### 金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

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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|利率建模代写Interest Rate Modeling代考|Evaluation of Stochastic Integrals

We now consider the evaluation of stochastic integrals. Suppose that we know the anti-derivative of a function, $f(t)$, such that
$$\frac{\mathrm{d} F(t)}{\mathrm{d} t}=f(t) .$$
Could there be
$$\int_{0}^{t} f(W(s)) \mathrm{d} W(s)=F(W(t))-F(W(0)) ?$$
The answer to this question is no. Consider, for example, $f(t)=W(t)$. If Equation $1.30$ was correct, then there would be
$$\int_{0}^{t} W(s) \mathrm{d} W(s)=\frac{1}{2}\left[W^{2}(t)-W^{2}(0)\right]=\frac{1}{2} W^{2}(t) .$$
Taking expectations on both sides and applying the first property of the stochastic integrals, we would obtain
$$0=E\left[\int_{0}^{t} W(s) \mathrm{d} W(s)\right]=E\left[\frac{1}{2} W^{2}(t)\right]=\frac{1}{2} t$$
which is a contradiction. This result suggests that general rules in deterministic calculus is not applicable to stochastic integrals.

As a showcase of integral evaluation, we try to work out the integral of $f(t)=W(t)$ according to its definition. Let $t_{j}=j t / n$ and denote $W_{j}$ for $W\left(t_{j}\right), j=0, \ldots, n$. Start from the partial sum as follows:
\begin{aligned} S_{n} &=\sum_{j=0}^{n-1} W\left(t_{j}\right) \Delta W\left(t_{j}\right)=\sum_{j=0}^{n-1} W_{j}\left(W_{j+1}-W_{j}\right) \ &=\sum_{j=0}^{n-1} W_{j} W_{j+1}-W_{j}^{2} \ &=\sum_{j=0}^{n-1}-W_{j+1}^{2}+2 W_{j+1} W_{j}-W_{j}^{2}+W_{j+1}^{2}-W_{j+1} W_{j} \ &=\sum_{j=0}^{n-1}-\left(W_{j+1}-W_{j}\right)^{2}+W_{j+1}^{2}-W_{j}^{2}-W_{j} \Delta W_{j} \ &=-\left[\sum_{j=0}^{n-1}\left(\Delta W_{j}\right)^{2}\right]+W_{n}^{2}-W_{0}^{2}-S_{n} \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Differentials and Ito’s Lemma

In this section, we study the differentials of functions of other stochastic processes. In stochastic calculus, the so-called Ito’s process is most often used as the basic stochastic process.

Definition 1.3.1. Ito’s process is a continuous stochastic process of the form:
$$X(t)=X_{0}+\int_{0}^{t} \sigma(s) d W(s)+\int_{0}^{t} \mu(s) d s,$$

where $\sigma(s)$ and $\mu(s)$ are adaptive functions satisfying
$$E\left[\int_{0}^{t}\left(\sigma^{2}(s)+|\mu(s)|\right) d s\right]<\infty, \quad \forall t .$$
The corresponding differential of Ito’s process is
$$d X(t)=\sigma(t) d W(t)+\mu(t) d t .$$
We call $\sigma(t)$ and $\mu(t)$ the volatility and drift of the $S D E$, respectively.
We now consider a function of $X(t), Y(t)=F(X(t), t)$. The next lemma describes the SDE satisfied by $Y(t)$.

Lemma 1.3.1 (Ito’s Lemma). Let $X(t)$ be Ito’s process with drift $\mu(t)$ and volatility $\sigma(t)$, and let $F(x, t)$ be a smooth function with bounded second-order derivatives. Then $Y(t)=F(X(t), t)$ is also Ito’s process with drift
$$N(t)=\frac{\partial F}{\partial t}+\frac{1}{2} \sigma^{2}(t) \frac{\partial^{2} F}{\partial x^{2}}+\mu(t) \frac{\partial F}{\partial x}$$
and volatility
$$\Sigma(t)=\sigma(t) \frac{\partial F}{\partial x} .$$
Proof: By Taylor’s expansion,
\begin{aligned} \Delta Y\left(t_{i}\right)=& F\left(X\left(t_{i}+\Delta t\right), t_{i}+\Delta t\right)-F\left(X\left(t_{i}\right), t_{i}\right) \ =& F_{x} \Delta X+F_{t} \Delta t+\frac{1}{2} F_{x x}(\Delta X)^{2}+F_{x t} \Delta X \Delta t+\frac{1}{2} F_{t t}(\Delta t)^{2} \ &+\text { higher order terms. } \end{aligned}
Because
$$\Delta W(t)=\sqrt{\Delta t} \cdot \varepsilon, \quad \varepsilon \sim N(0,1)$$
we generally have
$$E\left[|\Delta W|^{p} \Delta t^{q}\right] \propto \Delta t^{(p / 2)+q} .$$
Here ” $\propto$ ” means “of the order of.” Based on Equation 1.46, we know that the order of magnitude of both the cross term and the higher-order terms in Equation $1.45$ is $O\left(\Delta t^{3 / 2}\right)$, and thus we can rewrite Equation $1.43$ as
\begin{aligned} \Delta Y\left(t_{i}\right)=& F_{x} \Delta X+F_{t} \Delta t+\frac{1}{2} F_{x x} \sigma^{2}\left(t_{i}\right)\left(\Delta W\left(t_{i}\right)\right)^{2}+O\left(\Delta t^{3 / 2}\right) \ =& F_{x} \Delta X+F_{t} \Delta t+\frac{1}{2} F_{x x} \sigma^{2}\left(t_{i}\right) \Delta t \ &+\frac{1}{2} F_{x x} \sigma^{2}\left(t_{i}\right)\left(\Delta W^{2}\left(t_{i}\right)-\Delta t\right)+O\left(\Delta t^{3 / 2}\right) . \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|Multi-Factor Ito’s Process

