Probability Puzzle Page
Just for fun, here are some technical puzzles in stochastic analysis.
Maybe the answers are already out there in the literature (or may just
be obvious). Or maybe you have to do some work to solve them.
The problem of implicit Itô
This problem arises from financial mathematics. Suppose we have
a Markov diffusion X_t, driven by (say) a Brownian motion W_t. There
is also a convex function h, such that h(X_T) is integrable and we
can form the martingale M_t by taking the conditional expectation of
h(X_T) up to time t. As X_t is Markov, this martingale must equal
some function g(X_t,t) of X and time. Need this function satisfy
Itô's formula?
Good and better martingales
A martingale M_t, has to be L1. It might additionally be uniformly
integrable (UI). Further it might be in the Hardy space H1 (its maximum
value is L1). It might additionally be Lp-bounded (some p>1). Are there
some good examples of sequences of martingales which progressively have
more of these properties. That is, can you find a martingale which is not UI,
a UI martingale which is not H1, an H1 martingale which is not Lp-bounded
and so on?
A partial solution
Driftless bridges
It is possible for a driftless local martingale to hit a particular value
at a pre-nominated time. For instance the process with SDE
dX_t = (X_t / (T-t) ) dW_t, hits level 0 at deterministic time T. Is it
possible to do the same thing for a (random) stopping time T?
My thanks to David Hobson (Bath) and Mark Yor (Paris) for discussion and
suggestion of these problems.
Any comments or answers to
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