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katastrofa
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### Differential equation with a random walk forcing

I have an differential equation of the following form:
$\partial P(x,t)/\partial t= D \nabla^2 P(x,t) + \rho P(x,t) \delta(x-Q(t))$,
where $Q(t)$ is the Wiener process and $\delta$ is the Dirac delta (or some another function concentrated on $Q(t)$).
In the original problem that I'm trying to solve $Q(t)$ is discrete and on a lattice.

How can one solve it analytically? I've been staring at the Ito calculus and Kolmogorov forward / backward equations, but no epiphany.
Who if not you should be able to help with this kind of maths?

Alan
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### Re: Differential equation with a random walk forcing

The most I can come up with is two terms of a formal perturbation solution  -- in powers of $\rho$.

I'll suppose the spatial domain is the whole line and there is an initial function given at t=0, but no boundary conditions. If $\rho = 0$, suppose the PDE solution to that initial value problem is $P_0(x,t)$, a deterministic function. Then,

$P(x,t) = P_0(x,t) + \rho \, P_1(x,t) + O(\rho^2)$,

where $P_1(x,t) = \int_0^t \frac{1}{\sqrt{4 \pi D (t -s)}} \, e^{ - \frac{(x - Q(s))^2}{4 D (t -s)}} \, P_0(Q(s),s) \, ds$.

How do I get that? There is a standard solution to the inhomogenous heat equation with an initial condition, partly found here. At Wikipedia, the inhomogenous term (as it appears in the solution) is denoted $f(y,s)$. Just substitute
$f(y,s) = \rho \, P(y,s) \, \delta(y - Q(s)) = \rho \, P_0(y,s) \, \delta(y - Q(s)) + O(\rho^2)$, and then use the Dirac delta to do the spatial integral wrt $dy$ in the standard solution.

The solution at each (x,t) is a random function -- dependent upon the whole Brownian path from 0 to t. I suppose you could generate such paths via Monte Carlo, and then, for each path, do the integral above and get an answer. Each P(x,t), t > 0,  will then have a probability distribution that you can (approximately) develop with the above.

There is also a literature on stochastic PDE's. Beyond knowing that there's a literature, that's all I know about SPDE's.

Cuchulainn
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### Re: Differential equation with a random walk forcing

Maybe the stochastic heat equation is a good start. Eqs. 2.5 and 2.6 look similar to Alan's formula.

http://www.hairer.org/notes/SPDEs.pdf

This one may give some ideas as well

qu.edu.iq/journalcm/index.php/journalcm/article/view/646/477
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Alan
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### Re: Differential equation with a random walk forcing

Just thinking out loud, the next issue that interests me is: how to write a numerical solver for the full problem? (To get solutions accurate even for large $\rho$).

I would start by thinking of $Q(t)$ as just some given, single-valued, continuous function that starts at $Q(0) = 0$. Draw an example picture of one and turn it around so t is the vertical axis. Now, one wants to solve the heat equation with an initial function + an internal (moving) heat source that vanishes everywhere in the space-time domain -- except along the example curve, where it is an impulse-type function.

The tricky thing is that the intensity of the heat source depends (linearly!) upon the solution $P(x,t)$ at the heat source curve. This makes it trickier than (2.5)-(2.6) at Daniel's link.

At this point, the issue becomes a good question for Paul. Given the above approach, what are the boundary/continuity conditions on the solution as one crosses the internal heat-source boundary-curve at each level t? Once you pin those down, you can probably write a nice PDE solver that accepts an arbitrary initial function + an arbitrary, single-valued, continuous $\{ Q(t): 0 \le t \le T\}$. Suppose you've accomplished this.

Then, to get the full solution, run a combined Monte Carlo + solver routine:
For each of N trials:
{ generate a Brownian path from t=0 to t=T;
run the PDE solver, giving it the latest Brownian path;
collect the results $P(x,T)$;
}

katastrofa
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### Re: Differential equation with a random walk forcing

Thanks for the ideas! I will try them out and see what they change compared to my numerical solution.
(I solved it on a lattice by a simple split-step method, but I want to understand the properties of the solutions better.)

