The Question

I am reading Shreve's Stochastic Calculus for Finance, Volume II. On page 145, definition (4.4.20), he defines an integral with respect to an Itô process.

Definition 4.4.5. Let $X(t) = X(0)+\int_0^t\Delta(u)\mathrm{d}W(u)+\int_0^t \Theta(u)\mathrm{d}u $, $t \geq 0$, be an Itô process as described in Definition 4.4.3, and let $\Gamma(t)$, $t \geq 0$, be an adapted process. We define the integral with respect to an Itô process

$$\int_0^t\Gamma(u)\mathrm{d}X(u) = \int_0^t\Gamma(u)\Delta(u)\mathrm{d}W(u) + \int_0^t \Gamma(u) \Theta(u) \mathrm{d}u\text{.}\tag{1}$$

This definition is useful for a student who is asked to compute the LHS, $\int_0^t\Gamma(u)\mathrm{d}X(u)$. However, there is a second more useful definition of the integral. Let $\Pi = (k_1 = 0, k_2, \ldots, k_n=t)$ be a partition of $[0, t]$ and $|\Pi|$ be the maximum step size of the partition.

$$\int_0^t\Gamma(u)\mathrm{d}X(u) = \lim_{|\Pi|\to 0} \sum_{i=0}^{n-1} \Big(\Gamma(u) \big(X(k_{i+1})-X(k_i)\big)\Big)\text{.}\tag{2} $$

How do you prove that these two definitions, (1) and (2), are equivalent?

Motivating Example

You can define a discrete portfolio process $X$ recursively by $$X(k+1)-X(k) = \Delta(k)\big(S(k+1)-S(k)\big)+r\big(X(k)-\Delta(k)S(k)\big)\text{.}\tag{3}$$

You can write the value of the portfolio at time $X(t)$ as

$$X(t) = \sum_{k=0}^{t-1}\Big(\Delta(k)\big(S(k+1)-S(k)\big)\Big) +\sum_{k=0}^{t-1}\Big(r\big(X(k)-\Delta_k S(k)\big)\Big) + X(0)\text{.}\tag{4} $$

You may then want to take the limit of this process as $|\Pi|\to 0$, which should give you Itô integrals on the RHS of $(4)$ using definition $(2)$ that you then want to rewrite using definition $(1)$.


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