A multiple-factor Ito’s process takes the form
\begin{aligned} \mathrm{d} X_{i}(t) &=\mu_{i}(t) \mathrm{d} t+\sum_{j=1}^{n} \sigma_{i j}(t) \mathrm{d} W_{j}(t) \ &=\mu_{i}(t) \mathrm{d} t+\sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t} \end{aligned} where $$\mathbf{W}(t)=\left(\begin{array}{c} W{1}(t) \ W_{2}(t) \ \vdots \ W_{n}(t) \end{array}\right)$$
is a vector of independent Brownian motion, and
$$\boldsymbol{\sigma}{i}(t)=\left(\begin{array}{c} \sigma{i 1}(t) \ \sigma_{i 2}(t) \ \vdots \ \sigma_{i n}(t) \end{array}\right)$$
is called the volatility vector. Let $\mathbf{X}(t)=\left(X_{1}(t), X_{2}(t), \ldots, X_{n}(t)\right)^{\mathrm{T}}$. In integral form, the multi-factor Ito’s process is
$$\mathbf{X}{t}=\mathbf{X}{0}+\int_{0}^{t} \boldsymbol{\mu}{s} \mathrm{~d} s+\int{0}^{t} \boldsymbol{\Sigma}(s) \mathrm{d} \mathbf{W}{s},$$ where $$\boldsymbol{\mu}{t}=\left(\begin{array}{c} \mu_{1}(t) \ \mu_{2}(t) \ \vdots \ \mu_{n}(t) \end{array}\right)$$

is the vector of drifts, and
$$\boldsymbol{\Sigma}(t)=\left(\begin{array}{c} \boldsymbol{\sigma}{1}^{\mathrm{T}}(t) \ \boldsymbol{\sigma}{2}^{\mathrm{T}}(t) \ \vdots \ \boldsymbol{\sigma}{n}^{\mathrm{T}}(t) \end{array}\right)$$ is the volatility matrix. Note that both $\boldsymbol{\mu}(t)$ and $\sigma{i}(t), i=1, \ldots, n$ are $\mathcal{F}{t^{-}}$ adaptive processes, and they satisfy $$E\left[\int{0}^{t}\left(\sum_{j=1}^{n}\left|\boldsymbol{\sigma}{j}(s)\right|^{2}+\left|\boldsymbol{\mu}{s}\right|{1}\right) \mathrm{d} s\right]<\infty, \quad \forall t$$ Namely, $\sigma{j}(s), j=1, \ldots, n$ are square integrable and $\mu_{s}$ has bounded variation.

## 金融代写|利率建模代写Interest Rate Modeling代考|Evaluation of Stochastic Integrals

dF(吨)d吨=F(吨).

∫0吨F(在(s))d在(s)=F(在(吨))−F(在(0))?

∫0吨在(s)d在(s)=12[在2(吨)−在2(0)]=12在2(吨).

0=和[∫0吨在(s)d在(s)]=和[12在2(吨)]=12吨

## 金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Differentials and Ito’s Lemma

X(吨)=X0+∫0吨σ(s)d在(s)+∫0吨μ(s)ds,

dX(吨)=σ(吨)d在(吨)+μ(吨)d吨.

ñ(吨)=∂F∂吨+12σ2(吨)∂2F∂X2+μ(吨)∂F∂X

Σ(吨)=σ(吨)∂F∂X.

Δ是(吨一世)=F(X(吨一世+Δ吨),吨一世+Δ吨)−F(X(吨一世),吨一世) =FXΔX+F吨Δ吨+12FXX(ΔX)2+FX吨ΔXΔ吨+12F吨吨(Δ吨)2 + 高阶项。

Δ在(吨)=Δ吨⋅e,e∼ñ(0,1)

Δ是(吨一世)=FXΔX+F吨Δ吨+12FXXσ2(吨一世)(Δ在(吨一世))2+○(Δ吨3/2) =FXΔX+F吨Δ吨+12FXXσ2(吨一世)Δ吨 +12FXXσ2(吨一世)(Δ在2(吨一世)−Δ吨)+○(Δ吨3/2).

## 金融代写|利率建模代写Interest Rate Modeling代考|Multi-Factor Ito’s Process

dX一世(吨)=μ一世(吨)d吨+∑j=1nσ一世j(吨)d在j(吨) =μ一世(吨)d吨+σ一世吨(吨)d在吨在哪里

σ一世(吨)=(σ一世1(吨) σ一世2(吨) ⋮ σ一世n(吨))

X吨=X0+∫0吨μs ds+∫0吨Σ(s)d在s,在哪里

μ吨=(μ1(吨) μ2(吨) ⋮ μn(吨))

Σ(吨)=(σ1吨(吨) σ2吨(吨) ⋮ σn吨(吨))是波动率矩阵。请注意，两者μ(吨)和σ一世(吨),一世=1,…,n是F吨−自适应过程，它们满足

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