User pierre - MathOverflow

Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

2012-09-12T20:27:49Z

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special case. Removing all unnecessary information, I try to solve the following problem.

Given a given a martingale $H_t$ and a semimartingale $X_t$, let $y(t,T)$ be the process solving

$ y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t] $

if a solution exists. I wouldn't even call this an SDE. I was able to come up with a solution in case of a deterministic $X_t$, but got stuck otherwise. It looks very similar to a OU-type SDE and looking at the solution of this kind of SDE, I thought

$y(t,T)=\mathbb{E}\left[ \mathcal{E}(X)_t\left( \frac{H_t}{\mathcal{E}(-X)_T}+\int_t^T\mathcal{E}(X)^{-1}_s(dH_s-d\langle H,X\rangle_s) \right)|\mathcal{F}_t\right]$

might work. This was some kind of educated guess, and for deterministic $X$ works fine (the integral vanishes). Also since $H$ is a martingale the integral simplifies. I'd like to solve for general $X$ (or at least an Ito-process).

Can standard SDE-Theory be applied here in any way? Is there a general method to solve such problems? Results about existence and uniqueness? I couldn't find anything in the literature. I'd be grateful for any hints.

Calculate $\mathbb{E}[\int_o^T N_{t-}dS_t]$ - what went wrong?

2012-04-05T16:00:18Z

First note, I had asked a similar question here, but the thread seems to have died, so I'll revive it here with more details. As a simplification of my real problem, I want to compute $\mathbb{E}[\int_o^T N_{t-}dS_t]$, where $S_t$ is a $(\mu,\sigma)$-geom. Brownian motion, and $N_t$ an independent Poisson process with const. intensity $\lambda$. Hence $\mathbb{E}[\int_o^T N_{t-}dS_t] = \mathbb{E}[\int_o^T N_{t-}S_t(\mu dt + \sigma dW_t)]$. Ito formula shows that the latter part of the integral should be square integrable, with integrable quadratic variation, so it is zero in expectation. So we should get

$\mathbb{E}[\int_o^T N_{t-}dS_t] = \mu\int_0^T\underbrace{\mathbb{E}[N_{t-}]}_{=\mathbb{E}[N_t-\mathbb{1}_{\Delta N_t \neq 0}]=\lambda t}\mathbb{E}[S_t]dt = \int_0^T \mu\lambda t e^{\mu t}dt$

$N_{t-}$ is a simple, left-continuous process, so I tried to confirm that numerically by calculating (MC)

$\mathbb{E}\left[ \sum_{i=0}^{N_t} i(S_{\tau_{i+1}}-S_{\tau_i})\right]$,

with $\tau_0=0$, $\tau_{N_t+1}=T$ and $\tau_i$ the exp-$(\lambda)$-distributed jump times. But values don't seem to converge to the analytic solution above. Is anything wrong?

Minimum-Distance estimation of mixed/mixture distributions

2012-01-28T23:44:58Z

Please note: I posted this first on Mathoverflow. As it might fit better on stats.stackexchange I reposted it there. Here's the link: Post on stats.stackexchange.com/

For my thesis, I currently have to fit some heavy-tailed data. As the fitted (positive, continuous) distributions will be used in a numerical integration procedure that involves Fourier transformation, I am restricted to distributions with analytic characteristic function. Some first analysis (Hill Plot etc) showed that the tails can be fitted by a stable distribution quite well. However, closer to zero this is not the case. So I played a little with a mixture (or mixed – a term that seems to be heavily overloaded in statistics) of stable and an exponential distribution, i.e.: $f(x)=c_1 f_{\text{exp}}(x) + c_2 f_{\text{stable}}(x)$, where $c_1 + c_2=1$. This seems to improve the fit significantly. The question remains, how to fit the mixed distribution. From what I've read, it seems reasonable to consider minimum-distance estimators, like Anderson-Darling to achieve a maximum-goodness of fit. I did not find any implemented algorithms for the minimum-distance procedure. So I wanted to use some numerical optimization algorithm that allows constraints (which I need) and implement it myself.

Does this approach make sense? Recommendations for the optimization method? I do not have an analytic Jacobian, of course. Is there any tested, implemented method? Should I use a different approach? MLE is “involved” as there is no distribution function of the stable distribution

Remark: Computational effort is not relevant. However, as this is only a minor issue of the thesis, I'd rather spend not too much time on it. Parameter estimation of heavy-tailed distributions is a vast field, combined with a lack of experience this can end in a disaster. So I'd be happy if someone points me into the right direction.

Answer by Pierre for Normality tests

2012-01-28T23:27:46Z

The Anderson-Darling test is considered one of the best tests for normality, I think.

