Showing the Difference Between an Ito Integral & Riemann Integrals.

[mathfied]

Hi Everyone,
A problem I have here. I am trying to solve a problem involving Ito Integrals and Riemann interals.

Homework Statement

Prove

$$\int^{T}_{0} tdW(t) = TW(T) -\int^{T}_{0} W(t)dt$$

Homework Equations

I want to solve this question WITHOUT using Ito's Lemma directly.

The Attempt at a Solution

OK I know that in general, the Ito integral and the Riemann integral are going to be slightly different. The Ito Integral will have an extra term.

So in my question, I know if I was to generally just integrate the Riemann integral, I would get:
$$\int^{T}_{0} tdW(t) = TW(T)$$
But I know the Ito integral adds the extra:
$$-\int^{T}_{0} W(t)dt$$
But how does this arise? Is it possible to show how using the mean square error ? I keep reading the mean square error when reading about this but don't really see the connection. Maybe someone could kindly help out please?

Thanks

 

[mighty2000]

I would like to offer some guidance on how to approach this problem. Firstly, it is important to understand the fundamental differences between the Ito integral and the Riemann integral. The Ito integral is a stochastic integral, meaning that it is defined as the limit of a sequence of Riemann sums. This means that it takes into account the randomness of the process being integrated.

To solve this problem without using Ito's Lemma directly, you can use the definition of the Ito integral and the properties of stochastic integrals. Firstly, recall that the Ito integral is defined as a limit of Riemann sums:

\int^{T}_{0} f(t)dW(t) = \lim_{n\to\infty} \sum_{i=1}^{n} f(t_{i})\left(W(t_{i}) - W(t_{i-1})\right)

where t_{i} = \frac{iT}{n}. From this definition, we can see that the extra term in the Ito integral arises from the fact that the increments of the Brownian motion, W(t_{i}) - W(t_{i-1}), are random variables. Therefore, when taking the limit, we need to account for the randomness by including the term -\int^{T}_{0} f(t)dt.

To prove the desired result, you can start by expanding the Ito integral using the definition above and then manipulating the terms to show that it equals TW(T) -\int^{T}_{0} W(t)dt. You can also use the properties of stochastic integrals, such as linearity and the fact that the Ito integral of a deterministic function is equal to the Riemann integral.

I hope this helps in solving the problem. Good luck!