BoundaryCrossingProbabilities.jl

This is a julia package for computing accurate approximations of the time-dependent boundary crossing probability for a general diffusion process. The implemented algorithm is based on a Brownian bridge corrected Markov chain algorithm proposed in:

Liang, V. and Borovkov, K.: On Markov chain approximations for computing boundary crossing probabilities of diffusion processes. J. Appl. Probab. 60 1386–1415 (2023).

For $T>0,$ $x_0 \in \mathbb{R},$ let $X$ be a solution to the following stochastic differential equation:

$$ \begin{cases} dX_t = \mu(t,X_t)dt +\sigma(t,X_t)dW_t, \quad t \in (0,T),\\ X_0 = x_0. \end{cases}$$

where we assume $\mu$ and $\sigma$ satisfy the usual sufficient for the existence and uniqueness of a strong solution to the above SDE.

For two continuous functions $g_-$ and $g_+,$ this package computes an approximation of the non-crossing probability

$$ F(g_-,g_+) := \mathbf{P}(g_-(t) < X_t < g_+(t) , t\in [0,T]). $$

More generally, the tool can compute expressions of the form

$$ v(t,x) = \mathbf{E}[e^{-\int_t^TV(s,X_s),ds}\psi(X_T);g_{-}(s) < X_s < g_+(s), s \in [t,T] | X_t = x], $$

and

$$ u(t,x) = \frac{\partial}{\partial x}\mathbf{E}[e^{-\int_0^tV(s,X_s),ds} ; X_t \leq x, g_{-}(s) < X_s < g_+(s), s \in [0,t]|X_0 = x_0], $$

which are known to be probabilistic solutions to the Dirichlet problem for the following parabolic PDEs:

$$ Lv = 0, \quad v(t,g_{\pm}(t)) =0, \quad v(T,x)= \psi(x). $$

$$ L^*u = 0, \quad u(t,g_{\pm}(t)) =0, \quad u(0,x)= \delta_{x_0}(x) $$

where

$$L f(s,x) := \dot{f}(s,x) - V(s,x)f(s,x) + \mu(s,x) f'(s,x) + \frac{1}{2}\sigma^2(s,x)f''(s,x).$$

It was proved by Liang and Borovkov (2024) that the boundary non-crossing probability functional $F(g) := F(-\infty,g)$ is Gateaux differentiable and the derivative admits the following representation:

$$ \nabla_h F(g) := \lim_{\varepsilon \to 0}\frac{F(g+\varepsilon h) -F(g) }{\varepsilon} = -\int_0^T h(t)v'(t,g(t))f_{\tau}(t),\quad h \in H, \quad g \in C^2.$$

The Markov chain approximation can be used to obtain approximations for $v'(t,g(t))$ and $f_{\tau}(t)$ for all $t \in [0,T].$

To install and import the BoundaryCrossingProbabilities package, run the following at the Julia REPL

using Pkg
Pkg.add(url = "https://github.com/pika-pikachu/BoundaryCrossingProbabilities.jl")
using BoundaryCrossingProbabilities

To set up the boundary crossing probability algorithm, we need to specify parameters of the diffusion process (initial position, drift, diffusion and potential), and the time interval. Let's take the Brownian motion example ($\mu \equiv 0,$ $\sigma \equiv 1$) with a complex potential term $V(t,x):= ix^2$.

x0 = 0 # Initial condition
μ(t,x) = 0 # Drift coefficient
σ(t,x) = 1 # Diffusion coefficient
V(t,x) = (1im)*x^2  # Potential
T = 1.0 # Terminal Time

Then we set up a Julia Type called MeshParams, which takes in all the diffusion process parameters.

p = BoundaryCrossingProbabilities.MeshParams(
    x0, # x0 Initial condition
    T, # Terminal time	
    μ, # Drift coefficient
    σ, # Diffusion coefficient
    V, # Potential
    false, # no target set
    [1.2,3], # Target set X_T \in [a,b]
    "Brownian", #bridge correction,
    false, # one sided boundary
    25, # number of time steps 
    0, # δ, 1/2 + δ is the space step power before the final time
    1, # pn power of the space step at the final time
    2, # γ, constant space scaling
    "trapezoidal" # integration scheme
	);

Then we define the upper and lower boundaries

gU(t) = 4 - t^2
gL(t) = -4 + t^2

Now we can obtain the solution to the problem.

plotFlag = true
interpolationFlag = false

non_crossing_probability, v = BoundaryCrossingProbabilities.BKE(p, t -> gU(t), t -> gL(t), interpolationFlag, plotFlag);
#soltn_FKE, u = BoundaryCrossingProbabilities.FKE(p, t -> gU(t), t -> gL(t), interpolationFlag, plotFlag);