You cannot define a Lévy process by the individual distributions of its increments, except in the trivial case of a deterministic process Xt − X0 = bt with constant b. In fact, you can't identify it by the n-dimensional marginals for any n.
1) Let X be a nondeterministic Lévy process with X0 = 0 and n be any positive integer. Then, there is a cadlag process Y with a different distribution to X, but such that (Yt1,Yt2,…,Ytn) has the same distribution as (Xt1,Xt2,…,Xtn) for all times t1,t2,…,tn.
Taking n = 2 will give a process whose increments have the same distribution as for X.
The idea (as in my answer to this related question) is to reduce it to the finite-time case. So, fix a set of times 0 = t0 < t1 < t2 < … < tm for some m > 1.
We can look at the distribution of X conditioned on the ℝm-valued random variable U ≡ (Xt1,Xt2,…,Xtm). By the Markov property, it will consist of a set of independent processes on the intervals [tk−1,tk] and [tm,∞), where the distribution of {Xt }t ∈[tk−1,tk] only depends on (Xtk−1,Xtk) and the distribution of {Xt }t ∈[tm,∞) only depends on Xtm. By the disintegration theorem, the process X can be built by first constructing the random variable U, then constructing X to have the correct probabilities conditional on U. Doing this, the distribution of X at any one time only depends on the values of at most two elements of U (corresponding to Xtk−1,Xtk). The distribution of X at any set of n times depends on the values of at most 2n values of U.
Choosing m > 2n, the idea is to replace U by a differently distributed ℝm-valued random variable for which any 2n elements still have the same distribution as for U. We can apply a small bump to the distribution of U in such a way that the m − 1 dimensional marginals are unchanged. To do this, we can use the following.
2) Let U be an ℝm-valued random variable with probability measure μ. Suppose that there exist (non-trival) measures μ1,μ2,…,μm on the reals such that μ1(A1)μ2(A2)…μm(Am) ≤ μ(A1×A2×…×Am) for all Borel subsets A1,A2,…,Am ⊆ ℝ.
Then, there is an ℝm-valued random variable V with a different distribution to U, but with the same m − 1 dimensional marginal distributions.
By 'non-trivial' I mean that μk is a non-zero measure and does not consist of a single atom.
By changing the distribution of U in this way, we construct a new cadlag process with a different distribution to X, but with the same n dimensional marginals.
Proving (2) is easy enough. As μk are non-trivial, there will be measurable functions ƒk on the reals, uniformly bounded by 1 and such that μk(ƒk) = 0 and μk(|ƒk|) > 0. Replacing μk by the signed measure ƒk·μk, we can assume that μk(ℝ) = 0.
Then
$$
\mu_V = \mu + \mu_1\times\mu_2\times\cdots\times\mu_m
$$
is a probability measure different from μ. Choosing V with this distribution gives
$$
{\mathbb E}[f(V)]=\mu_V(f)=\mu(f)={\mathbb E}[f(U)]
$$
for any function ƒ: ℝm → ℝ+ independent of one of the dimensions. So, V has the same m − 1 dimensional marginals as U.
To apply (2) to U = (Xt1,Xt2,…,Xtm), consider the following cases.
X is continuous. In this case, X is just a Brownian motion (up to multiplication by a constant and addition of a constant drift). So, U is joint-normal with nondegenerate covariance matrix. Its probability density is continuous and strictly positive so, in (2), we can take μk to be a multiple of the uniform measure on [0,1].
X is a Poisson process. In this case, we can take μk to be a multiple of the (discrete) uniform distribution on {2k,2k + 1} and, as X can take any increasing nonnegative integer-valued path on the times tk, this satisfies the hypothesis of (2).
If X is any non-continuous Lévy process, case 2 can be used to change the distribution of its jump times without affecting the n dimensional marginals: Let ν be its jump measure, and A be a Borel set such that ν(A) is finite and nonzero. Then, X decomposes as the sum of its jumps in A (which occur according to a Poisson process of rate ν(A)) and an independent Lévy process. In this way, we can reduce to the case where X is a Lévy process whose jumps occur at a finite rate, with arrival times given by a Poisson process.
In that case, let Nt be the Poisson process counting the number of jumps in intervals [0,t]. Also, let Zk be the k'th jump of X. Then, N and the Zk are all independent and,
$$
X_t=\sum_{k=1}^{N_t}Z_k.
$$
As above, the Poisson process N can be replaced by a differently distributed cadlag process which has the same n dimensional marginals. This will not affect the n dimensional marginals of X but, as its jump times no longer occur according to a Poisson process, X will no longer be a Lévy process.
I have TA'ed a "Mathematics for Future Elementary School Teachers" course. The point of the course is to develop a deep understanding of elementary school math (read: An actual understanding, rather than a knowledge of how to do computations). The book we used was Sybilla Beckmann's "Mathematics for Elementary Teachers".
At the end of the course, most students could really explain why 2/3 of 4/5 of a cup of milk was 8/15 of a cup of milk, and could draw a picture which showed why it was true. Ditto for the addition of fractions, and the algorithms for addition, multiplication, and division. I had many students who were flabbergasted that no one had ever shown them why these things were true before. Of course, I didn't actually show them: Sybilla's book is geared toward activities which help students to discover why these things work on their own or in small groups. The role of the teacher is to direct and clarify.
The reason that this course works, though, is because the students (at least initially) think that they are only learning how to explain these things to elementary school students. You never come right out and say "You do not understand addition, and I am going to show you". So it is a unique circumstance. Even then there are many students who resist the course because they feel like they don't have to put in any work to understand such "basic concepts". A lot of these students turn around when they realize that they do not really understand, and see that they are doing poorly on examinations and homework. Some of them do not ever feel comfortable enough to face their ignorance, and these people generally do not do so well in the course. A teacher must be humble enough to realize when they do not understand something, so it is a good thing that this course is a requirement for future teachers.
If you are serious about starting a course focused on elementary school math at the college level, which I think is a GREAT idea, I would use Beckmann's book. It is really fantastic. If you want more info, like an actual plan for a quarter's worth of work, I could email one to you.
Best Answer
I think the situation in which you are is not so uncommon with the proliferation of Financial Mathematics Master's programs...
One widely adapted solution is Steve Shreve's "Stochastic Calculus for Finance II: Continuous time Models". Whereas the book develops much of the more advanced techniques in close relationship with option prices, the chapter's 1-4 are an excellent introduction into Brownian motion and Ito calculus.
A bit on the lighter side I like Thomas Mikosch's "Elementary Stochastic Calculus with Finance in View", whereas on the more rigorous side I recommend Kuo's "Introduction to Stochastic Integration". In my classes I use the letter one, but only after an introduction into measure-theoretic probability. Its main shortcoming for financial applications is imho the treatment of the Feynman-Kac formula, which is best substitute from other sources (as Shreve).