Cuchulainn
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### Re: Differential equation with a random walk forcing

How many space dimensions there?
What about MOL (Method of Lines) in space to get a systems of Random ODEs? Just sayin'

simple split-step method,
a) deterministic/stochastic
b) Lie-Trotter
?
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katastrofa
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### Re: Differential equation with a random walk forcing

I dunno which split-step method it is I just solved it in two steps.

If I could use your know-how once more, how do you deal with fractional derivates numerically on the lattice? On a 2D lattice, can I simply add the vertical and horizontal fractional derivatives like in the case of Gaussian diffusion (after all, I'm still in the alpha-stable family)?

Cuchulainn
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### Re: Differential equation with a random walk forcing

In space or time?

I don't have much savvy about the Fractional Calculus but this caught my eye a while back

https://www.ijert.org/research/analysis ... S04012.pdf

It defines what a fd is and applied fractional Euler to SIR ODE (it feels like the factorial n! for non-integral n).

And Fractional Laplace etc.??

https://arxiv.org/abs/1311.7691

As John von Neumann said, maths is just getting used to the symbols. There's lots of intro stiff out there.

https://en.wikipedia.org/wiki/Fractional_calculus
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katastrofa
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### Re: Differential equation with a random walk forcing

In space.
Cool papers! Thank you. I only don't understand why fractional derivates / the Levy flight models memory effects (from the paper about fractional SIR). Levy flight flight (specifically the clustering) is like Brownian motion + memory? (Sorry if I'm talking nonsense.)

Cuchulainn
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### Re: Differential equation with a random walk forcing

Levy processes model non-local (memory?) conditions an occur in finance (see book by Cont and Tankov). Eventually it is a PDE + IE equation which can be solved numerically.
I haven't see fractional Levy, but why not?

I would say that integrals model memory.

Alan?
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Alan
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### Re: Differential equation with a random walk forcing

I don't know much about fBm, but  --- I think a lot of the motivation has to do with the decay of the autocorrelation function of the stochastic process.

Typically (but not always), processes that are (noise) driven by standard Brownian motion increments have exponentially decaying autocorrelation functions. But, many real processes seem to decay more slowly, say with a power law. This is called "long memory".

Apparently, that's what you can get when the driver is  fractional Brownian motion or discrete time analogs. But, I have a vague recollection that the long memory property depends on whether the Hurst parameter H is greater than or less than 1/2.

In finance, volatility was thought to have long memory. Maybe it does. But, Jim Gatheral and colleagues looked carefully at fitting fBm-type models to volatility time series ("rough volatility"). His estimates for H came in on the wrong side of 1/2. The researchers liked the fBm models for other reasons, and so abandoned the long-memory requirement as a likely "statistical artifact" of short samples or some such. (details here). I tell this story because the same issue could occur in other applications.

If the idea is to get long memory, you don't have to have fBm. I remember cooking up some version of the Heston/square-root model for volatility that had this property -- with standard Brownian motion.  This was unpublished work a long time ago -- unfortunately,  I personally don't have long memory, so am vague on the details.

Cuchulainn
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### Re: Differential equation with a random walk forcing

Is all this related to hysteresis/materials with memory?

https://en.wikipedia.org/wiki/Materials_with_memory

and heard it on the grapevine, retarded SIR models from savvy armchair epidemiologists (delay ODE), maybe an idea

http://www.inase.org/library/2013/venic ... MCM-07.pdf

And delayed SIR

https://www.researchgate.net/profile/Fa ... ion_detail

These are solutions to problems, not sure if it's your problem (there are no requirements  )
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Cuchulainn
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### Re: Differential equation with a random walk forcing

One of my students is doing an MSc thesis on rough Heston model and Machine Learning. JIm Gatheral and colleagues is working in this area.

// Seems that fractional calculus was first mentioned by Leibnitz in 1695!
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Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: Differential equation with a random walk forcing

One of my students is doing an MSc thesis on rough Heston model and Machine Learning. JIm Gatheral and colleagues is working in this area.

// Seems that fractional calculus was first mentioned by Leibnitz in 1695!
Update: the student is also finished with the pricing of rough Heston models. It will be published in a number of weeks.
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