Humedad Relativa

The property of “moving along rays”, implied by Proposition2.8, suggests considering test functions on the plane which have good properties only along every ray emanating from the origin. The following definitions and results formalize this idea.

Definition 2.9. Let D be the class of Borel-measurable functions g : R² → R , such that

(i) for every θ ∈ [0, 2π) , the function r 7→ gθ(r) := g(r, θ) is differentiable on [0, ∞), and the derivative r 7→ g_θ⁰(r) is absolutely continuous on [0, ∞);

(ii) the function θ 7→ g⁰_θ(0+) is bounded; and

(iii) there exist a real number η > 0 and a Lebesgue-integrable function c : (0, η] → [0, ∞) such that |g⁰⁰_θ(r)| ≤ c(r) holds for all θ ∈ [0, 2π) and r ∈ (0, η].

Definition 2.10. For every given function g ∈ D , we set for every x ∈ R²\ {0} :

g⁰(x) := g⁰_θ(r), g⁰⁰(x) := g_θ⁰⁰(r) , where x = (r, θ) in polar coordinates.

Proposition 2.11. Every function g ∈ D has the following properties:

(i) The mappings (r, θ) 7→ g_θ⁰(r) and (r, θ) 7→ g_θ⁰⁰(r) are Borel-measurable on R²\{0} , and the mapping θ 7→ g_θ⁰(0+) is Borel-measurable on [0, 2π). Also, for the constant η > 0 in Definition 2.9(iii) for g , we have

sup

0<r≤η θ∈[0,2π)

|g⁰_θ(r)| < ∞ .

(ii) The function g is continuous in the tree-topology.

Proof: (i) The first claim is a consequence of the measurability of the function g and of Definition 2.9(i), because of the definition of derivatives as limits. The second comes from the fact g_θ⁰(r) − g_θ⁰(0+) = Rr

0 g⁰⁰_θ(y) dy and the requirements (ii) and (iii) in Definition 2.9.

(ii) By the fact g_θ(r) − g(0) = Rr

0 g_θ⁰(y) dy and the second claim of (i), the function g is continuous at the origin in the tree-topology. The continuity at other points is equivalent to the continuity of the functions r 7→ g_θ(r) for all θ ∈ [0, 2π) , and this is implied by (i) of Definition2.9.

We remark that, in Definition 2.9 we do not even assume local boundedness for derivatives of functions in the class D ; we assume only some boundedness near the origin. The reason why this will suffice for the development of a stochastic calculus for semimartingales on rays, is provided by the following two preliminary results. These supply the keys to the main result of this section, Theorem2.15.

Lemma 2.12. Let f : R² → [0, ∞) be Borel-measurable, and with the following properties:

(i) For every θ ∈ [0, 2π) , the function r 7→ f (r, θ) is locally integrable on [0, ∞) . (ii) There exist a real number η > 0 and a Lebesgue-integrable function c : [0, η] → [0, ∞) such that f (r, θ) ≤ c(r) holds for all θ ∈ [0, 2π) and r ∈ [0, η] .

Then for any semimartingale on rays X(·) in the context of Definition2.1, we have

P

Lemma 2.13. In the context of Definitions 2.1, 2.9 and 2.10, we have, for every semimartingale X(·) on rays and for every function g ∈ D, the properties

(i) P

Remark 2.14. Lemma2.12can be thought of as an analogue of the Engelbert-Schmidt 0-1 law (cf. [9] and Section 3.6.E of [27]), as it gives a condition guaranteeing the finiteness of some integral functional of the process X(·). In contrast to the necessary and sufficient condition of local integrability considered in the Engelbert-Schmidt 0-1 law, the condition here is only sufficient, due to the difficulty in dealing with the

“roundhouse singularity” at the origin.

Proof of Lemma 2.12: By condition (ii) of Lemma 2.12, we have f (x) ≤ c(kxk) whenever x ∈ R² and kxk ≤ η . In conjunction with (1.4) and (2.1), we have on the

c(kX(t)k) 1{kX(t)k≤η}dhkXki(t) = 2 Z η

0

c(r)L^kXk(T, r)dr ;

whereas, by Theorem1.4, the mapping r 7→ L^kXk(T, r, ω) is RCLL, hence bounded on [0, η], for P−a.e. ω ∈ Ω . Thus, the integrability of c gives the P−a.s. finiteness of

the last expression above. We have used (2.1) and Proposition 2.8 for the first equality, and (1.4) for the second.

The last equality follows from the theory of semimartingale local time.

We claim that the last expression above is a.s. finite. Indeed, r 7→ L^kXk(T, r, ω) is bounded on [0, M (T, ω)] for a.e. ω ∈ Ω , just as before; thus, by condition (i) of Lemma 2.12, each integral in the last expression is a.s. finite. Moreover, the set {` : τ<sub>2`+1</sub>^η < T } is a.s. finite; for otherwise the continuity of the path t 7→ kX(t, ω)k would be violated. The validity of this finiteness claim follows.

With all the considerations above, Lemma2.12is seen to have been established.

