## 经济代写|随机微积分代写Stochastic calculus代考|MA451A

2022年10月11日

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## 经济代写|随机微积分代写Stochastic calculus代考|Stochastic Integration

In this chapter we consider processes $X$ that are good integrators: i.e.
$$J_X(f)(t)=\int_0^t f d X$$
can be defined for a suitable class of integrands $f$ and the integral has some natural continuity properties. We will call such a process a stochastic integrator. In this chapter, we will prove basic properties of the stochastic integral $\int_0^t f d X$ for a stochastic integrator $X$.

In the rest of the book, $(\Omega, \mathcal{F}, \mathrm{P})$ will denote a complete probability space and $\left(\mathcal{F}\right.$.) will denote a filtration such that $\mathcal{F}_0$ contains all null sets in $\mathcal{F}$. All notions such as adapted, stopping time, martingale will refer to this filtration unless otherwise stated explicitly.

For some of the auxiliary results, we need to consider the corresponding right continuous filtration $\left(\mathcal{F}^{+}\right)=\left{\mathcal{F}t^{+}: t \geq 0\right}$ where $$\mathcal{F}_t^{+}=\bigcap{s>1} \mathcal{F}_s .$$
We begin with a discussion on the predictable $\sigma$-field.

## 经济代写|随机微积分代写Stochastic calculus代考|The Predictable σ-Field

Recall our convention that a process $X=\left(X_t\right)$ is viewed as a function on $\widetilde{\Omega}=$ $[0, \infty) \times \Omega$ and the predictable $v$-field $\mathcal{P}$ has been defined as the 0 -field on $\widetilde{\Omega}$ generated by $\mathbb{S}$. Here $\mathbb{S}$ consists of simple adapted processes:
$$f(s)=a_0 1_{[0}}(s)+\sum_{k=0}^m a_{k+1} 1_{\left[s_k, s_{k+1}\right]}(s)$$

where $0=s_0<s_1<s_2<\ldots<s_{m+1}<\infty, a_k$ is bounded $\mathcal{F}{s{k-1}}$ measurable random variable, $1 \leq k \leq(m+1)$, and $a_0$ is bounded $\mathcal{F}0$ measurable. $\mathcal{P}$ measurable processes have appeared naturally in the definition of the stochastic integral w.r.t. Brownian motion and play a very significant role in the theory of stochastic integration with respect to general semimartingales as we will see. A process $f$ will be called a predictable process if it is $\mathcal{P}$ measurable. Of course, $\mathcal{P}$ depends upon the underlying filtration and would refer to the filtration that we have fixed. If there are more than one filtration under consideration, we will state it explicitly. For example $\mathcal{P}(\mathcal{G}$.) denotes the predictable $\sigma$-field corresponding to a filtration $(\mathcal{G}$.) and $\mathbb{S}(\mathcal{G}$. $)$ denotes simple predictable process for the filtration $(\mathcal{G}$.). The following proposition lists various facts about the $\sigma$-field $\mathcal{P}$. Proposition 4.1 Let $(\mathcal{F}$.) be a filtration and $\mathcal{P}=\mathcal{P}(\mathcal{F}$.). (i) Let $f$ be $\mathcal{P}$ measurable. Then $f$ is $(\mathcal{F}$.) adapted. Moreover, for every $t<\infty$, $f_t$ is $\sigma\left(\cup{s<t} \mathcal{F}s\right)$ measurable. (ii) Let $Y$ be a left continuous adapted process. Then $Y$ is $\mathcal{P}$ measurable. (iii) Iet $\mathbb{A}$ he the class of all hounded adapted continuous processes. Then $\mathcal{P}=\sigma(\mathbb{A})$ and the smallest $\mathrm{bp}$-closed class that contains $\mathbb{A}$ is $\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$. (iv) For any stopping time $\tau, U-1{[0, \tau]}$ (i.e. $U_t-1_{[0, \tau]}(t)$ ) is $\mathcal{P}$ measurable.
(v) For an r.c.l.l. adapted process $Z$ and a stopping time $\tau$, the process $X$ defined by
$$X_t=Z_\tau 1_{(\tau, \infty)}(t)$$
is predictable.
(vi) For a predictable process $g$ and a stopping time $\tau, g_\tau$ is a random variable and $h$ defined by
$$h_t=g_\tau 1_{(\tau, \infty)}(t)$$
is itself predictable.

# 随机微积分代考

## 经济代写|随机微积分代写Stochastic calculus代考|随机积分

$$J_X(f)(t)=\int_0^t f d X$$

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