# Propagating randomness

Ricardo: This post will be in english, for we have a very special guest today. Alexander Dobrinevski, author of the blog inordinatum.wordpress.com, is a good friend of mine from Belarus. Having finished his graduation at LMU, he followed the Advanced Studies in Mathematics program from Cambridge University to end up his doctoral thesis at the École Normale, where I had the pleasure to share with him the office for a few months. He is addicted to coffee, exotic movies and Laplace transforms. Today’s post is also very hardcore, to the benefit of our dear specialist in the field, and to mine, I who have learned so much and still do every time I get to talk to this dearest friend of mine.

# Introduction

I guess anybody doing statistical physics or probability theory has played around with Brownian Motion (by which I mean, here and in the following, the Wiener process, and not the physical phenomenon) at some time or another. It is used e.g. for modelling the price of a stock, the position of a particle diffusing in a gas or liquid, or the pinning force on an elastic interface in a disordered medium.

Being Markovian, time evolution of Brownian motion is completely determined by its propagator, i.e. the probability (density) to arrive at $x_1$ at time $t_1$ starting from $x_0$ at time $t_0$. This is, of course, known to be a Gaussian. However, for practical applications, one often needs to restrict the Wiener process (in general, with drift) to a half-line or an interval, with some imposed boundary conditions (absorbing or reflecting). In the example of a stock price, these would describe call or put options on the stock. In this post I will derive, hopefully in a pedagogical way, the propagator of Brownian motion with drift and linear boundaries.

# Propagators without drift

Let us first consider standard Brownian motion $W(t)$ without drift. Without loss of generality, let us assume it starts at $W(0)=0$. It satisfies the Langevin equation

$\displaystyle \dot{W}(t) = \xi(t)$,

where $\xi(t)$ is Gaussian white noise with correlation

$\displaystyle \overline{\xi(t)\xi(t')} = 2\sigma \delta(t-t')$.

With these conventions, the free propagator (i.e. the propagator without any boundaries) $P(x,t)$ is given by the solution of the Fokker-Planck equation

$\displaystyle \partial_t P(x,t) = \sigma \partial_x^2 P(x,t)$

with initial condition $P(x,0) = \delta(x)$. This PDE, also known as the heat equation, is easily solved by taking a Fourier transform. The solution is given by

$\displaystyle P(x,t) = \frac{1}{\sqrt{4\pi\sigma t}}e^{-\frac{x^2}{4\sigma t}}$.

Now, let us determine the propagator with an absorbing boundary at $x=b > 0$. In the Fokker-Planck equation, this is equivalent to the boundary condition $P(b,t)=0$, which makes applying Fourier transforms difficult. However, we can use the method of images to find the solution: $P(b,t)=0$ is enforced automatically if we add a negative source at $x=2b$ (the position of the original source, reflected at $b$), i.e. take the initial condition $P(x,0) = \delta(x)-\delta(x-2b)$. The final propagator with an absorbing boundary at $x=b$ is thus

$\displaystyle P^{(b)}(x,t) = \frac{1}{\sqrt{4\pi\sigma t}}\left[e^{-\frac{x^2}{4\sigma t}}-e^{-\frac{(x-2b)^2}{4\sigma t}}\right]$.

Similarly one can treat the case of two absorbing boundaries, one at $x=b>0$, one at $x=a<0$. One then needs an infinite series of images, and obtains the propagator as a series which can be rewritten in terms of Jacobi Theta functions.

# Propagators with drift

Now let us generalize to the Brownian motion with drift $\mu$. Then the Langevin equation for $W(t)$ becomes

$\displaystyle \dot{W}(t) = \mu + \xi(t)$.

The free propagator is obtained from the Fokker-Planck equation just as above:

$\displaystyle P(x,t) = \frac{1}{\sqrt{4\pi\sigma t}}e^{-\frac{(x-\mu t)^2}{4\sigma t}}$.

Let us now introduce again a constant absorbing boundary at $x=b$. Applying the method of images is not so straightforward anymore. Due to the drift, a path which goes from $x=0$ to the boundary $x=b$ will not have the same weight as the reflected path which goes from $x=2b$ to the boundary $x=b$. However, for the case of constant drift considered here, the weights of Brownian paths with and without drift have a simple relationship. In my view, the easiest way to see it is using path integrals. The propagator is given by

$\displaystyle P(x_f,t) = \int_{x(0)=0}^{x(t)=x_f} \mathcal{D}[x]e^{-\int_0^t \mathrm{d}s\, \frac{1}{4\sigma}\left(\dot{x}(s)-\mu\right)^2}$

Now, expanding the “action” in the exponent, using the fact that our drift $\mu$ is constant, and using our boundary conditions, this is equal to

$\displaystyle P(x_f,t) = e^{-\frac{\mu}{2\sigma}x_f+ \frac{\mu^2}{4\sigma}t}\int_{x(0)=0}^{x(t)=x} \mathcal{D}[x]e^{-\int_0^t \mathrm{d}s\, \frac{1}{2\sigma}\left(\dot{x}(s)\right)^2}$

We thus get a simple weight depending on the final position, but the remaining path integral is taken over a drift-less Brownian motion, and there we know the solution already, both with and without the boundary! In mathematical literature, you will often find this manipulation under the name of the Cameron-Martin-Girsanov theorem, but I find the path integral explanation much clearer for somebody coming from physics. Note that in the case where the drift $\mu$ is a function of time, we cannot pull the weight out of the path integral, because it involves the whole trajectory and not just the final point. This shows why non-constant drift with absorbing boundaries is a much more complicated problem (although the free propagator is still trivial to write down!).

The final formula for the propagator of the Brownian motion with drift $\mu$ and an absorbing boundary at $x=b$ (also known as Bachelier-Levy formula) is thus

$\displaystyle P^{(b,\mu)}(x,t) = \frac{1}{\sqrt{4\pi\sigma t}}e^{-\frac{\mu}{2\sigma}x+ \frac{\mu^2}{4\sigma}t}\left[e^{-\frac{x^2}{4\sigma t}}-e^{-\frac{(x-2b)^2}{4\sigma t}}\right]$

This is now all that is required to compute things like first-passage times, survival probabilities, etc. The generalization to two absorbing boundaries follows from the solution in the driftless case by multiplying with the same weight as here.

I hope you see that with the right tools, obtaining these propagators is nothing miraculous. If you have any questions or comments, I’d be glad to hear them! If people are interested, at some point I may write a continuation of this blog post, possibly on generalizations of the methods discussed here to Ornstein-Uhlenbeck processes, Bessel Processes, or more complicated boundaries.

Have fun, and thanks for reading!

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