# Tagged Questions

14 views

33 views

40 views

### Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
23 views

78 views

### Approximation of stochastic processes in Protter

I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes ...
34 views

### Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
33 views

### Stochastic Differential Equation- When martingale?

Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others?
37 views

### If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set?

Given a $X$ semi-martingale on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$ I am trying to prove: For any $B\in\mathcal F_\infty$ and processes $a_1,a_2$ such that ...
46 views

### The uniqueness of solution for stochastic differential equation involved with sign function.

When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following: There is a unique strong solution ...
53 views

### Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
39 views

### Distribution of Stopped Brownian motion at hitting time of another Brownian motion.

Suppose $B_t$ and $W_t$ are two independent Brownian motions and $\tau$ is the first hitting time of $B_t$ to some $a >0$. Compute the distribution of $W_{\tau}$. We can try the characteristic ...
30 views

### A few questions about Stochastic Processes and Numerical Methods

I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ...
I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...