## Posts Tagged ‘**diffusion**’

## Fokker-Planck equation for a jump diffusion process

One of the simplest stochastic processes is a Brownian motion with drift, or a **diffusion process**:

(1)

Here, is Gaussian white noise with mean zero and variance . Its integral is a Brownian motion.

Continuous ItÃ´ stochastic processes such as eq. (1) are insufficient for applications where the random variable may jump suddenly (such as in avalanches). A natural extension of (1) for modelling this is a so-called **jump-diffusion process**. Let us suppose that our jump sizes are positive, independent and identically distributed with density . Then, the jump diffusion process is

(2)

where are i.i.d. jump sizes as above, and is the number of jumps encountered up to time . For simplicitly, let us assume that jumps occur independently with rate , i.e. that the probability to have a jump in a time interval is . Then, is a Poisson process with rate .

It is well-known that the diffusion process in (1) is equivalently described by a partial differential equation for the distribution of , the **Fokker-Planck equation** (FPE)

. (3)

This representation is useful e.g. for first-passage problems: they correspond to various boundaries introduced in the PDE (3). **So how does one generalise the Fokker-Planck (3) to the case of the jump-diffusion process in (2)? ** I will explain in the following that the answer is

, (4)

and then discuss a specific example.

## 1. Deriving the jump-diffusion FPE

Let us consider a time step from to . The probability for a jump to occur during this interval is , so

, (5)

where denotes averaging over all realizations of the Brownian motion , and denotes averaging over the distribution of the jump size . Since the jump term is already multiplied by the jump probability , the drift and noise contributions there are of higher order in and were dropped.

The averaging over the jump size in (5) yields a convolution with the jump size distribution :

.

The average over the noise in (5) is the same as for standard diffusion. During the interval , the increment of the noise term in (2) is

,

where the last equality is a definition for .

Since is a Brownian motion, is normally distributed:

.

Thus, the average over in (5) is

.

In the last line we used the fact that the normal distribution has mean zero and variance 1, and dropped all terms of higher order than . Inserting all this into (5), we obtain the jump-diffusion FPE (4).

## 2. Example: Exponential jumps

Now let us consider a simple example, where the jump sizes are distributed exponentially:

.

We will compute the distribution of , starting from .

For this, we solve the jump-diffusion FPE (4) with the initial condition . Let us take a Fourier transform of (4) from to :

.

The different spatial modes decouple, and instead of a partial differential equation we obtain an ordinary differential equation for each value of the Fourier variable .

For the above exponential form of the jump size distribution ,

.

Furthermore, the initial condition gives . Hence, the solution $\tilde{P}$ reads

.

While it does not seem feasible to invert this Fourier transform to obtain a closed expression for (but if anyone has an idea, let me know!), is already enough to determine the moments of . Taking derivatives, we obtain for example

.

Similarly, solving the FPE (4) with an absorbing boundary allows computing first-passage times (or at least their moments) for our jump-diffusion process.

Have fun, and do let me know if you have any comments!