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lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Perturbation of a stochastic differential equation

Ho, I was just wondering where troll katastrofa with a wounded ego is.   You are stalking me with a vendetta, aren't you? You had zilch to say after the question had been posed for several days and when the mathematics was discussed but simply could not bear passing up the opportunity to talk trash once the opportunity presented itself. Haha, so pathetic. I understand you just cannot bear to see your twin go down alone. "Slow and balky"? Apt self-portrait of your ineptitude in "high school" linear algebra demonstrated to the hilt a mere few weeks ago. Do I need to exhibit the links here for all to see?

If you have anything mathematically meaningful to say, say it with mathematical rigor and some semblance of professionalism. Stop behaving like an infant throwing temper tantrum. Act like an adult in case you are one. Don’t make people think your psychological age is stunted at three. You don’t see yourself, but the immaturity you have displayed here is stunningly embarrassing.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Perturbation of a stochastic differential equation

Critter noise notwithstanding, I think I have found the answer to be affirmative and the convergence is pathwise almost surely. I have constructed a proof which will be written up later. The method can be extended to a more general setting.

katastrofa
Posts: 7236
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Perturbation of a stochastic differential equation

Pathwise and in distribution are two different things, genius.
Last edited by katastrofa on May 23rd, 2018, 7:38 am, edited 1 time in total.

ISayMoo
Posts: 1455
Joined: September 30th, 2015, 8:30 pm

### Re: Perturbation of a stochastic differential equation

I tried to explain it to him, but he just wouldn't listen.

katastrofa
Posts: 7236
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Perturbation of a stochastic differential equation

He understands only himself. And only himself understands him.

ISayMoo
Posts: 1455
Joined: September 30th, 2015, 8:30 pm

### Re: Perturbation of a stochastic differential equation

The problem with pathwise convergence is the same as with pointwise convergence of functions - it's a weak form of convergence. The rate of convergence will depend on what path you're on. So you don't know if the prices (or other expectations) computed for one process will converge to the prices computed for the other process, even if the processes convergence pathwise.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Perturbation of a stochastic differential equation

1. I meant to type "the answer to be affirmative and the convergence is in probability, possibly pathwise almost surely". In fact, the convergence in probability is uniform in every time interval.

2. The original question is:

Question: Does $y\rightarrow x_1$ with respect to the trajectories in some sense, e.g. in probability or distribution, as $\epsilon\rightarrow0$?

It is asking whether there is a convergence in a sense as yet to be determined. It is an open question. What are your objections to the question?

3. Your, ISayMoo's, previous posts demand to know the distribution of $x(t)$'s and $y(t)$ at every $t$. It is not clear what your objection is. Are you asking for the explicitly written expression of the distribution of $x(t)$'s and $y(t)$ at every $t$? What is it for?

Do you object to what I said as follows? Given the initial conditions $x(t=0), x_0(t=0), y(t=0), x_1(t=0)$, $x(t), x_0(t), y(t), x_1(t)$ are all uniquely determined for every sample point and so the distributions of those variables at every $t$.

4.
The problem with pathwise convergence is the same as with pointwise convergence of functions - it's a weak form of convergence.
Weak compared to what form? The pointwise convergence implies convergence in probability. The gist of the proof is that the set of the sample point whose distance to the target exceeds the positive $\epsilon$ infinitely many times in the sequence is a subset of the point-wise divergent set which is of measure zero. Therefore the former is stronger than the latter which in turn is stronger than convergence in distribution.  I can write out the proof if you want to see it.

More importantly, how is this an objection to my original question, which asks for if there is a convergence and if there is, in what sense the convergence is. How can you object to a question. My upcoming answer provides one solution. What is the objection in that?
The rate of convergence will depend on what path you're on.
This is true. Nevertheless the rate of convergence is irrelevant. However varied the rate is, the random variables at those sample points will eventually converge. The set of all those converging sample points can not be in the set $U(\epsilon)$ the point of which remains further than any given positive $\epsilon$ away from the target infinitely many times in the sequence. Since the measure of point-wise convergent set is $1$, the measure of  $U(\epsilon)$ has to be zero. Therefore the almost sure point-wise convergence implies convergence in probability.
So you don't know if the prices (or other expectations) computed for one process will converge to the prices computed for the other process, even if the processes convergence pathwise.
This is a non sequitur. The convergence of distribution is equivalent to the convergence of the expectation of all bounded continuous functions.

lovenatalya
Topic Author
Posts: 287
Joined: December 10th, 2013, 5:54 pm

### Re: Perturbation of a stochastic differential equation

Pathwise and in distribution are two different things, genius.
Good job for retaining something from the lesson. But do you have a point to make? Did your elementary school teach you how to read?

Gamal
Posts: 2188
Joined: February 26th, 2004, 8:41 am

### Re: Perturbation of a stochastic differential equation

It would be much funnier in person. You should meet and post here a short film.

Paul
Posts: 8995
Joined: July 20th, 2001, 3:28 pm

### Re: Perturbation of a stochastic differential equation

+1

katastrofa
Posts: 7236
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: Perturbation of a stochastic differential equation

I take the suggestion of meeting lovenatalya as an insult.

Paul
Posts: 8995
Joined: July 20th, 2001, 3:28 pm

### Re: Perturbation of a stochastic differential equation

This is a job for peacemaker Donald Trump.

Cuchulainn
Posts: 58395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Perturbation of a stochastic differential equation

I wanted to study physical mathematics, but they didn't offer that at Leiden University.
What's "physical mathematics"??

Cuchulainn
Posts: 58395
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:

### Re: Perturbation of a stochastic differential equation

It would be much funnier in person.
I'm not so sure about that. The bark might be louder than the bite One of them will hide in the corner of the bar.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: Perturbation of a stochastic differential equation

I wanted to study physical mathematics, but they didn't offer that at Leiden University.
What's "physical mathematics"??
The design and use of a slide rule or abacus?

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