The conditions in the definition of Brownian motion

Question

A (standard, real-valued) Brownian motion $W = \{W(t): t \geq 0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t \geq 0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost surely, the function $t \mapsto W(t)$ is continuous.

As is well known, the above set of conditions can be reduced to 2), 3') for all $t \geq 0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0 \leq s \leq t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]

Answer 1

No, it is not true that a process W satisfying the properties (1), (3) and (4) has to be a Brownian motion. We can construct a counter-example as follows.

This construction is rather contrived, and I don't know if there's any simple examples. Start with a standard Brownian motion W. The idea is to apply a small bump to its distribution while retaining the required properties. I will do this by first reducing it to the discrete-time case. So, choose a finite sequence of times 0 = t₀ < t₁ < ... < t_n. Then define a piecewise linear process X by X_tk = W_tk (k = 0,1,...,n) and such that X is linearly interpolated across each of the intervals [t_k-1,t_k] and constant over [t_n,∞).

Then, Y = W - X is a continuous process independent from X. In fact, Y is just a sequence of Brownian bridges across the intervals [t_k-1,t_k] and is a standard Brownian motion on [t_n,∞). Also by linear interpolation, for any time t ≥ 0, X_t is a linear combination of at most two of the random variables X_t1,...,X_tn. The increments of W, $$ W_t-W_s = X_t-X_s + Y_t-Y_s, $$ are then a linear combination of at most 4 of the random variables X_t1,...,X_tn plus an independent term. So, choosing n ≥ 5, if it is possible to replace (X_t1,...,X_tn) by any other ℝⁿ-valued random variable without changing the joint-distribution of any 4 elements, then the distributions of the increments W_t - W_s will be left unchanged. So, properties (1), (3), (4) will still be satisfied but the new process for W will not be a standard Brownian motion. It is possible to change the distribution in this way:

Let X = (X₁,X₂,...,X_n) be an ℝⁿ-valued random variable with a continuous and strictly positive probability density p_X: ℝⁿ → ℝ. Then, there exists a random variable Y = (Y₁,Y₂,...,Y_n) with a different distribution than X but for which the projection onto any n - 1 elements has the same distribution as for X.

That is, for any k₁,k₂,...,k_n-1 in {1,...,n}, (Y_k1,Y_k2,...,Y_kn-1) has the same distribution as (X_k1,X_k2,...,X_kn-1).

We can construct the probability density p_Y of Y by applying a bump to the probability distribution of X, $$ p_Y(x)=p_X(x)+\epsilon f(x_1)f(x_2)\cdots f(x_n). $$ Here, ε is a fixed real number and f: ℝ → ℝ is a continuous function of compact support and zero integral, $\int_{-\infty}^\infty f(x)\,dx=0$. Then, $\int_{-\infty}^\infty p_Y(x)\,dx_k=\int_{-\infty}^\infty p_X(x)\,dx_k$ for each k. So, the integral of p_Y over ℝⁿ is 1 and, by choosing ε small, p_Y will be positive. Then it is a valid probability density function. Finally, as the integral along the k^th direction (any k) agrees for p_X and p_Y, the projection of X and Y onto ℝ^n-1 along the k^th direction give the same distribution.

Answer 2

Hi,

No you can't, because (1-3-4) implies that $W_t$ is a continuous martingale with quadratic variation equals to $t$ so by a very well known Lévy's Theorem, it has to be a Brownian Motion.

Regards