经济代写|随机微积分代写Stochastic calculus代考|MATH530A

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

Let us fix an r.c.l.l. $(\mathcal{F}$.) adapted stochastic process $X$.
Recall, $S$ consists of the class of processes $f$ of the form
$$f(s)=a_0 1_{[0}}(s)+\sum_{j=0}^m a_{j+1} 1_{\left(s_j, s_{j+1}\right]}(s)$$
where $0=s_0<s_1<s_2<\ldots<s_{m+1}<\infty, a_j$ is bounded $\mathcal{F}{s{j-1}}$ measurable random variable, $1 \leq j \leq(m+1)$, and $a_0$ is bounded $\mathcal{F}_0$ measurable.

For simple predictable $f \in \mathbb{S}$ given by $(4.2 .1)$, let $J_X(f)$ be the r.c.l.1. process defined by
$$J_X(f)(t)=a_0 X_0+\sum_{j=0}^m a_{j+1}\left(X_{s_{j+1} \wedge t}-X_{s_j \wedge t}\right)$$
One needs to verify that $J_X$ is unambiguously defined on $\mathbb{S}$. That is, if a given $f$ has two representations of type (4.2.1), then the corresponding expressions in (4.2.2) agree. This as well as linearity of $J_X(f)$ for $f \in \mathbb{S}$ can be verified using elementary algebra. By definition, for $f \in \mathbb{S}, J_X(f)$ is an r.c.l.l. adapted process. In analogy with the Ito’s integral with respect to Brownian motion discussed in the earlier chapter, we wish to explore if we can extend $J_X$ to the smallest $b p$-closed class of integrands that contain $\mathbb{S}$. Each $f \in \mathbb{S}$ can be viewed as a real-valued function on $\widetilde{\Omega}=[0, \infty) \times \Omega$. Since $\mathcal{P}$ is the $\sigma$-field generated by $\mathbb{S}$, by Theorem $2.66$, the smallest class of functions that contains $\mathbb{S}$ and is closed under $b p$-convergence is $\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$.

When the space, filtration and the probability measure are clear from the context, we will write the class of adapted r.c.l.l. processes $\mathbb{R}^0(\Omega,(\mathcal{F}$. $), \mathrm{P})$ simply as $\mathbb{R}^0$.

经济代写|随机微积分代写Stochastic calculus代考|Properties of the Stochastic Integral

First we note linearity of $(f, X) \mapsto \int f d X$.
Theorem 4.27 Let $X, Y$ be stochastic integrators, $f, g$ be predictable processes and $\alpha, \beta \in \mathbb{R}$
(i) Suppose $f, g \in \mathbb{L}(X)$. Let $h=\alpha f+\beta g$. Then $h \in \mathbb{L}(X)$ and
$$\int h d X=\alpha \int f d X+\beta \int g d X$$
(ii) Let $Z=\alpha X+\beta Y$. Then $Z$ is a stochastic integrator. Further, if $f \in \mathbb{L}(X)$ and $f \in \mathbb{L}(Y)$. then $f \in \mathbb{L}(Z)$ and $$\int f d Z=\alpha \int f d X+\beta \int f d Y$$
Proof We will begin by showing that (4.3.1) is true for $f . g$ bounded predictable processes. For a bounded predictable process $f$, let
$$\mathbb{K}(f)=\left{g \in \mathbb{B}(\tilde{\Omega}, \mathcal{P}): \int(\alpha f+\beta g) d X=\alpha \int f d X+\beta \int g d X, \forall \alpha, \beta \in \mathbb{R}\right}$$
If $f \in \mathbb{S}$, it easy to see that $\mathbb{S} \subseteq \mathbb{K}(f)$ and Theorem $4.22$ implies that $K(f)$ is $b p$ closed. Hence invoking Theorem $2.66$, it follows that $\mathbb{K}(f)=\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$.

Now we take $f \in \mathbb{B}(\widetilde{\Omega}, \mathcal{P})$ and the part proven above yields $\mathbb{S} \subseteq \mathbb{K}(f)$. Once again, using that $K(f)$ is $b p$-closed we conclude that $\mathbb{K}(f)=\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$. Thus $(4.3 .1)$ is true when $f, g$ are bounded predictable process.

Now let us fix $f, g \in \mathbb{L}(X)$. We will show $(|\alpha f|+|\beta g|) \in \mathbb{L}(X)$, let $u^n$ be bounded predictable processes converging to $u$ pointwise and
$$\left|u^n\right| \leq(|\alpha f|+|\beta g|) ; \forall n \geq 1 .$$
Let $v^n=u^n 1_{\langle|\alpha f| \leq| \beta g |}$ and $w^n=u^n 1_{\langle|\alpha f|>|\beta g|}}$. Then $v^n$ and $w^n$ converge pointwise to $v=u 1_{{|\alpha f| \leq|\beta g| \mid}$ and $w=u 1_{\lfloor|\alpha f|>|\beta g| \mid}$, respectively, and further
$$\begin{gathered} \left|v^n\right| \leq 2|\beta g| \ \left|w^n\right| \leq 2|\alpha f| . \end{gathered}$$

随机微积分代考

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

.

$$f(s)=a_0 1_{[0}}(s)+\sum_{j=0}^m a_{j+1} 1_{\left(s_j, s_{j+1}\right]}(s)$$
where $0=s_0<s_1<s_2<\ldots<s_{m+1}<\infty, a_j$ 是有界的 $\mathcal{F}{s{j-1}}$ 可测量随机变量， $1 \leq j \leq(m+1)$，以及 $a_0$ 是有界的 $\mathcal{F}_0$

$$J_X(f)(t)=a_0 X_0+\sum_{j=0}^m a_{j+1}\left(X_{s_{j+1} \wedge t}-X_{s_j \wedge t}\right)$$

经济代写|随机微积分代写随机积分代考|随机积分的性质

(i)假设 $f, g \in \mathbb{L}(X)$。让 $h=\alpha f+\beta g$。然后 $h \in \mathbb{L}(X)$ 和
$$\int h d X=\alpha \int f d X+\beta \int g d X$$
(ii)让 $Z=\alpha X+\beta Y$。然后 $Z$ 是一个随机积分器。此外，如果 $f \in \mathbb{L}(X)$ 和 $f \in \mathbb{L}(Y)$。然后 $f \in \mathbb{L}(Z)$ 和 $$\int f d Z=\alpha \int f d X+\beta \int f d Y$$我们将首先证明(4.3.1)对…是正确的 $f . g$ 有界可预测过程。对于有界可预测过程 $f$，让
$$\mathbb{K}(f)=\left{g \in \mathbb{B}(\tilde{\Omega}, \mathcal{P}): \int(\alpha f+\beta g) d X=\alpha \int f d X+\beta \int g d X, \forall \alpha, \beta \in \mathbb{R}\right}$$

$$\left|u^n\right| \leq(|\alpha f|+|\beta g|) ; \forall n \geq 1 .$$

$$\begin{gathered} \left|v^n\right| \leq 2|\beta g| \ \left|w^n\right| \leq 2|\alpha f| . \end{gathered}$$

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