# 金融代写|金融数值计算代写Market Risk, Numerical Analysis for Finance代考|FIM549

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

Definition 3.14. Let $V$ be an open set in $\mathbb{R}^n$, let $\boldsymbol{a} \in V$ and suppose that $f: V \rightarrow \mathbb{R}$. Then:
(i) $f(\boldsymbol{a})$ is called a local minimum of $f$ if and only if there exists $r>0$ such that $f(\boldsymbol{a}) \leq f(\boldsymbol{x})$ for all $\boldsymbol{x} \in B_r(\boldsymbol{a})$, an open ball neighbourhood of $\boldsymbol{a}$ (recall Definition 1.13);
(ii) $f(\boldsymbol{a})$ is called a local maximum of $f$ if and only if there exists $r>0$ such that $f(\boldsymbol{a}) \geq f(\boldsymbol{x})$ for all $\boldsymbol{x} \in B_r(\boldsymbol{a})$
(iii) $f(\boldsymbol{a})$ is called a local extremum of $f$ if and only if $f(\boldsymbol{a})$ is a local maximum or a local minimum of $f$.

Remark 3.15. If the first-order partial derivatives of $f$ exist at $\boldsymbol{a}$, and if $f(\boldsymbol{a})$ is a local extremum of $f$, then $\nabla f(\boldsymbol{a})=\mathbf{0}$.
In fact, the one-dimensional function:
$$g(t)=f\left(a_1, \ldots, a_{j-1}, t, a_{j+1}, \ldots, a_n\right)$$
has a local extremum at $t=a_j$ for each $j=1, \ldots, n$. Hence, by the onedimensional theory:
$$\frac{\partial f}{\partial x_j}(\boldsymbol{a})=g^{\prime}\left(a_j\right)=0 .$$
As in the one-dimensional case, condition $\nabla f(\boldsymbol{a})=\mathbf{0}$ is necessary but not sufficient for $f(\boldsymbol{a})$ to be a local extremum.

Example 3.16. There exist continuously differentiable functions satisfying $\nabla f(\boldsymbol{a})=\mathbf{0}$ and such that $f(\boldsymbol{a})$ is neither a local maximum nor a local minimum.
Consider, for instance, in the case $n=2$, the following function:
$$f(x, y)=y^2-x^2 .$$
It is easy to check that $\nabla f(\mathbf{0})=\mathbf{0}$, but the origin is a saddle point, as shown in Figure 3.1.
Let us give a formal definition to such a situation.

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

To establish sufficient conditions for optimization, we introduce the notion of Hessian $^6$ matrix.

Definition 3.18. Let $V \subset \mathbb{R}^n$ be an open set and let $f: V \rightarrow \mathbb{R}$ be a $\mathcal{C}^2$ function. The Hessian matrix of $f$ at $\boldsymbol{x} \in V$ (or, simply, the Hessian) is the symmetric square matrix formed by the second-order partial derivatives of $f$, evaluated at point $\boldsymbol{x}$ :
$$H(f)(\boldsymbol{x}):=\left[\frac{\partial^2 f}{\partial x_i \partial x_j}(\boldsymbol{x})\right], \quad \text { for } \quad i, j=1, \ldots, n .$$
Tests for extrema and saddle points, in the simplest situation of $n=2$, are stated in Thenrem $3.19$.

Theorem 3.19. Let $V$ be open in $\mathbb{R}^2$, consider $(a, b) \in V$, and suppose that $f: V \rightarrow \mathbb{R}$ satisfies $\nabla f(a, b)=0$. Suppose further that $f \in \mathcal{C}^2$ and set:
$$D:=f_{x x}(a, b) f_{y y}(a, b)-f_{x y}^2(a, b) .$$
(i) If $D>0$ and $f_{x x}(a, b)>0$, then $f(a, b)$ is a local minimum.
(ii) If $D>0$ and $f_{x x}(a, b)<0$, then $f(a, b)$ is a local maximum.

(iii) If $D<0$, then $(a, b)$ is a saddle point.
Notice that $D$ is the determinant of the Hessian of $f$ evaluated at $(a, b)$ :
$$D=\operatorname{det}[H(f)(a, b)] .$$

# 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|极大值和极小值

3.14.

(i) $f(\boldsymbol{a})$ 的局部极小值 $f$ 当且仅当存在 $r>0$ 如此这般 $f(\boldsymbol{a}) \leq f(\boldsymbol{x})$ 为所有人 $\boldsymbol{x} \in B_r(\boldsymbol{a})$附近的开放舞会 $\boldsymbol{a}$ (回想定义1.13);
(ii) $f(\boldsymbol{a})$ 的局部最大值 $f$ 当且仅当存在 $r>0$ 如此这般 $f(\boldsymbol{a}) \geq f(\boldsymbol{x})$ 为所有人 $\boldsymbol{x} \in B_r(\boldsymbol{a})$
(iii) $f(\boldsymbol{a})$ 的局部极值 $f$ 当且仅当 $f(\boldsymbol{a})$ 的局部极大值还是局部极小值 $f$.

$$g(t)=f\left(a_1, \ldots, a_{j-1}, t, a_{j+1}, \ldots, a_n\right)$$

$$\frac{\partial f}{\partial x_j}(\boldsymbol{a})=g^{\prime}\left(a_j\right)=0 .$$

$$f(x, y)=y^2-x^2 .$$

## 金融代写|金融数值计算代写市场风险，数值分析的金融代考|充分条件

$$H(f)(\boldsymbol{x}):=\left[\frac{\partial^2 f}{\partial x_i \partial x_j}(\boldsymbol{x})\right], \quad \text { for } \quad i, j=1, \ldots, n .$$

$$D:=f_{x x}(a, b) f_{y y}(a, b)-f_{x y}^2(a, b) .$$
(i)如果$D>0$和$f_{x x}(a, b)>0$，则$f(a, b)$是局部极小值
(ii)如果$D>0$和$f_{x x}(a, b)<0$，则$f(a, b)$是局部极大值

(iii)如果$D<0$，那么$(a, b)$是一个鞍点。

$$D=\operatorname{det}[H(f)(a, b)] .$$

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