This is somehow lengthy, but I think you will better understand how Girsanov actually works. The theorem you stated are more an application of Girsanov. The motivation behind Girsanov is the following: You are interested in how semimartingales behave under a change of measure. Since the finite variation part do not change, the question reduces to how local martinagles behave under a change of measure. As I was taught, Girsanov answers this question:
Suppose you have $Q\approx P$ and assume for simplicity that the density process $Z$ is continuous. If you have a continuous local martingale $M$ null at zero (wrt $P$), i.e. $M\in \mathcal{M}_{0,loc}^c(P)$, then $$\bar{M}=M-\int\frac{1}{Z}d\langle Z, M\rangle = M-\langle L,M\rangle \in \mathcal{M}_{0,loc}^c(Q)$$
where we write $Z=Z_0\mathcal{E}(L)$.
This is what I would refer to Girsanov's theorem. Note that this implies that in particular $M$ is $Q$-Semimartingale. Of course there are generalizations of this theorem ($Z$ general etc.).
Both of your theorems are the same. It is a special case of Girsanov. Take $M=W$, where $W$ is $P$ Brownian Motion. As an application of Girsanov you get:
If $W$ is a $P$-Brownian Motion and $Q\approx P$ with density process of the form $Z=\mathcal{E}(\int \Theta_s dW_s)$, for a predictable process $\Theta$. Then $W$ is under $Q$ a Brownian Motion with drift, i.e.$$ W=\bar{W}+\int\Theta_s ds$$ for a $Q$-Brownian Motion $\bar{W}$.
This is an immediate consequence of Girsanov and the proof is straight forward using Lévy's characterization of Brownian Motion. However in most cases you have to go the other way around: Usually you do not have $Q\approx P$. This means you have a probability measure and want to construct an equivalent probability measure $Q$ such that the density process $Z$ is a stochastic exponential. Hence you start with $L\in\mathcal{M}_{0,loc}^c(P)$ and define $Z:=\mathcal{E}(L)$. You hope that $Z$ can be used to define an equivalent probability measure $Q$, as $\frac{dQ}{dP}=Z_\infty$. We have $\mathcal{E}(L)=Z$, hence $Z$ is a local martingale and strictly positive. Therefore it is a supermartingale on $[0,\infty)$ (use Fatou to prove that)! By the supermartingale convergence theorem $Z_t$ converges $P-a.s.$ to $Z_\infty$. The problem is, $Z_\infty$ can be $0$ or $E[Z_\infty]<1$ (or both together). As already mentioned you want do define $\frac{dQ}{dP}:=Z_\infty$. You want at least that this $Q$ is absolutely continuous w.r.t $P$, i.e. $Q\ll P$. Hence you need at least
- $Z_\infty >0$
- $E[Z_\infty]=1$.
A priori, as said before, $Z_\infty=0$ and/or $E[Z_\infty]<1$. Hence we must find some conditions, such that $1.$ and $2.$ are true. For $1.$ we must have $\langle L\rangle_\infty < \infty$ (by definition of $Z_\infty=e^{L_\infty -\frac{1}{2}\langle L \rangle_\infty}$). For $2.$ you can use: $E[Z_\infty]=1 $ if and only if $Z$ is a uniformly integrable $P$ martingale on $[0,\infty]$. Now there is a famous condtion, called Novikov's condition, which gives a sufficient condition of $Z=\mathcal{E}(L)$ to be a uniformly integrable martingale on $[0,\infty]$.
Looking at your question. The theorem from Wikipedia is exactly my second statement with $X:=\int\Theta_s dW$. Note that $[W,X]=\langle W,X\rangle = \langle W,\int \Theta_s dW \rangle = \int \Theta_s d\langle W,W\rangle = \int\Theta_s ds$. Furthermore $Z:=\mathcal{E}(X)$. The whole difference between the theorem from Wikipedia and Shreve is in specifying the process $X$ further. Shreve assumes that $X$ has a particular form, i.e. an integral w.r.t to a Brownian Motion. That is the only difference.