Regular Conditional Probability given a natural filtration of a stochastic process

2011-07-24T17:27:32Z

OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot.

Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, (\mathcal{F}^\circ_t)_{t \leq T}, \mathbb{P}). \mathcal{F}^\circ_t=\sigma(X_s;s\leq t)$ is the uncompleted, natural filtration generated by $X_t$. Unfortunately $X_t$ neither has independent increments, nor is it markov. Since $\Omega$ is a Polish space, $\mathcal{F}^\circ_T$ and also $\mathcal{F}^\circ_t$ are countably generated, so we know, there exists a regular version of the conditional probability of $\mathbb{P}$ for any fixed $t$ for $\mathbb{P}$-a.a. $\omega$, i.e. for fixed $t$, $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$ is a prob. measure f.a.a. $\omega$.

Hence we know, that for all $t\in [0,T]\cup \mathbb{Q}$, we find a regular conditional probability f.a.a. $\omega$, depending on $t$. In words: Given almost any path of the process up to time $t$, we can deduce the probablity of events, taking that information into account.

On the remaining $\omega$'s, define some meaningless measure, so we have a measure $\forall \omega$. How can I extend this to all $t$ in a reasonable way? Reasonable means: There is one Null set $N$, so that $\forall t$ $\mathbb{P}(\cdot|\mathcal{F}_t)(\omega)$, $\omega\in N^c$, is a measure Anybody seen anything like this?

I read something like this only for Markov and Feller processes using infinitesimal generators, but this cannot be carried over one to one, because we do not have a transition semigroup.

Maybe I have a deep misunderstanding here. Grateful for any objections, hints and comments.

Fourier transform of distributions with non-standard test functions

2011-05-27T19:59:18Z

This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:

Given a Schwartz function $f \in S$ that is the density of some prob. measure, one can perfectly write for the Heaviside function $\theta$.

$\int \theta \operatorname{d\mathbb{P}}(x) = <\theta,f> = <\theta, F^{-1}Ff> = $

where $F$ and $F^{-1}$ denote the (inverse) Fourier transform.

Now $F^{-1}\theta$ is well known:

$\frac{1}{2\pi i}[P.V.(\frac{1}{x}) + i\pi \delta(x)]$

But: Smoothness is a way too restrictive property. The distribution $F^{-1}\theta$ is a well defined functional for test functions that have a bounded derivative in a neighbourhood $E$ around 0, and where $\sup_{x \in R} |x^\alpha (Ff)(x)|$ is finite (hence it is in $L^p$, $p<\infty$). $\alpha$ and $E$ are fixed! One can define a norm on that space (let's call it $C_\alpha$) using infty norm and infty norm of the derivative in E. This is also the standard standard proof for $F^{-1}\theta$ being in $S'$, the dual of the Schwartz functions.

I want to weaken the restrictions of test functions $f$, but I need to justify

$<\theta,f> = < F^{-1} \theta, Ff>$

it holds if f is a:

Schwartz function
$L^1$ function, $Ff$ also $L^1$ and in $C_\alpha$

However it is not clear (to me), if it holds if the space of test functions is not closed under Fourier transform. I'd like it to hold for as much prob. densities as possible for which $Ff \in C_\alpha$.

Of course Schwartz functions are dense in $L^1$, so I thought about using density arguments. $L^1$-convergence of the sequence (say $\psi_n$) of Schwartz functions implies only pointwise convergence under fourier transform. Too weak. I need some form of uniform convergence also under Fourier transform. I could achieve it if I assume that $|f(x)-\psi_n(x)| < b_n(x)$ outside a compact set where b_n is some sequence of $L^1$ functions controlling the "tail behaviour" of the pointwise convergence. $b_n$ goes to zero pointwise for n to infty and $b_n < B$; $B \in L^1$. However, I think this is again a too strong assumption - it rules out many densities (the fat tailed I am interested in).

I am not sure how to "save" the equation $<\theta,f> = < F^{-1} \theta, Ff>$.

Best Numerical Method for Evaluating a Hilbert transform

2011-07-09T21:41:15Z

I have to evaluate a Hilbert transform for some $\mathcal{L}^p(\mathbb{R},\mathbb{C})$-function ($1\leq p<\infty$). I know there are a number of algorithms out there to do it, but I don't have a full literature overview. I am aware of one of Stenger, which is based on Sinc approximation of analytic functions. But that is restricted to $p=1$ or $p=2$.

Short question: Any other favorable methods? Thanks for dropping some names and papers.

