### 金融代写|波动率模型代写Market Volatility Modelling代考|COVID19

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|波动率模型代写Market Volatility Modelling代考|Infinitesimal Generators and Associated Martingales

For simplicity we first consider a time-homogeneous diffusion process $\left(X_t\right)$ that solves the stochastic differential equation
$$d X_t=\mu\left(X_t\right) d t+\sigma\left(X_t\right) d W_t .$$
Let $g$ be a twice continuously differentiable function of the variable $x$ with bounded derivatives, and define the differential operator $\mathcal{L}$ acting on $g$ according to
$$\mathcal{L}{\mathcal{G}}(x)-\frac{1}{2} \sigma^2(x) g^{\prime \prime}(x)+\mu(x) g^{\prime}(x) .$$ In terms of $\mathcal{L}$, Itô’s formula (1.16) gives $$d g\left(X_t\right)=\mathcal{L} g\left(X_t\right) d t+g^{\prime}\left(X_t\right) \sigma\left(X_t\right) d W_t,$$ which shows that $$M_t=g\left(X_t\right)-\int_0^t \mathcal{L} g\left(X_s\right) d s$$ defines a martingale. Consequently, if $X_0=x$, we obtain $$\mathbb{E}\left{g\left(X_t\right)\right}=g(x)+\mathbb{E}\left{\int_0^t \mathcal{L} g\left(X_s\right) d s\right} .$$ Under the assumptions made on the coefficients $\mu$ and $\sigma$ and on the function $g$, the Lebesgue dominated convergence theorem is applicable and gives \begin{aligned} \left.\frac{d}{d t} \mathbb{E}\left{g\left(X_t\right)\right}\right|{t=0} & =\lim {t \downarrow 0} \frac{\mathbb{E}\left{g\left(X_t\right)\right}-g(x)}{t} \ & =\lim {t \downarrow 0} \mathbb{E}\left{\frac{1}{t} \int_0^t \mathcal{L} g\left(X_s\right) d s\right}=\mathcal{L} g(x) . \end{aligned}
The differential operator $\mathcal{L}$ given by (1.61) is called the infinitesimal generator of the Markov process $\left(X_t\right)$.

## 金融代写|波动率模型代写Market Volatility Modelling代考|Application to the Black-Scholes Partial Differential Equation

In the previous section we assumed the existence, uniqueness, and regularity of the solution of the partial differential equation (1.66) in order to apply Itô’s formula.. A sufficient condition for this is that the coefficients $\mu$ and $\sigma$ are regular enough and that the operator $\mathcal{L}_t$ is uniformly elliptic, meaning (in this one-dimensional situation) that there exists a positive constant $A$ such that
$$\sigma^2(t, x) \geq A>0 \quad \text { for every } t \geq 0 \text { and } x \in \mathcal{D},$$
so that the diffusion coefficient $\sigma^2(t, x)$ cannot become too small. Here $\mathcal{D}$ is the domain of the process $\left(X_t\right)$, which may be natural (e.g., $\mathcal{D}={x>0}$ for the geometric Brownian motion) or imposed externally from other modeling considerations.

When $\mu(t, x)=r x$ and $\sigma(t, x)=\sigma x$ in (1.66), we have the Black-Scholes partial differential equation (1.35) on the domain ${x>0}$. The ellipticity condition (1.68) is clearly not satisfied, since the diffusion coefficient $\sigma^2 x^2$ goes to zero as the state variable approaches zero. We get around this difficulty here (and also in more general situations) with the change of variable $P(t, x)=u(t, y=\log x)$, so that equation (1.35) becomes
$$\frac{\partial u}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 u}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial u}{\partial y}-r u=0$$
to be solved for $0 \leq t \leq T, y \in \mathbb{R}$, and with the final condition $u(T, y)=h\left(e^y\right)$. The operator
$$\mathcal{L}=\frac{1}{2} \sigma^2 \frac{\partial^2}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial}{\partial y}$$
is the infinitesimal generator of the (nonstandard) Brownian motion
$$Y_t=\left(r-\frac{1}{2} \sigma^2\right) t+\sigma W_t^{\star},$$
where $\left(W_t^{\star}\right)$ is a standard Brownian motion under $P^{\star}$. We use here the same notation as in the equivalent martingale measure context, but the only important fact is that $W^$ is a standard Brownian motion with respect to the probability used to compute the expectation in the Feynman-Kac formula (1.67). Applying this formula to $Y_t$ yields $$u(t, y)=\mathbb{E}^\left{e^{-r(T-t)} h\left(e^{y+\left(r-\sigma^2 / 2\right)(T-t)+\sigma\left(W_T^-W_t^\right)} \mid Y_t=y\right}\right.$$ which is indeed the same as (1.57) by undoing the change of variable $e^y=x$.

# 波动率模型代考

## 金融代写|波动率模型代写Market Volatility Modelling代考|Infinitesimal Generators and Associated Martingales

$$d X_t=\mu\left(X_t\right) d t+\sigma\left(X_t\right) d W_t .$$

$$\mathcal{L} \mathcal{G}(x)-\frac{1}{2} \sigma^2(x) g^{\prime \prime}(x)+\mu(x) g^{\prime}(x)$$

$$d g\left(X_t\right)=\mathcal{L} g\left(X_t\right) d t+g^{\prime}\left(X_t\right) \sigma\left(X_t\right) d W_t$$

$$M_t=g\left(X_t\right)-\int_0^t \mathcal{L} g\left(X_s\right) d s$$

## 金融代写|波动率模型代写Market Volatility Modelling代考|Application to the Black-Scholes Partial Differential Equation

$$\sigma^2(t, x) \geq A>0 \quad \text { for every } t \geq 0 \text { and } x \in \mathcal{D},$$

$$\frac{\partial u}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 u}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial u}{\partial y}-r u=0$$

$$\mathcal{L}=\frac{1}{2} \sigma^2 \frac{\partial^2}{\partial y^2}+\left(r-\frac{1}{2} \sigma^2\right) \frac{\partial}{\partial y}$$

$$Y_t=\left(r-\frac{1}{2} \sigma^2\right) t+\sigma W_t^{\star},$$

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