# stochastic-processes

### time series - Fractional Brownian motion - probability density function of the increments

There is a full derivation of the conditional distribution of fBM in Fink et al: "Conditional distributions of processes related to fractional Brownian motion", J. Appl. Probab. Volume 50, Number 1 (... Read More

### stochastic processes - How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?

Some basic details $\quad$ The Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily... Read More

### black scholes - Why should we expect geometric Brownian motion to model asset prices?

To provide a straight forward answer: It is not a good model. It never was, it never will be. Until we all do not come up with a better model that provides better modeling accuracy while it is equal... Read More

### stochastic processes - Variance of Multi-Dimensional OU process

This interesting question provides excellent links to Dynamic Nelson-Siegel Term Structure Models for interest rates for No Arbitrage and exposes key formulation in an interesting way. Appendix in p3... Read More

### stochastic processes - How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Your notations are really hard to follow as you define $\mathbb{P}$ twice at the beginning. The notation $\mathbb{P} = \mathbb{\hat{P}}$ and $\mathbb{P} =\mathbb{\tilde{P}}$ is not meaningful as the... Read More

### stochastic calculus - The distribution of jump gaps for Levy processes

This is a good shorter reference: http://www.impan.pl/CZM/tankov.pdf. Cont and Tankov have also written a longer book about modelling with Levy processes that I think is really good. There's going to... Read More

### stochastic processes - Ito`s Lemma problem

write down Ito's lemma for the function X: dX=\frac{\partial X}{\partial Y}dY+\frac{1}{2}\frac{\partial^2 X}{\partial Y^2}(dY)^2+\frac{\partial X}{\partial c}dc+\frac{1}{2}\frac{\partial^2 X}{\part... Read More

### interest rates - Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time. If you have a simpler model (like interpolation of these total variances from your vol... Read More

### stochastic processes - Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

IMHO the problem isn't stated correctly indeed, in the sense that the Radon-Nikodym derivative provided as the "solution" is not the unique way to define a measure $\mathbb{Q}$ equivalent to \$\mathbb... Read More