Category: Unsolved problems

Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample

f(t,x)
Figure 1: The function f, convex in x and decreasing in t

Here, I attempt to construct a counterexample to the hypotheses of the earlier post, Do convex and decreasing functions preserve the semimartingale property? There, it was asked, for any semimartingale X and function {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} such that {f(t,x)} is convex in x and right-continuous and decreasing in t, is {f(t,X_t)} necessarily a semimartingale? It was explained how this is equivalent to the hypothesis: for any function {f\colon[0,1]^2\rightarrow{\mathbb R}} such that {f(t,x)} is convex and Lipschitz continuous in x and decreasing in t, does it decompose as {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t. This is the form of the hypothesis which this post will be concerned with, so the example will only involve simple real analysis and no stochastic calculus. I will give some numerical calculations suggesting that the construction below is a counterexample, but do not have any proof of this. So, the hypothesis is still open.

Although the construction given here will be self-contained, it is worth noting that it is connected to the example of a martingale which moves along a deterministic path. If {\{M_t\}_{t\in[0,1]}} is the martingale constructed there, then

\displaystyle C(t,x)={\mathbb E}[(M_t-x)_+]

defines a function from {[0,1]\times[-1,1]} to {{\mathbb R}} which is convex in x and increasing in t. The question is then whether C can be expressed as the difference of functions which are convex in x and decreasing in t. The example constructed in this post will be the same as C with the time direction reversed, and with a linear function of x added so that it is zero at {x=\pm1}. Continue reading “Do Convex and Decreasing Functions Preserve the Semimartingale Property — A Possible Counterexample” →

Do Convex and Decreasing Functions Preserve the Semimartingale Property?

Some years ago, I spent considerable effort trying to prove the hypothesis below. After failing at this, I spent time trying to find a counterexample, but also with no success. I did post this as a question on mathoverflow, but it has so far received no conclusive answers. So, as far as I am aware, the following statement remains unproven either way.

Hypothesis H1 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and right-continuous and decreasing in t. Then, for any semimartingale X, {f(t,X_t)} is a semimartingale.

It is well known that convex functions of semimartingales are themselves semimartingales. See, for example, the Ito-Tanaka formula. More generally, if {f(t,x)} was increasing in t rather than decreasing, then it can be shown without much difficulty that {f(t,X_t)} is a semimartingale. Consider decomposing {f(t,X_t)} as

\displaystyle f(t,X_t)=\int_0^tf_x(s,X_{s-})\,dX_s+V_t, (1)

for some process V. By convexity, the right hand derivative of {f(t,x)} with respect to x always exists, and I am denoting this by {f_x}. In the case where f is twice continuously differentiable then the process V is given by Ito’s formula which, in particular, shows that it is a finite variation process. If {f(t,x)} is convex in x and increasing in t, then the terms in Ito’s formula for V are all increasing and, so, it is an increasing process. By taking limits of smooth functions, it follows that V is increasing even when the differentiability constraints are dropped, so {f(t,X_t)} is a semimartingale. Now, returning to the case where {f(t,x)} is decreasing in t, Ito’s formula is only able to say that V is of finite variation, and is generally not monotonic. As limits of finite variation processes need not be of finite variation themselves, this does not say anything about the case when f is not assumed to be differentiable, and does not help us to determine whether or not {f(t,X_t)} is a semimartingale.

Hypothesis H1 can be weakened by restricting to continuous functions of continuous martingales.

Hypothesis H2 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and continuous and decreasing in t. Then, for any continuous martingale X, {f(t,X_t)} is a semimartingale.

As continuous martingales are special cases of semimartingales, hypothesis H1 implies H2. In fact, the reverse implication also holds so that hypotheses H1 and H2 are equivalent.

Hypotheses H1 and H2 can also be recast as a simple real analysis statement which makes no reference to stochastic processes.

Hypothesis H3 Let {f\colon{\mathbb R}_+\times{\mathbb R}\rightarrow{\mathbb R}} be such that {f(t,x)} is convex in x and decreasing in t. Then, {f=g-h} where {g(t,x)} and {h(t,x)} are convex in x and increasing in t.

Continue reading “Do Convex and Decreasing Functions Preserve the Semimartingale Property?” →

The Gaussian Correlation Conjecture 2

Update: This conjecture has now been solved! See A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions by T. Royen, and Royen’s proof of the Gaussian correlation inequality by Rafał Latała, and Dariusz Matlak.

We continue investigating the Gaussian correlation conjecture in this post. This states that if {\mu_n} is the standard Gaussian measure on {{\mathbb R}^n} then

\displaystyle \mu_n(A\cap B)\ge \mu_n(A)\mu_n(B) (1)

for all symmetric and convex sets {A,B}. In this entry, we consider a stronger `local’ version of the conjecture, which has the advantage that it can be approached using differential calculus. Inequality (1) can alternatively be stated in terms of integrals,

\displaystyle \mu_n(fg)\ge\mu_n(f)\mu_n(g). (2)

This is clearly equivalent to (2) when {f,g} are indicator functions of convex symmetric sets. More generally, using linearity, it extends to all nonnegative functions such that {f^{-1}([a,\infty))} and {g^{-1}([a,\infty))} are symmetric and convex subsets of {{\mathbb R}^n} for positive {a}. A class of functions lying between these two extremes, which I consider here, are the log-concave functions. Continue reading “The Gaussian Correlation Conjecture 2” →

The Gaussian Correlation Conjecture

Update: This conjecture has now been solved! See A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions by T. Royen, and Royen’s proof of the Gaussian correlation inequality by Rafał Latała, and Dariusz Matlak.

The Gaussian correlation conjecture is a simple looking inequality which, nevertheless, remains unproven. The standard n-dimensional Gaussian measure is

\displaystyle \mu_n(A) = (2\pi)^{-n/2}\int_A\exp(-\frac{1}{2}x^2)\,dx

for measurable subsets {A} of n-dimensional space {\mathbb{R}^n}. The conjecture states that

\displaystyle \mu_n(A\cap B)\ge \mu_n(A)\mu_n(B) (1)

for all convex and symmetric sets {A,B}. This inequality can be shown to be equivalent to the following statement. If {\{X_i\}_{i=1}^n} are jointly Gaussian random variables with mean zero, and {1\le k\le n}, then

\displaystyle {\mathbb P}\left(\max_{i\le n}|X_i|\le 1\right)\ge {\mathbb P}\left(\max_{i\le k}|X_i|\le 1\right){\mathbb P}\left(\max_{k<i\le n}|X_i|\le 1\right). (2)

A less general version of this inequality was initially stated in 1955 in a paper by Dunnet and Sobel, and the full version as stated above was conjectured in 1972 by Das Gupta et al.

Various special cases of the conjecture are known to be true. In 1967, Khatri and Šidák independently proved (2) for {k=1} or, equivalently, (1) in the case where {A} is a symmetric slab (the region between two parallel hyperplanes). The two dimensional case was proven in 1977 by Pitt, and the case where {A} is an arbitrary centered ellipsoid was proven in 1999 by Hargé.

In this entry we shall discuss the following interesting partial results due to Schechtman, Schlumprecht and Zinn in 1998.

  1. There is a positive constant {c} such that the conjecture is true whenever the two sets are contained in the Euclidean ball of radius {c\sqrt{n}}.
  2. If, for every {n}, the conjecture is true whenever the sets are contained in the ball of radius {\sqrt{n}}, then it is true in general.

Continue reading “The Gaussian Correlation Conjecture” →