Conditional Means of Multivariate Normals Under Inequality Constraints

Summary

The document presents an integral method for finding the conditional expectation of one component of a trivariate normal vector when two other components are restricted by upper bounds. It first obtains the conditional density of the target variable by integrating the joint density over the allowed region for the constrained variables, then normalizes by the probability of that region. The desired expectation is the integral of the target variable against this conditional density.

The joint density is specified through the covariance matrix and the multivariate normal quadratic form. This gives a general formulation that can be evaluated analytically in special cases or numerically. The material does not provide a closed-form simplification, computational procedure, or worked numerical example, and it does not connect the calculation to a particular trading application. It is useful as a probability and statistics tool for conditioning on joint threshold events.

Key ideas

• The target conditional mean can be computed by integrating the target variable against its conditional density.
• The conditional density is obtained by integrating the joint density over the constrained variables and dividing by the event probability.
• The event probability is an integral of the trivariate normal density over the specified region.
• The covariance matrix determines the joint normal density used in the calculation.

Tags

# How to compute conditional expectation of multivariate normal


# How to compute conditional expectation of multivariate normal












$(x_1, x_2, x_3)$~$N(0, \Sigma(\sigma_{ij}))$

then how to calculate $$E[x_2| x_1\leq a, x_3\leq b]$$

## Answer by msitt (score 1)

https://quant.stackexchange.com/a/31930

Your expectation is given by $$ \begin{align*} E[x_2 \:|\: x_1 \leq a, x_3 \leq b] &= \int_{-\infty}^\infty x_2 f(x_2 \:|\: x_1 \leq a, x_3 \leq b) \:dx_2 \end{align*} $$

To solve this problem you first need the pdf of $x_2 \:|\: x_1 \leq a, x_3 \leq b$. This is given by $$ f(x_2 \:|\: x_1 \leq a, x_3 \leq b) = \frac{\int_{-\infty}^a\int_{-\infty}^bf_x(x_1,x_2,x_3) \:dx_3dx_1}{P[x_1 \leq a, x_3 \leq b]} $$ where $$ P[x_1 \leq a, x_3 \leq b] = \int_{-\infty}^a\int_{-\infty}^\infty\int_{-\infty}^bf_x(x_1,x_2,x_3)\:dx_3dx_2dx_1 $$ and the joint probability distribution $f_x$ is given by $$ f_x(x_1,x_2,x_3) = \frac{\mathrm{exp}\left(-\frac{1}{2}x^T\Sigma^{-1}x\right)}{\sqrt{(2\pi)^3|\Sigma|}} $$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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