| Abstract: |
| In the talk, let $X$ be a solution to one-dimensional stochastic differential equations (SDEs), and consider discrete and continuous time maximum of the solution which are denoted as $M^{n}_{T}$ and $M_{T}$, respectively.
We will show some important properties of $p_{M^{n}_{T}}$ and $p_{M_{T}}$, where $p_{M^{n}_{T}}$ and $p_{M_{T}}$ denote probability density functions of $M^{n}_{T}$ and $M_{T}$, respectively.
In particular, we shall obtain expressions, upper bounds for $p_{M^{n}_{T}}$ and $p_{M_{T}}$
and that $p_{M^{n}_{T}}$ converges to $p_{M_{T}}$ as $n\to\infty$.
The key of the proofs is to show integration by parts formulas for $M^{n}_{T}$ and $M_{T}$ by means of the Malliavin calculus.
If time permits, we will consider some other properties of the density functions which have been obtained recently. |
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