The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
569 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
1
vote
2answers
185 views

Random diffusion coefficient in the Fourier equation

I'm stuck on the following simple problem: It's given the Fourier equation: $$\partial_t{u(x,t)}=\partial_x[k(t)\partial_xu(x,t)]$$where the diffusion coefficient $k(t)$ is a random variable with a ...
2
votes
1answer
89 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
1
vote
1answer
269 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
4
votes
1answer
74 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
2
votes
1answer
343 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...