j-1}, t, x_{j+1}, \ldots, x_n\right)$$
by
$$f\left(x_1, \ldots, x_{j-1}, \cdots, x_{j+1}, \ldots, x_n\right) \text {. }$$

\begin{aligned} f x_j\left(x_1, \ldots, x_{j-1}, t_0, x_{j+1}, \ldots, x_n\right) &:=\frac{\partial f}{\partial x_j}\left(x_1, \ldots, x_{j-1}, t_0, x_{j+1}, \ldots, x_n\right) \ &:=g^{\prime}\left(t_0\right) . \end{aligned}

$$\frac{\partial f}{\partial x_j}(\boldsymbol{a}):=\lim _{h \rightarrow 0} \frac{f\left(\boldsymbol{a}+h \boldsymbol{e}_j\right)-f(\boldsymbol{a})}{h} .$$ 高于一阶的偏导数由迭代定义

## 金融代写|金融数值计算代写市场风险，金融数值分析代考|可微性

(1) $f$ 将在 $\boldsymbol{a}$
(2)所有一阶偏导数 $f$ 会存在于 $\boldsymbol{a}$.

$$\lim {h \rightarrow 0} \frac{f(x+h)-f(x)}{h}:=f^{\prime}(x) .$$ 以上定义等价于: $f$ 是可微的 $x \in \mathbb{R}$ 如果存在的话 $\alpha \in \mathbb{R}$ 一个函数 $\omega:(-\delta, \delta) \rightarrow \mathbb{R}$，与 $\omega(0)=0$ 和 $\lim {h \rightarrow 0} \frac{\omega(h)}{h}=0$，则:
$$f(x+h)=f(x)+\alpha h+\omega(h) h .$$

$$\lim _{\boldsymbol{h} \rightarrow 0} \frac{f(\boldsymbol{a}+\boldsymbol{h})-f(\boldsymbol{a})-\boldsymbol{d} \cdot \boldsymbol{h}}{|\boldsymbol{h}|}=0$$
$\boldsymbol{d}$ 叫做的全导数 $f$ 在 $\boldsymbol{a}$

(i) $f$ 是连续的 $\boldsymbol{a}$
(ii)的所有一阶偏导数 $f$ 存在于 $\boldsymbol{a}$
(iii) $\boldsymbol{d}=\nabla f(\boldsymbol{a}):=\left(\frac{\partial f}{\partial x_1}(\boldsymbol{a}), \ldots, \frac{\partial f}{\partial x_n}(\boldsymbol{a})\right)$.
$\nabla f(\boldsymbol{a})$ 叫做渐变(或nabla) $f$ 在 $\boldsymbol{a}$.

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