Support theorem in p-variation and the positivity of the density

Yasushi ISHIKAWA

[Department of Mathematics, Ehime University (Japan)]

Recently, there is an expansive work by Simon [3] on the support theorem in pvariation space, where “rough paths” driven by Lévy processes live.

Let 1 ≤ p < 2. For an R^{d}-valued function f defined on I = [0, T], we put

∥f∥_{(p)} = v_{p}(f)^{(1/p)} = (sup_{{t_{i}}} ∑_{i=1}^{n} |f(t_{i}) − f(t_{i−1})|^{p})^{1/p}.  (1)

Here {t_{i}} denotes the collection of all finite partitions of I. We put W_{p} = {f; v_{p}(f) < +∞}, W_{p}(0) = {f; v_{p}(f) < +∞, f(0) = 0}. Further we introduce the following norms. ∥f∥_{p} = ∥f∥_{(p)} + ∥f∥_{0,∞}, where ∥f∥_{0,∞} = ∥f − f(0)∥_{∞}.

Lemma 1 ([2]) The space (W_{p}, ∥.∥_{p}) is complete, hence it is a Banach space.

We put D_{p} = D ∩ W_{p}. We introduce on D_{p} a metric d_{p}, called p-Skorohod distance, by

d_{p}(x, y) = inf_{λ∈Λ}{∥λ∥^{∘} ∧ ∥x − yλ∥_{p}}.  (2)

Here Λ is as above and ∥λ∥^{∘} denotes ∥λ∥^{∘} = sup_{s,t} | log \frac{λt−λs}{t−s}|. We call (D_{p}, d_{p}) the p-Skorohod space. We can show easily that (D_{p}, d_{p}) is complete and separable.

We would like to make a pathwise approch to the trajectories of the Lévy process. The following result is due to Bretagnolle.

Theorem 2 ([1]) Assume the index β of the Lévy measure is in [1, 2]. If p > β then

P (∥z∥_{p} < +∞) = 1.  (3)

Due to this theorem we can make use of the relation between the space of trajectories of index-β Lévy process and the W_{p} space.

Theorem 3 ([3]) For a Lévy process z ∈ W_{p}(0), for all ϵ > 0, all T > 0 we have

P (∥z̃∥_{p} < ϵ) > 0.  (4)

Here z̃ denotes the decomposed process of z. (see below)

Let Y_{t} be a canonical process. Then the process Y can be rewritten as

Y_{t} = x + ∫_{0}^{t} ∑_{j=1}^{m} f_{j}(Y_{s−})dz_{j}(s) + ∑_{s≤t} g(Y_{s−}, (Δz_{j}(s))),

where |g(x, z)| ≤ K|z|^{2} for |z| > 0 small.

We state a support theorem in the p-Skorohod topology along the line of Simon [3]. First we construct skeltons.

Let

K = {x; ∫_{|z|≤1} | < x, z > |μ(dz) < +∞},  (5)

and L = K^{⊥}, so that R^{d} = K ⊕ L. We put

U_{1} = {u = (u_{i}) = ((t_{i}, ζ_{i})); 0 < t_{1} < t_{2} < ... < T, ζ_{i} ∈ supp μ \ {0}}  (6)

and

α_{μ} = − ∫_{|ζ|≤1} ζ_{K}μ(dζ).  (7)

We transpose the process z into the decomposed process z̃

z̃_{t} = ∫_{0}^{t} ∫_{|ζ|≤1} ζ_{K}N(dsdζ) + ∫_{0}^{t} ∫_{|ζ|≤1} ζ_{L}Ñ(dsdζ) + ∫_{0}^{t} ∫_{|ζ|>1} ζN(dsdζ),  (8)

where ζ_{K}, ζ_{L} are orthonormal projection of ζ into the spaces K, L respectively.

For every ϕ_{L} : [0, T] → L, ϕ_{L} ∈ W_{p}, we put ϕ^{L} = ϕ_{L}(t) + α_{μ}t. Define ψ_{t} by

ψ_{t} = x + ∫_{0}^{t} f(ψ_{s})dϕ_{s}^{L} + ∑_{t_{i}≤t} g_{f}(ψ_{t_{i}−}, ζ_{i})  (9)

where

g_{f}(x, ζ) = f(x)ζ + g(x, ζ).

We put

supp _{p}Y_{.} = {φ ∈ D; for all ϵ > 0 P (d_{p}(φ, Y) < ϵ) > 0},  (10)

and

S_{p} = {ψ_{.} : ψ_{t} is as above u ∈ U_{1}}.  (11)

Then Theorem 3 implies the following

Theorem 4 ([3]) We have

supp _{p}Y_{.} = d_{p} − closure{S_{p}},  (12)

where the closure in R.H.S. is with respect to d_{p}-topology.

We put for t > 0

S_{t} = {ψ_{t} ∈ R^{d}; ψ ∈ S_{p}}.

Admitting the existence of the density fnction p_{t}(x, y) of Y_{.}, we can deduce from theorem 4 that

supp p_{t}(x, .) ⊃ S̄_{t},  (13)

where ¯ means the closure in Euclidian norm. Hence the remaining question will be to verify the positivity of p_{t}(x, y) in S_{t}, that is, to identify the set

S′_{t} ≡ {y ∈ R^{d}; p_{t}(x, y) > 0}.  (14)

In a very special case, we can show S′_{t} = R^{d} for all t > 0 by using the stochastic calculus. More precisely, we assume

the driving Lévy process z_{j} has a Lévy measure μ(dζ_{j}) = g(ζ_{j})dζ_{j}, j = 1, ..., m, where g(ζ) = η(ζ)|ζ|^{−1−β}, for some β ∈ (1, 2).

Here η ∈ C_{0}^{∞}(R), 0 ≤ η ≤ 1, η is symmetric, supp η = {|ζ| ≤ 1} and η(ζ) ≡ 1 in {|ζ| ≤ 1/2}. Then we have

Proposition 5

S′_{t} = R^{d} for all t > 0.  (15)

References

[1] J. Bretagnolle, p-variation de fonctions alatoires I , II, Sminaire de Probabilits, VI (Univ. Strasbourg, anne universitaire 1970-1971), Lecture Notes in Math., Vol. 258, Springer, Berlin, 1972, pp. pp. 51–63, 64–71.

[2] R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.

[3] Th. Simon, Small deviations in p-variation for multidimensional Lévy processes, Preprint, 2002.