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# 金融代写|固定收益与信贷代写Fixed Income and Credit代考|BUSN7770 Stochastic Calculus and the Itô Formula

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## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Stochastic Calculus and the Itô Formula

Let $X$ be a stochastic process and suppose that there exists a real number $a$ and two adapted processes $\mu$ and $\sigma$ such that the following relation holds for all $t \geq 0$.
$$X_t=a+\int_0^t \mu_s d s+\int_0^t \sigma_s d W_s$$
where $a$ is some given real number. As usual $W$ is a Wiener process. To use a less cumbersome notation we will often write eqn (4.16) in the following form:
\begin{aligned} d X_t & =\mu_t d t+\sigma_t d W_t, \ X(0) & =a . \end{aligned}
In this case we say that $X$ has a stochastic differential given by (4.17) with an initial condition given by (4.18). It is important to note that the formal string $d X_t=\mu_t d t+\sigma_t d W_t$ has no independent meaning. It is simply a shorthand version of the expression (4.16) above. From an intuitive point of view the stochastic differential is, however, a much more natural object to consider than the corresponding integral expression. This is because the stochastic differential gives us the “infinitesimal dynamics” of the $X$-process, and as we have seen in Section $4.1$ both the drift term $\mu_t$ and the diffusion term $\sigma_t$ have a natural intuitive interpretation.

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Multidimensional Itô Formula

Let us now consider a vector process $X=\left(X^1, \ldots, X^n\right)^{\star}$, where the component $X^i$ has a stochastic differential of the form
$$d X_t^i=\mu_t^i d t+\sum_{j=1}^d \sigma_t^{i j} d W_t^j$$
and $W^1, \ldots, W^d$ are $d$ independent Wiener processes.
Defining the drift vector process $\mu$ by
$$\mu_t=\left[\begin{array}{c} \mu^1 \ \vdots \ \mu^n \end{array}\right],$$
the $d$-dimensional vector Wiener process $W$ by
$$W=\left[\begin{array}{c} W^1 \ \vdots \ W^d \end{array}\right]$$
and the $n \times d$-dimensional diffusion matrix process $\sigma_t$ by
$$\sigma=\left[\begin{array}{cccc} \sigma^{11} & \sigma^{12} & \ldots & \sigma^{1 d} \ \sigma^{21} & \sigma^{22} & \ldots & \sigma^{2 d} \ \vdots & \vdots & \ddots & \vdots \ \sigma^{n 1} & \sigma^{n 2} & \ldots & \sigma^{n d} \end{array}\right],$$
we may write the $X$-dynamics as
$$d X_t=\mu_t d t+\sigma_t d W_t$$

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|Stochastic Calculus and the Itô Formula

$$X_t=a+\int_0^t \mu_s d s+\int_0^t \sigma_s d W_s$$

$$d X_t=\mu_t d t+\sigma_t d W_t, X(0)=a .$$

## 金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Multidimensional Itô Formula

$$d X_t^i=\mu_t^i d t+\sum_{j=1}^d \sigma_t^{i j} d W_t^j$$

$$\mu_t=\left[\mu^1 \vdots \mu^n\right]$$

$$W=\left[W^1: W^d\right]$$

$$d X_t=\mu_t d t+\sigma_t d W_t$$

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