Posted on Categories:Uncertainty Quantification, 金融代写, 风险估值理论

# 金融代写|风险估值理论代写Uncertainty Quantification代考|ACM41000 Recession Approach for Linear Variational Inequalities

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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Recession Approach for Linear Variational Inequalities

Let $H$ be a Hilbert space, $a: H \times H \rightarrow \mathbb{R}$ be a positive and continuous bilinear form, $K \subset H$ be a nonempty, closed, and convex set, and $f \in H^*$.
We continue to consider the variational inequality: Find $u \in K$ such that
$$a(u, v-u) \geq\langle f, v-u\rangle, \quad \text { for every } v \in K .$$
Let $\mathcal{S}(a, f, K)$ be the set of solutions of (4.26).
We formulate the following:
Assumption (A): There are two linear maps $P_0: H \rightarrow H$ and $P_1: H \rightarrow H$ such that the set $P_0(K)$ is bounded, $P_1$ is compact, and there is a constant $\alpha>0$ such that
$$a(v, v)+\left|P_0 v\right|^2+\left|P_1 v\right|^2 \geq \alpha|v|^2, \quad \text { for every } v \in H .$$
Note that due to (4.27), for continuous and positive bilinear form $a(\cdot, \cdot)$, the term $\left[a(v, v)+\left|P_0 v\right|^2+\left|P_1 v\right|^2\right]^{1 / 2}$ defines a norm which is equivalent to the original norm. Evidently, Assumption (A) is satisfied trivially if either $a(\cdot, \cdot)$ is elliptic or $K$ is bounded.
The following notion of a recession cone will play a central role:
Definition 4.2.1 Let $H$ be a Hilbert space and $C \subset H$ be a closed, and convex set. The recession cone of $C$, denoted by $C_{\infty}$, is defined, for any fixed $c_0 \in C$, by
$$C_{\infty}:=\bigcap_{t>0} t\left(C-c_0\right) .$$

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Existence Results and Stability of Solutions

Let $H$ be a Hilbert space with inner product $\langle\cdot, \cdot\rangle$ and $|\cdot|, K \subset H$ be closed and convex, $F: H \rightarrow H$ be a given map, and $f \in H$.
We consider the nonlinear variational inequality of finding $u \in K$ such that
$$\langle F u-f, v-u\rangle \geq 0, \quad \text { for every } v \in K .$$
Let $\mathcal{S}(F, f, K)$ be the solution set of (4.34).
By using the indicator function, (4.34) can be written as a generalized equation:
$$f \in F u+\partial I_K u,$$
or equivalently, by using the normal cone of $K$ at $u$ :
$$f-F u \in N_K u .$$
In Theorem 4.2.1, we characterized linear variational inequality (4.10) as an equivalent optimization problem. A similar equivalence exists for nonlinear variational inequalities, albeit under more restrictive conditions, as shown next:

Theorem 4.3.1 Let $H$ be a Hilbert space, $K \subset H$ be closed, and convex, and $f \in$ H. Let $F: K \rightarrow H$ be monotone, continuous, and potential. Then variational inequality (4.34) is equivalent to the minimization problem: Find $u \in K$ such that
$$\Phi(u) \leq \Phi(v), \quad \text { for every } v \in K$$

where, for some fixed $\tilde{v} \in K, \Phi: H \rightarrow \mathbb{R}$ is a convex function defined by
$$\Phi(v)=\int_0^1\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle d \tau-\langle f, v\rangle+\Phi_0$$
with $\Phi_0=\Phi(\tilde{v})+\langle f, \tilde{v}\rangle$, and it satisfies the following identity:
$$\nabla \Phi(u)=F u-f, \quad \text { for every } u \in K$$
Proof. Since $F$ is potential, there is a functional $\Psi$ such that $F=\nabla \Psi$. Therefore, $\left.\frac{d}{d t} \Psi(\tilde{v}+t(v-\tilde{v}))\right|_{t=\tau}=\langle\nabla \Psi(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle=\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle$, and by integrating in $\tau$ from 0 to 1 , we obtain
$$\Psi(v)-\Psi(\tilde{v})=\int_0^1\langle F(\tilde{v}+\tau(v-\tilde{v})), v-\tilde{v}\rangle d \tau .$$
Defining $\Phi(\cdot)=\Psi(\cdot)-\langle f, \cdot\rangle$, we obtain (4.36) or equivalently (4.37). Since $F$ is monotone, $\nabla \Phi$ is monotone, which is equivalent to the convexity of $\Phi$. The equivalence of (4.34) and (4.35) is now quite evident.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Recession Approach for Linear Variational Inequalities

$$a(u, v-u) \geq\langle f, v-u\rangle, \quad \text { for every } v \in K \text {. }$$

$$C_{\infty}:=\bigcap_{t>0} t\left(C-c_0\right) .$$

## 金融代写川风险估值理论代写Uncertainty Quantification代考|Existence Results and Stability of Solutions

$$\langle F u-f, v-u\rangle \geq 0, \quad \text { for every } v \in K .$$

$$f \in F u+\partial I_K u$$

$$f-F u \in N_K u .$$

$$\Phi(u) \leq \Phi(v), \quad \text { for every } v \in K$$

$$\Phi(v)=\int_0^1\langle F(\bar{v}+\tau(v-\bar{v})), v-\bar{v}\rangle d \tau-\langle f, v\rangle+\Phi_0$$

$$\nabla \Phi(u)=F u-f, \quad \text { for every } u \in K$$

$$\Psi(v)-\Psi(\bar{v})=\int_0^1\langle F(\bar{v}+\tau(v-\bar{v})), v-\bar{v}\rangle d \tau$$

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