I have two questions which are quite popular. I would like to ask how to train myselfto answer this kind of questions? if X, Y, and Z are 3 random variables such that X and Y are 90% correlated,Y and Z are 80% correlated, what is the minimum and maximum correlation thatX and Z can have?Suppose that x is a Brownian motion with drift m and unit variance, i.e. dx =m dt + dz. If x starts at 0, what is the probability that x hits 3 before hitting -5? any help will be highly appreciated.

A hint : the correlations you are talking about are the cosines of angles in the Hilbert space L²(Omega,P).

For the latter part, here are my thoughts: start with our measure P.Find a measure Q (by Girsanov) so that X(t) is brownian motion under Q. Let A be the event that X hits 3 before -5.If T is the stopping time inf{ t : X(t) = 3 or X(t) = -5 }, then X_{min(t,T)} is a martingale and we should get0 = E^Q[X_T] = 3 Q(A) - 5(1 - Q(A)) = 8Q(A) - 5, so Q(A) = 5/8So this tells us that if say m = 0 then the probability would be 5/8.Now, here is where I get confused because I want to use Girsanov's theorem, but over the random interval [0,T] where T is my stopping timeusing the form of the Girsanov density,P(A) = E^P[I_A]= E^Q[I_A * L_T^{-1} ] = E^Q[I_A * exp( 0.5*m^2 + mW(T)]= E^Q[I_A * exp( 0.5*m^2 + m*3] = Q(A) * exp(0.5*m^2 + 3m]= 5/8 * exp(0.5*m^2 + 3m)I feel like this is right but I get a bit confused once working with the density change.

how do u answer the first one

Well, you know all elements of the correlation matrix of X, Y, Z but for two (well, actually one since it is symmetric). Then the question is for what range of values of corr(X,Z) is the correlation matrix positive definite?

For the first one, see that X Y and Z are vectors in a 3-d space (generated by X, Y,Z).The question can be re formulated as follows : knowing that the cosine between X and Y is 0.9, and the one between Y and Z is 0.8, we'd like to kniw the max and min cosines (X,Z).So X is on a cone surrounding Y, and you would like to estimate the max and min cosine of angle (X,Z).This can be seen graphically : the max angle (corresponding to min correlation) is the sum of the angles (X,Y) and (Y,Z), and the min is the difference angle(Y,Z)-angle(X,Y). Hence you may calculate these cosine values by simple trigonometric manipulations.