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Exact solutions for the probability density of various conditioned processes with an entrance boundary

Alain Mazzolo

Abstract

The probability density is a fundamental quantity for characterizing diffusion processes. However, it is seldom known except in a few renowned cases, including Brownian motion and the Ornstein-Uhlenbeck process and their bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In this paper, we utilize Girsanov's theorem, along with a variation of the method of images, to derive the exact expression of the probability density for diffusions that have one entrance boundary. Our analysis encompasses numerous families of conditioned diffusions, including the Taboo process and Brownian motion conditioned on its growth behavior, as well as the drifted Brownian meander and generalized Brownian excursion.

Exact solutions for the probability density of various conditioned processes with an entrance boundary

Abstract

The probability density is a fundamental quantity for characterizing diffusion processes. However, it is seldom known except in a few renowned cases, including Brownian motion and the Ornstein-Uhlenbeck process and their bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In this paper, we utilize Girsanov's theorem, along with a variation of the method of images, to derive the exact expression of the probability density for diffusions that have one entrance boundary. Our analysis encompasses numerous families of conditioned diffusions, including the Taboo process and Brownian motion conditioned on its growth behavior, as well as the drifted Brownian meander and generalized Brownian excursion.
Paper Structure (9 sections, 77 equations, 4 figures)

This paper contains 9 sections, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Probability densities of the conditioned processes
  3. Conditioned process with generator $\mathcal{L}_{II}. = -\mu \coth\left[\mu(a-x)\right] \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$
  4. Conditioned process with generator $\mathcal{L}_{I}. = - \frac{1}{(a-x)} \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$ (Taboo process)
  5. Conditioned process with generator $\mathcal{L_{\alpha \beta}}. = \alpha \left(1 - \coth\left[\alpha(\alpha t + \beta -x)\right] \right) \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$
  6. Conditioned process with generator $\mathcal{L^*_{\alpha \beta}}. = \left(\alpha - \frac{1}{\alpha t + \beta -x} \right) \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$
  7. Conditioned process with generator $\mathcal{L}_{e}. = \frac{1}{T-t}\left( X \coth \left(\frac{x X}{T-t}\right) -x \right) \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$ (generalized Brownian excursion)
  8. Conditioned process with generator $\mathcal{L}_{m}. = \left( \mu + \frac{ 2 \sqrt{\frac{2}{\pi (T-t)}} e^{-\frac{(x+\mu (T-t))^2}{2 (t-T)}} +2 \mu e^{-2 \mu x} \mathop{\mathrm{erfc}}\nolimits\left(\frac{x - \mu (T-t)}{\sqrt{2 (T-t)}}\right)}{1 + \mathop{\mathrm{erf}}\nolimits\left(\frac{x + \mu (T-t)}{\sqrt{2 (T-t)}}\right)-e^{-2 \mu x} \mathop{\mathrm{erfc}}\nolimits\left(\frac{x -\mu (T-t)}{\sqrt{2 (T-t)}}\right)} \right) \frac{\partial .}{\partial x} + \frac{1}{2} \frac{\partial^2 .}{\partial x^2}$ (Drifted Brownian meander)
  9. Conclusion

Figures (4)

  • Figure 1: $\tilde{p}_{II}(x,t)$ profile (blue curve) and the profile given by of its image $\tilde{p}_{II}(2 a - x,t)$ (blue dotted curve). The sum of both contributions (thick black curve) is the density probability function of the process $X_{II}(x,t)$ with drift $\mu_{II}(x) = -\mu \coth\left[\mu(a-x)\right]$ in the physical region $]-\infty,a[$. The parameters are $a=1$ (vertical line) and $\mu = -1$.
  • Figure 2: A sample of 100 diffusions for the conditioned process $X_{\alpha \beta}(t)$ with parameters $\alpha = 1/2$ and $\beta = 1$. The time step used in the discretization is $dt = 10^{-3}$. All trajectories generated with different noise histories are statistically independent. The thick black curve is the average profile of the stochastic process given by Eq.\ref{['Xmeantype_alpha_beta']}. Observe that at large times the process no longer feels the boundary and evolves almost freely.
  • Figure 3: A sample of 100 diffusions for the conditioned process $X_{\alpha \beta}(t)$ with parameters $\alpha = -1/2$ and $\beta = 1$. The time step used in the discretization is $dt = 10^{-4}$. All trajectories generated with different noise histories are statistically independent. The thick black curve is the average profile of the stochastic process given by Eq.\ref{['Xmeantype_alpha_beta']}.
  • Figure 4: A sample of 100 diffusions for the conditioned process $X^*_{\alpha \beta}(t)$ with parameters $\alpha = -1/2$ and $\beta = 1$. The time step used in the discretization is $dt = 10^{-4}$. All trajectories generated with different noise histories are statistically independent. The thick black curve is the average profile of the stochastic process given by Eq.\ref{['Xmeantype_alpha_beta*']}. Observe that the process $X^*_{\alpha \beta}(t)$ resulting from Doob conditioning remains globally closer to the boundary than process $X_{\alpha \beta}(t)$ studied in Sec.\ref{['subsec4']} and presented in Fig. \ref{['fig3']}.