8
votes
Accepted
Convergence of random functions
No. Take $f^n$ all independent to be $0$ with probability $1/\sqrt{n}$ and consisting of a bump of height $1$ and width $1/n$ at a uniformly distributed location otherwise. Clearly $f^n \to 0$ in law (...
- 10.3k
4
votes
Accepted
Designing an SDE satisfied by $\frac{B(t)}{1+t}$
The Brownian motion $dX=Adt+DdW$ transforms for $F=f(X,t)$ as
$$dF=\frac{\partial f}{\partial t}(X,t)dt+\frac{\partial f}{\partial x}(X,t)dX+\tfrac{1}{2}D^2\frac{\partial^2 f}{\partial x^2}(X,t)dt.$$
...
- 188k
2
votes
Accepted
What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?
Let $\mu_0$ be the law of a Brownian motion $B$. Let $\mu$ be any measure equivalent to $\mu_0$. Then by a converse version of Girsanov there exists a progressively measurable $F$ whose sample paths ...
- 1,904
2
votes
Accepted
Self-adjointness of generator and semigroup of an SDE
Let's at least elaborate on why $P_{s,t} $ in general is not self-adjoint, even if $L_t $ is. $P_{s,t} $ can be composed from infinitesimal time evolutions,
$$
P_{s,t} = \lim_{N\rightarrow \infty } \...
- 6,696
1
vote
Accepted
On the stationarity of Gaussian processes
Suppose that $(X_t)_{t\in\Bbb R}$ is a Gaussian process stationary in the wide sense, so that $m(t):=EX_t=m$ and $Cov\,(X_s,X_t)=g(s-t)$ for some real $m$, some real-valued function $g$, and all real $...
- 128k
1
vote
Accepted
Reconstruction of law of diffusion process from call option values
For any random variable $X$ with $E\max(X,0)<\infty$, you can determine the distribution of $X$ if you know the values of
$$g(c):=E\max(X,c)$$
for all real $c$.
Indeed, take any real $c$ and any ...
- 128k
1
vote
Accepted
What happens to an SDE conditional on the underlying Brownian motion being close to $f \in C[0, T]$?
It is not the case that the support is a singleton in general, and in fact I believe the support will be generically full when the dimension is high enough. I think the full picture is a bit subtle, ...
- 3,669
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