B*******t
发帖数: 135 | 1 不好意思，不是我偷懒，做了好半天的，实在是才疏学浅没做出来。
据说这里牛人多，啥题目都能解决，所以来这里请教一下。
Steele 书(Stochastic Calculus and Financial Applications, J. Michael Steele)
上的 7.2 & 7.3
7.2
Show that if X_t is any continuous martingale and \phi is any convex
function, then Y_t=\phi(X_t) is always a local submartingale. Give an
example that shows Y_t need not be an honest submartingale.
7.3
Show that if X_t is a continuous local submartingale such that
E(sup_{0<=s<=T} |X_s|) < inf,
then {X_t:0<=t<=T} is an honest submartingale. Show h | f****e
发帖数: 590 | 2 7.2
by Jensen's inequality, Y_t is a local submartingale
the only thing that could make it not a true submartingale is integrability.
Try to construct a martingale which is in L1 but not in L2, and let \phi(x)
=x^2.
7.3
hint: use Fatou's lemma for conditional expectation.
good luck.
Steele)
【在 B*******t 的大作中提到】
: 不好意思，不是我偷懒，做了好半天的，实在是才疏学浅没做出来。
: 据说这里牛人多，啥题目都能解决，所以来这里请教一下。
: Steele 书(Stochastic Calculus and Financial Applications, J. Michael Steele)
: 上的 7.2 & 7.3
: 7.2
: Show that if X_t is any continuous martingale and \phi is any convex
: function, then Y_t=\phi(X_t) is always a local submartingale. Give an
: example that shows Y_t need not be an honest submartingale.
: 7.3
: Show that if X_t is a continuous local submartingale such that
| B*******t
发帖数: 135 | 3 Thank you for your reply!
integrability.
x)
It is very obvious that Jensen is the key to use since convex function is
the issue here.
However, what is tricky is how to find the localizing sequence. I considered
about the following
\tau_n = inf{t: sup|\phi(X_t)-\phi(X_0)| >= n, t >= n}
But I am stuck in proving
\tau_n -> inf almost surely.
I actually solved this problem by DCT instead of Fatou's lemma. Did you
actually mean DCT?
I have not seen how Fatou is involved here.
But still thank you for
【在 f****e 的大作中提到】
: 7.2
: by Jensen's inequality, Y_t is a local submartingale
: the only thing that could make it not a true submartingale is integrability.
: Try to construct a martingale which is in L1 but not in L2, and let \phi(x)
: =x^2.
: 7.3
: hint: use Fatou's lemma for conditional expectation.
: good luck.
:
: Steele)
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