On p. 157 of the book Applied Stochastic Differential Equations, there is an algorithm presented which claims to allow "exact" sampling from an SDE$^1$ $${\rm d}x=f(x){\rm d}t+{\rm d}\beta\tag1,$$ where $f=\nabla\psi$ for some scalar function $\psi$ and $\beta$ is a vector of standard Brownian motions. However, I don't understand how I implement this in practice.
The derivation is as follows: Let $$M_t:=\exp\left(\int_0^tf^\ast(\beta_s)\:{\rm d}\beta_s-\frac12\int_0^t\|f(\beta_s)\|^2\:{\rm d}s\right)\;\;\;\text{for }t\ge0.$$
Let $P_x$ and $P_\beta$ denote the measures of the processes $x$ and $\beta$, respectively. (Most probably, they mean the measures on $C([0,T];\mathbb R^d)$ (which is a metric space) for some fixed time $T>0$ and where $d$ is the space dimension.)
Then, by a suitable version of the Girsanov theorem, $$\frac{{\rm d}P_x}{{\rm d}P_\beta}\propto M_T.\tag2$$ Now, by the Ito formula, $$M_t=\exp\left(\psi(\beta_t)-\psi(\beta_0)-\frac12\int_0^t\|f(\beta_s)\|^2+(\nabla\cdot f)(\beta_s)\:{\rm d}s\right)\tag3$$ for all $t\ge0$ almost surely.
Up to this point, everything is clear to me (though the rigorous conditions on the objects need to be checked). However, now they say the following:
I don't even get what is meant. I never heard the term "biased" Brownian motion. And what do they mean with "final point"? Do they mean that the distribution of $\tilde\beta_T$ should be the normalized version of $$h(y):=\exp\left(\psi(y)-\frac1{2T}\|y\|^2\right)\tag4?$$ And if that's the case: (a) How is such a "biased" Brownian motion constructed? (b) How is it simulated in a computer program? (c) In which sense is this still a Brownian motion?
The final algorithm then reads as follows:
My open questions are: (d) How do I determine $M$ and $\phi$ in practice? Do I need to know them analytically? Or can we numerically obtain them? In my situation, I know almost nothing about $\psi$ (though I can assume that it is bounded from below and above). (e) What do they man simulation the "skeleton" of $\tilde\beta$? (f) How do I sample from a Brownian bridge?
$^1$ I've adopted he notation of the book in order to prevent confusion.