1) Prove that the sum of two martingale is also a martingale

2) Show that the quadratic covariation is a bilinear map

3) Let X(t) and Y(t) be two continuous time stochastic processes. Find [X,Y] in terms of [X+Y] and [X-Y]

4) Let M(t) be a continuous time martingale and A(t) an increasing continuous time process adapted to the same filtration. Prove that the stochastic process X(t)=M(t)-A(t) is a supermartingale.

5)Suppose the amount of time X(n) till the n^th customer brings a claim to a cars insurance company is exp(0.5). Show that the aggregate amount of time S(n)=X(1)+....+X(n) increases linearly fast with a probability decreasing exponentially fast to 0 as the number of customers tends to infinity. i.e there are constants a,c>0 such that P[S(n) na] e^-nc for large enough n