Filtration and measure change

Solution 1:

You have $$\widetilde{W}_t=W_t+\int\Theta(u)du$$ which is in general not a Brownian motion, because it has a drift component.

But 5.3.1 states

$$M_t=M_0+\int \Gamma(u)dW_u\tag{5.3.1}$$

, which holds only for a Brownian motion $W$ (and $M_t$ martingale).

So one cannot trivially replace $W_t$ and $W_t+\int\Theta(u)du=\widetilde{W}_t$ in 5.3.2 aswell by setting

$$\widetilde M_t = \widetilde M_0 + \int_0^t \widetilde \Gamma(u) d \widetilde W_u\tag{5.3.2}$$

(because $\widetilde W_t$ is not in general a Brownian motion).

5.3.2 holds only under the special change of measure defined as $$Z_t = \exp\left\{ - \int_0^t \Theta(u) d W(u) - \frac{1}{2} \int_0^t \Theta^2(u) d u \right\}$$

Then $\widetilde M$ is a martingale, and $\widetilde W$ becomes a Brownian motion (proof is not trivial).

But still the filtration of $W_t$ and $\widetilde{W}_t=W_t+\int\Theta(u)du$ is obviously not the same.

Solution 2:

First of all, a filtration $( \mathscr{F}_t )_{t \geq 0 }$ is a "set" of sigma algebras indexed usually by time t that are increasing. That is, for every $t>0$, $\mathscr{F}_t$ is a sigma algebra and $\mathscr{F}_t \subseteq \mathscr{F}_T$ for all $0\leq t \leq T$. The canonical example, is the filtration generated by a process, say Brownian Motion $W$: The filtration $( \mathscr{F}_t )_{t \geq 0 }$ is such that $\mathscr{F}_0$ is the minimum $\sigma$-algebra such that $W_0$ is measurable with respect to it (that is if $W_0$ is $\mathscr{G}$-measurable, then $\mathscr{F}_0 \subset \mathscr{G}$ ) and $\mathscr{F}_t$ is the minimum $\sigma$-algebra such that all increments of $W$ up to time $t$ are measurable with respect to it. Having said that, the interpretation of filtration is that of the flow of information: as time progresses you know at least as much information as before. In particular they are useful to define the concept of a martingale.

We say $(M_t)_{t\geq0}$ is a martingale if the conditional expectation at time $t$ given the information up to time $s$ is the process at time $s$; that is, the best we can say about the process at time $t$ with the information up tp time $s < t$ is the process itself at time $s$, but which information and how to define it formally, this is where the concept of filtration comes in play. On a filtered probability space (a probability space with a filtration ) $( \Omega, \mathscr{F}, \mathbb{P}, (\mathscr{F} )_{t \geq 0 } ) $, the process $( M_t )_{ t \geq 0 } $ adapted to the filtration $( \mathscr{F}_t )_{t\geq 0}$ is a martingale if $M_0$ is integrable ( $M_0 \in L^1(\mathbb{P})$ ), and $$ \mathbb{E} \left [ M_t \right \vert \left. \mathscr{F}_s \right] = M_s, \quad \text{ for every }s \leq t $$ In particular, a process might be martingale with respect to one filtration and not with respect to another. Also, it is not necessarily the case in which, say the filtration generated by Brownian Motion $W$, and the filtration generated by $$X_t = W_t + \int_0^t \theta_s ds$$ are the same. I will give a famous example below.

I think the comment that Shreve wants to make is that the process $\widetilde M$ is a martingale in the filtered probability space $( \Omega, \mathscr{F},\widetilde{ \mathbb{P} }, (\mathscr{F} )_{t \geq 0 } ) $ (hence with respect to the filtration generated by $W$ ) still, it admits a stochastic representation representation like in Theorem 5.3.1 with $\widetilde W$. As you mentioned if the filtrations generated by $W$ and $\widetilde W$ were the same, then the result would be trivial, but they are in general not the same as I will give an example next.

This is a famous example due to Ito, and it can be found on the stochastic integration book by Protter, on the second section of the last chapter. Consider brownian motion $W$ with its respective filtration $(\mathscr{F}_t)_{t\geq 0}$. Now for every $t>0$, consider the filtration $\mathcal{G}_t$ that is the minimum filtration such that $\mathcal{F}_t \subseteq \mathcal{H}_t$ and $W_1$ is $\mathscr{H}_t$-measurable. It is easy to see that $(\mathscr{H}_t)_{t \geq 0 }$ is a filtration different from $( \mathscr{F}_t )_{ t \geq 0 }$ (since $W$ is not a martingale with respect to $\mathscr{H}$ ). What Ito showed was that the process $$ \beta_t = W_t - \int_0^t \frac{W_1 - W_s}{1-s}ds, t \in [0,1]$$ is a martingale with respect to the $\mathscr{H}$ filtration, and in fact (due to Levy's theorem) a Brownian Motion. So there you have two completely different filtrations for processes related by a simple drift addition.