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Jun
30
comment Ito's formula for Jump process
No, it's just arithmetic...The poisson jumps by 1, so $B_t$ jumps correspondingly.
Jun
30
comment Ito's formula for Jump process
@Behrouz thoughts?
Jun
30
comment Reflection Principle
Use the fact that $\widetilde{W}_t := W_{t \vee \tau} - W_\tau$ is a Brownian Motion independent of $W_\tau$ by the strong markov property, and that subsequently so is $-\widetilde{W}_t$.
Jun
30
comment Ito's formula for Jump process
@Behrouz I think you can use the product rule of ordinary calculus. The jump term itself doesn't require calculus at all.
Jun
30
comment Ito's formula for Jump process
Ito's Lemma isn't really necessary, since Poisson processes have finite variation.
Apr
27
comment Why the expected return rate of a stock has nothing to do with its option price?
Because the pricing is by no arbitrage.
Oct
9
comment Show that $E[B_t|\mathscr{F}_s] = B_s$
Yep, totally correct.
Oct
9
comment Show that $E[B_t|\mathscr{F}_s] = B_s$
Technical point: the pde condition on f only guarantees that it will be a local martingale. Integrability has to be checked separately.
May
22
comment Existence of a hedging portfolio and martingale property
ok, so the supermartingale is the value process corresponding to an american option. if i understand your question, then at a theoretical level, the answer is yes: doob-meyer decomposition to split the supermartingale into a martingale and decreasing process, and then martingale representation to get a hedging strategy for the martingale part.
May
22
comment Existence of a hedging portfolio and martingale property
what do you mean by a super martingale price process?
May
18
comment Difference betweem martingale property and adapted filteration
Also, this is something I think about from time to time. You can definite martingality intrinsically, by taking the filtration to be the natural one generated by $X$. Does this mean that martingality is an intrinsic property?
May
18
comment Difference betweem martingale property and adapted filteration
Continuity and measurability are defined slightly differently than what you said. For a continuous function, inverse image of an open set is open, and similarly with measurable. If you unpack the $\epsilon-\delta$ definition of continuity, you'll see that this is what it's logically equivalent to.
Apr
22
comment Distribution of Brownian Bridge
The starting point doesn't matter. You're conditioning on the Brownian Motion being $a$ at time $T_1$, so there's no variance there. Just do the calculation on $[0, T_2 - T_1]$.
Apr
9
comment unique equivalent martingale measure in incomplete markets
yeah. you apply $f(x)$ pointwise.
Apr
8
comment unique equivalent martingale measure in incomplete markets
yeah, thanks. this was a while ago, i have to try and understand what i wrote.
Apr
8
comment How can the Wiener process be nowhere differentiable but still continuous?
Continuity is a weaker property than differentiability.
Apr
8
comment How do you calibrate a poisson arrival rate process?
Don't you know $\delta$? You have to estimate $A$, where $A \leq 1$?
Mar
31
comment backward Kolmogorov equations - Markov properties
Answer below looks correct to me. Also, it doesn't make sense to say that $a$ and $b$ are ito-integrable, as they're just real-valued functions.
Mar
24
comment Definition of orthogonality and independence for a stochastic processes
I've spent quality time with all of these. Williams: Probability w/ Martingales, Oksendahl, Karatzas/Shreve, Protter: Stochastic Integration, Revuz/Yor: Continuous Martingales and BM, Kallenberg: Foundations of Modern Probability
Mar
24
comment Definition of orthogonality and independence for a stochastic processes
I'm not sure where I learned them, that's just how I remember them. I think they're both pretty standard.