Weak*-continuity of regular conditional probabilities "in time"

2011-07-08T17:51:08Z

Let $(\Omega, F, (F_t)_{t\geq 0}, \mathbb{P})$ assume that $(X_t)_{t\leq T} $ is some cadlag, real valued stochastic process, not too bad: say something like a Brownian Motion and some Poisson finite activity jump process. The filtration is generated by $X_t$ in the usual way $F_t = \bigvee_{s\leq t} F_s$. We take the right continuous version of this filtration. If we include all Null sets of $F_T$ to get the usual assumptions, I am worried the filtration is then too large for my purpose (read on).

To have all the nice properties needed for Stochastic Analysis anyway and have a sufficiently small filtration at the same time, it should be possible to use the "natural" augmentation instead: sets which are contained in a countable union $(B_n)_{n\geq0}$ of sets of probability zero, such that $B_n \in F_{t_n}$ for all $n \geq 0$, where $t_n$ is some unbounded sequence in $\mathbb{R}^+$. (see Bichteler-Stochastic Integration with Jumps or Najnudel&Nikeghbali-A new kind of augmentation of filtrations)

Now, I want to know the probability of $X_T$ crossing a certain threshold $K$. Assume that we have a density, so for $t=0$ we have simply $\int_{[K,\infty]} f d\lambda$. I then applied some tricks to compute the integral. However, I would like to extend this for conditional expectations. First thought: Regular conditional probability. Assume that we can find one $\forall t>0$. We then have a measure for each level $x$ of $X_t$ for a.a. $x$. On the remaining x in the Null $N(t)$ set, one just defines some probability measure. Assuming then that each of these measures is abs. cont. w.r.t to the Lebesgue measure, I can apply the same trick as before. The Null sets $N(t)$ vary with t, so they are uncountable. So to apply the result using the density, I think I need one Null set $N$, independent of $t$, so I can compute the integral $\forall t$ for a.a. paths. Of course I have such an $N$ for a dense subset of $[0,T]$.

Let $P_{t,x}(\cdot)$ be the regular conditional probability in (t,x). Is it possible to establish weak*-continuity in $t$? Has anybody seen anything the like? I checked the internet for papers and books like Rao etc., though very elaborate, I didn't find a hint. So I am grateful for tips, or rectification. Maybe I come up with sth on the weekend...

Cheers

Answer by Pierre for Fourier transform of distributions with non-standard test functions

2011-06-19T08:46:08Z

OK, the proof doesn't seem so hard using density arguments and mollifiers! Needs a last check, but I think I got it.

Comment by Pierre

2012-09-13T19:48:20Z

Thank you! I'll check that paper out and let you know!

Comment by Pierre

2012-04-06T10:10:24Z

Well in this special case, as $H$ is constant between two jumps $\tau_i,\tau_{i+1})$, most of the terms would cancel out. However, can you give me a reference of the above theorem? Shouldn't this limit exist only for finite variation process $X$? - cf Protter p. 44 "Stochastic Integration and Differential Equations".

Comment by Pierre

2012-04-05T19:09:44Z

It probably isn't. But thank you anyway. I thought this requires only a glimpse. And it did. Indeed, there was only a $=$ missing. So again, thank you.

Comment by Pierre

2012-01-29T06:59:14Z

I posted it on the stats site: Here's the link: stats.stackexchange.com/questions/21900/…

Comment by Pierre

2011-07-24T21:30:23Z

You have control one a dense subset of [0,T], because then it is only a Nullset of $\Omega$, where you have to declare the nonsense measure. So I thought some kind of continuity (from the right?) might help, because we have cadlag paths.

Comment by Pierre

2011-07-24T21:20:10Z

I was not as precise, as I hoped I was: The situation is indeed much simpler. $\mathbb{P}$ is just the law of $X_T$. The question is, what can I say about the outcome of $X$ in T, given the path up to t. I want to compute $\mathbb{P}(\{f(X_T)\in \cdot \}|F_t)$ using $(\mathbb{P}|F_t)(\cdot,\omega)$ as a RCP. One may not need the path space $D$ here, though it might be the way to prove it. What I worry about is, that you always have to declare some nonsense measure on a Null set of $\Omega$ for each $t$ and you have no control which $\omega$ that concerns for some $t$.

Comment by Pierre

2011-07-10T20:33:49Z

Looks promising. Thanks.

Comment by Pierre

2011-07-10T20:25:13Z

Wow, that was a thorough answer! From first sight it looks a bit similar to what King has written in his book on the Hilbert transform in the chapter "The Hilbert transform via conjugate Fourier series". I'll have to take a closer look again tomorrow.

Comment by Pierre

2011-07-07T21:42:52Z

I'll do that soon.