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1
vote
2answers
169 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
56 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 ...
2
votes
1answer
151 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 $K=2j-1$,
$$f_{j,n}(t)=\left\{ ...
2
votes
1answer
274 views
Expectation of exponential martingale and indicator function.
Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$.
I want to evaluate
$$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