In finance you often work on finite time horizon, i.e. on $[0,T]$. You can easily extend everything to $[0,\infty)$ by setting everything equal zero outside $[0,T]$.
Stochastic calculus is to do with mathematics that operates on stochastic processes.
The best known stochastic process is the Wiener process used for modelling Brownian motion.
Other key components are Ito calculus & Malliavin calculus.
Stochastic calculus is used in finance where prices can be modelled to follow SDEs. In the Black-Scholes model, prices follow geometric Brownian motion.
The Ito integral is one of the major components in stochastic calculus. It is defined as the integral
\begin{equation*}
\int HdX
\end{equation*}
where $X$ is a semimartingale & $H$ is a locally bounded predictable process. Note that we cannot use the generalized Riemann-Stieltjes integral because strong bounded variation is assumed & Brownian motion is not of bounded variation.
Stochastic analysis is looking at the interplay between analysis & probability.
Examples of research topics include linear & nonlinear SPDEs, forward-backward SDEs, rough path theory, asymptotic behaviour of stochastic processes, filtering, sequential monte carlo methods, particle approximations, & statistical methods for stochastic processes.
Something I am interested in is how probabilistic techniques can be used to settle problems in in harmonic analysis, such as proving the $L^p$ boundedness of the Riesz transform. For $f\in C^1_K:$
\begin{equation*}
||R_jf||_p\leq c||f||_p,~1<p<\infty
\end{equation*}
for some constant $c.$
A technique is to give a probabilistic interpretation of the Riesz transform before proving $L^p$ boundedness:
\begin{equation*}
R_jf(x)=c\lim_{s\to\infty}\mathbb{E}^{(0,s)}_{x}\int^{\tau}_{0}A\bigtriangledown u(Z_r)\cdot
\end{equation*}
where $u$ is the harmonic extension of $f,~f\in C^{\infty}_K.$ Here $A$ is the $(d+1)\times (d+1)$ matrix with $A_{ik}$ zero unless $i=d+1,~k=j,$ in which case it is one. Here, $Z_t$ is Brownian motion in $\mathbb{R}^2.$
I hope this is okay. I am happy to expand on any points in more detail if you would like.
Best Answer
just to offer two cents on this (health warning is that this is from an ex-trader rather than quant, and based on personal experience through self-study rather than eg as part of a formal PhD programme):
Williams really is fantastic, learned the basics of measure theoretic probability from that as an undergrad, and it's stood the test of time and is still a classic.
After this, some natural canonical texts would be:
I think this really does contain a huge amount of material (in addition to containing the material from Probability with Martingales so in some sense is a natural transition). You'll certainly find the standard Ito/Girsanov/Radon-Nikodym material well presented therein.
A different tack would be:
This was very popular when I was reading up on SDEs, and has a somewhat less formal style than some of the other standard references.
This is a bit more encyclopaedic than Oksendal, but again was very popular when I was reading the material about 10 or so years ago. More heavy going than Oksendal, and possibly overkill if the ultimate aim is more finance than analysis orientated.
I didn't use this myself, (again I was reading for interest and as ancillary to finance rather than for embarking on a stochastic analysis PhD - am a number-theory/algebra nut at heart!), but this is I think a classic text, although more formal than the others I've mentioned.
Finally a rather pleasing book is:
This has a nice survey, as the title suggests, of some of the functional analytic underpinnings of measure-theoretic probability, and I found the exposition a delight to read.
Hope some of those help, these are not finance books (sounds like you've got that covered), definitely can offer some views on that side of things should you need depending on which area of finance might interest most (credit/rates etc..). Many of the finance books by authors such as Brigo are highly rigorous but much much better suited to assimilating the finance concepts and acquiring facility with actual problems that matter 'at the coal face', but again depends on the aim / perspective.
Good luck and cheers!