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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.