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3
votes
1answer
141 views

Solving SDE's on subsets of $R^n$.

It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n~~~~~~\sigma(\cdot,\cdot):[0,T] \times \mathbb{R}^n ...
1
vote
0answers
72 views

Langevin equation with uniform noise

Given the Langevin equation written in the form: $$\ddot{x}(t)+\lambda \dot{x}(t)=\mu(t)$$ if $\mu(t)$ is noise with gaussian $pdf$, the solution is well known in therms of the spectrum of the ...
2
votes
1answer
203 views

Reference request for Optimal Stopping (Stochastic Analysis)

I would like to start and get into the habit of reading some publications in different areas of mathematics, to get used to the writing style / mathematical sophistication etc. that is expected. In ...
1
vote
2answers
189 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 ...
3
votes
2answers
510 views

calculate all the equivalent martingale measure

Under the assumption of no arbitrage without vanish risk, in an incomplete market $(\Omega,{\cal F}, P)$, the set of equivalent martingale measure is NOT empty, i.e. ${\cal P} = \{Q: Q \sim P\}\neq ...
1
vote
1answer
195 views

basic application of Strong Law of Large Numbers

In $$ \sum_{j=0}^q {q\choose j}{1\over n}\sum_{i=1}^n X_i^j(-\bar X)^{q-j} \quad \overrightarrow{a.s.} \quad \sum_{j=0}^q {q\choose j} \mathbb{E}(X^j) (-\mathbb{E}(X))^{q-j} $$ using the Strong ...