8
votes
How can I learn stochastic process & stochastic calculus in two weeks?
This is impossible unless you are very intelligent with good memory-retention skills and already mathemathically proficient in the field of analysis and statistics (and no, a single course in basic ...
- 101
6
votes
Accepted
Quadratic Variation Of Mixed Brownian Motion and Poisson Process
Your attempt is correct.
The quadratic variation for a Poisson process is:
$$[N]_t=\lim_{\sup(t_{i+1}-t_i)\rightarrow0}\sum_{i:t_i\leq t}(N_{t_{i+1}}-N_{t_i})^2\tag{1}$$
for some partition $\Pi(t)=0\...
- 7,989
3
votes
Accepted
How do I learn the stochastic calculus of Poisson processes?
Summarizing the suggestions in comments:
Nicolas Privault's chapter Stochastic Calculus of Jump Processes [available online] provides only a very brief overview.
Chapter 11 of Shreve's II volume (...
- 10.9k
3
votes
Prove the jump times of a poisson process in a given interval are uniformly distributed
http://www.math.tau.ac.il/~uriy/Papers/encyc57.pdf - page 5 in this does that do it?
- 518
3
votes
Accepted
Second variation of a Brownian motion under jump-diffusion process
$$ X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$
$$ [X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$
(the author probably intended to use $0.5$ as jump size ...
- 5,008
2
votes
Accepted
Expected Value of Mean-Reverting Jump Process
First, we need to be careful about putting the condition at the right place:
\begin{align} e^{kt}\mathbb{E}[\mu_t] -\mu_0 &= \mathbb{E}\bigg[\sum_{m=1}^{N_t} e^{k\tau_m}\eta_m\bigg]\\
&= \...
- 136
1
vote
Accepted
Pure jump process in Duffie, Pan and Singleton's paper
Essentially yes - $Z_t$ is a compound Poisson process, except that the underlying counting process $N_t$ has intensity $\lambda(X_t)$. I.e
$$
N_t - N_s \sim Pois\bigg( \int_s^t \lambda(X_u) \mathrm{d}...
- 69
1
vote
Accepted
Separating jumps and diffusion
More some thoughts than a definite answer.
Even if you want to allow some form of price divergence between the two markets (because arbitrage oportunities are hard to monetise) it is probabily ...
- 2,003
1
vote
Accepted
Ito Lemma for Poisson Process
The basic Ito formula for a Poisson process is
$$ dY_t = \mu_t dt + g_t dN_t $$
$$ df(Y_t) = \mu_t f'(Y_t) dt + (f(Y_{t-}+g_t) - f(Y_{t-}))dN_t $$
(dropped $f$'s direct dependence on the time variable ...
- 5,008
1
vote
Poisson process under equivalent martingale measure
$N_t$ process comes with its own Poisson law (probability measure) $P$ defined via intensity $\lambda$. Under it, $N_t-\lambda t$ is a martingale wrt ${\cal F}_t =\sigma(N_u | u\in [0,t])$ (as $E^P[...
- 5,008
1
vote
Poisson process under equivalent martingale measure
Consider a radon nikodym derivative, the Random variable: $Z(w)= 1_{N(T,w)=1}+1-Pr(N(T)=1)$. It is admissible since it is always positive and has an expectation 1. This will lead us to the formation ...
- 1,875
1
vote
Accepted
Arbitrage free in a Black-Scholes/Poisson model
This is easy to answer with the meta theorem given in the same chapter. Here you have two sources of randomness (W and N), and one risky asset.
Q1: Arbitrage generally happens when you have more ...
- 6,204
1
vote
Marked poisson process vs compounded
The compound Poisson process isn't technically a marked process because we formulate the process with respect to $\sum_i D_i$ instead of $(\tau_i, D_i)$. However the compound process is constructed ...
- 111
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