Proof of Lemma 2.13: The claim (ii) is a direct consequence of Lemma 2.12 and of

Definition2.9. For the claim (i), we observe by Definition2.9 and Proposition 2.8that

Now we can state and prove the main result of this section, a generalized Freidlin-Sheu-type ([15]) identity for semimartingales on rays.

Theorem 2.15. A Change-of-Variable Formula: Let X(·) be a semimartingale on rays with driver U (·) , in the context of Definition 2.1.

(i) Then for every function g ∈ D , the process g(X(·)) is a continuous semimartin-gale and satisfies Here V_g^X(·) is a continuous process of finite variation on compact intervals, with

V_g^X(t₂)−V_g^X(t₁)

(iii) If X(·) is also a Walsh semimartingale with some angular measure ν , then for every function g ∈ D the decomposition (2.6) holds with

V_g^X(·) = Z 2π 0

g_θ⁰(0+) ν(dθ)

L^kXk(·) . (2.8)

Proof. (i) With N⁻¹ := N⁰ ∪ {−1} and the sequence of stopping times {τ_k^ε}_k∈N−1 defined as in (2.5), we have the decomposition

g X(T )

• Recalling the notation Θ(·) = arg(X(·)) , we write the first summand above as

X

For the second equality of this string, we have used Proposition2.8and the generalized Itˆo’s rule (cf. Problem 3.7.3 in [27]; although θ = Θ(T ∧ τ_2`+1^ε ) is a random variable, a careful look into the proof of the generalized Itˆo’s rule will justify its application here).

The third equality is valid because of (2.1), and of the fact that the process L^kXk(·) is flat off the set {0 ≤ t < ∞ : kX(t)k = 0} . Now with the help of Lemma2.13, we let ε ↓ 0 and obtain the convergence in probability

X

• By Definition 2.9 and Proposition2.11, the process g(X(·)) is adapted and is also adapted and continuous, and we have from (2.9), (2.10) the convergence in probability We also note that (2.6) follows trivially from (2.11), but it remains to establish (2.7).

• Let us concentrate now on the summand on the left-hand side of the above display (2.12). We recall the constant η > 0 and the function c in Definition 2.9 (iii), and note for 0 < ε ≤ η the decompositions

X

where we have used Proposition2.11(i) to obtain the term O(ε). We also have  prob-ability (the “downcrossings” representation of local time; here we are in fact using

“upcrossings”, since the numbers of up- and downcrossings differ by at most 1 .).

This, in conjunction with (2.12) and (2.13), gives the convergence

X

in probability. Together with (2.15), this last convergence in probability leads to the estimate (2.7), which in turn implies that the process V_g^X(·) is of finite variation on compact intervals. Thus the process g(X(·)) is a continuous semimartingale, and the last claim of (i) is justified. The claim (ii) follows readily.

(iii) Finally, we need to argue that the “partition of local time” property (2.3) leads to the representation (2.8). By virtue of (2.15), it suffices to show

X

Let us consider the function g^A in (ii). Applying the result of (i) gives

R_A^X(·) = g^A X(·)
= g^A X(0)
+

(2.7), and hence flat off {t : R^X_A(t) = 0} as well, we have

V_g^XA(·) = Z ·

0

1_{R^X

A(t)=0}dR^X_A(t) = L^RXA(·) = ν(A)L^kXk(·) ,

where the last equality is simply the requirement (2.3). Comparing this with (2.15), we obtain (2.16) with g replaced by g^A. We then note that the function θ 7→ g_θ⁰(0+) is bounded and Borel-measurable, and thus can be uniformly approximated by linear combinations of indicators 1_A(θ) = (g^A)⁰_θ(0+) , A ∈ B([0, 2π)) . Therefore, (2.16) holds for any g ∈ D , and the proof of Theorem2.15 is now complete.

We consider now functions g₁(r, θ) = r cos θ, g₂(r, θ) = r sin θ in D , and note that a planar process X(·) can be written as g₁(X(·)), g₂(X(·)
, in Euclidean coordinates.

Therefore, an application of Theorem2.15to functions g_i, i = 1, 2, gives the following.

Corollary 2.16. (i) For any semimartingale on rays X(·) = (X₁(·), X₂(·)) in Eu-clidean coordinates, both X₁ and X₂ are continuous semimartingales.

(ii) Let X(·) = (X1(·), X2(·)) , written in Euclidean coordinates, be a Walsh semi-martingale with driver U (·) and angular measure ν . Then we have

X₁(·) = X₁(0) + Z ·

0

1{X(t)6=0} cos arg(X(t))
dU (t) + γ₁L^kXk(·) , (2.17)

X2(·) = X2(0) + Z ·

0

1{X(t)6=0} sin arg(X(t))
dU (t) + γ2L^kXk(·) , (2.18) where

γ₁ :=

Z 2π 0

cos(θ) ν(dθ) and γ₂ :=

Z 2π 0

sin(θ) ν(dθ) .

The system (2.17)-(2.18) will be studied in Section 2.1.2, from a slightly different perspective, as a two-dimensional analogue of the equation (1.8) of Harrison & Shepp.

2.1.2 Two-dimensional Stochastic Equations of Harrison-Shepp