Joint Brownian Boundary Crossings and Endpoint Distributions

Summary

The document asks how to find the joint endpoint distribution of two correlated Brownian motions when each path is also required to cross its own lower boundary before a fixed horizon. It describes the ordinary bivariate normal distribution of the two endpoints, parameterized by the drifts, volatilities, time, and correlation. It also mentions estimating each process’s individual boundary-crossing probability with a Brownian bridge.

The central issue is whether the endpoint density can be weighted by those two separate crossing probabilities to obtain the desired conditional or stopped-path distribution. The document does not provide a solution or evidence that independent weighting is valid. Because the paths are correlated, crossing events can be dependent, and endpoint values also affect crossing probabilities. A correct construction would need to account for the joint path constraints rather than assume that correlation in the endpoints alone resolves their dependence.

Key ideas

• The two Brownian endpoints have a correlated bivariate normal distribution under the stated drift and volatility assumptions.
• A Brownian bridge can be used to assess boundary crossing conditional on specified endpoints.
• The separate crossing events may be dependent because the processes are correlated.
• Weighting the endpoint density by marginal crossing probabilities is not established as a valid joint distribution method.

Tags

# Joint distribution correlated Brownian motions weighted by stopping probabilities


# Joint distribution correlated Brownian motions weighted by stopping probabilities












I want to find the joint distribution of 2 correlated Brownian motions, $X$ and $Y$, at time $T$ after crossing boundaries $a<0$ and $b<0$ respectively with initial values $X_0=Y_0=0$, drifts $\mu_X>0, \mu_Y>0$ and volatilities $\sigma_X>0, \sigma_Y>0$.

For fixed $X_T$ and $Y_T$, I can obtain the joint distribution $$f_{X,Y}(x,y)=\frac{\exp(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})\mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{\mu}))}{\sqrt{2\pi}\det(\mathbf{\Sigma})}$$ with $$\mathbf{\Sigma}=\begin{pmatrix}\sigma_X^2T&\sigma_X\sigma_Y\rho T\\ \sigma_X\sigma_Y\rho T&\sigma_Y^2T\end{pmatrix}$$ and $\mathbf{\mu}=(\mu_XT,\mu_YT)^T$ incorporating correlation $\rho$ and for both $X_T$ and $Y_T$. Subsequently, I can also obtain the individual probabilities that for each Brownian motion, $X$ and $Y$, with $X_0=Y_0=0$, $X_T$ and $Y_T$ the boundaries $a$ and $b$ were crossed through construction of a Brownian Bridge, $\mathbb{P}(\inf_{0\leq t\leq T} X_t\leq a)$ and $\mathbb{P}(\inf_{0\leq t\leq T} Y_t\leq b)$.

Does weighting the joint distribution of $X_T$ and $Y_T$ by the individual probabilities of crossing boundaries $a$ and $b$ respectively for each Brownian motion, $X$ and $Y$, lead to the correct distribution as mentioned above? Can this weighting be performed individually and separately as the correlation is already incorporated in the joint distribution of $X_T$ and $Y_T$ or are these probabilities dependent?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.