Párrafo 6.8

6.1 Path-by-path uniqueness of individual trajectories We next consider equation (sDE). Its integral formulation is

Xt(ω) = x + Z t

0

b(s, Xs(ω))ds + σWt(ω)

and therefore, we may give a path-by-path meaning to it. Assume for some constant C > 0thatb : [0, T ] × R^d → R^d is a measurable locally bounded function (defined for all(t, x), not only a.e.). As before, let us assume thatW has continuous trajectories (everywhere). Givenω ∈ Ω, hence given the continuous functiont 7→ Wt(ω), consider all continuous functionsy : [0, T ] → R^dwhich satisfy the identity

y(t) = x + Z t

0

b(s, y(s))ds + σW_t(ω)

and callC(ω, x)the set of all such functions. Denote byCard(C(ω, x))the cardinality of the setC(ω, x).

Remark 6.1. Ifbis continuous with|b(t, x)| ≤ C(1 + |x|)for all(t, x) ∈ [0, T ] × R^{d, then} by classical deterministic argumentsC(ω, x)is non empty.

Definition 6.2. We say that the sDE satisfies path-by-path uniqueness if

P Card(C(ω, x)) ≤ 1for allx ∈ R^d
= 1,

namely if for a.e.ω ∈ Ω,C(ω, x)is at most a singleton for everyxinR^d.

To our knowledge, the only two results on path-by-path uniqueness are [26] and [18].

We present here a new strategy for this kind of results.

Lety ∈ C(ω, x)be a solution. Set

z^ω(t) := y(t) − σW_t(ω) solution of

z^ω(t) = x + Z t

0

b(s, z^ω(s) + σW_s(ω))ds = x + Z t

0

eb^ω(s, z^ω(s))ds,

where, as usual,˜b^ω(t, x) = b(t, x + σW_t(ω)). Consider the time-dependent Dirac measure µe_ω(t) = δ_zω(t) onR^d. For everyϕ ∈ C_c^∞(R^d), we writehµe^ω(t), ϕiforR

R^dϕdµe^ω(t), which, in this particular case, is simplyϕ(z^ω(t)).

Lemma 6.3. For allt ∈ [0, T ]andϕ ∈ C¹([0, T ]; C_c^∞(R^d)), we have

hµe^ω(t), ϕ(t)i = hµe^ω(0), ϕ(0)i + Z t

0

D

µe^ω(s), eb^ω(s) · ∇ϕ(s) + ∂tϕ(s)E ds.

Proof. We have to prove that

ϕ(t, z^ω(t)) = ϕ(0, z^ω(0)) + Z t

0

eb^ω(s, z^ω(s)) · ∇ϕ(s, z^ω(s)) + ∂_tϕ(s, z^ω(s))
ds, which is true by ordinary calculus.

We can now prove a central fact. For a bounded functionf and a Borel setE, denote withkf k0,E the supremum off overE; in general, this is not the essential supremum, unlessf is continuous.

Theorem 6.4. Let (b_ε)_ε∈(0,1) be a family in C_c^∞([0, T ] × R^d). Assume that, for every t_f ∈ [0, T ]andv₀∈ C_c^∞(R^d), we have

P − lim

ε→0

Z t_f 0

k(b − bε) · ∇v_ε^ωk0,B_Rds = 0 (6.1) for every positiveR, and where, for everyε ∈ (0, 1), vε is the smooth solution of the backward sPDEs (3.17) corresponding tobεandv0, withcε= 0(v^ω_ε denotesvε(·, ·, ω)as before). Then path-by-path uniqueness holds for (sDE).

Proof. Step 1: Identification of Ω0, independently ofx. By assumption (6.1), given tf ∈ [0, T ]andv0 ∈ C_c^∞(R^d), there exist a full measure setΩt_f,v₀ ⊂ Ωand a sequence ε_n→ 0such that

ev_ε^ω_nbelongs toC¹([0, T ]; C_c^∞(R^d))and satisfies (3.19) (withc = 0), for alln ∈ N^{, (6.2)}

n→∞lim Z t_f

0

(b^ω− b^ω_εn) · ∇v_ε^ω_{n 0,B}

Rds = 0 for allω ∈ Ωt_f,v₀. Hence also

n→∞lim Z t_f

0

(eb^ω− eb^ω_εn) · ∇ev_ε^ω_{n 0,B}

R−σWt(ω)ds = 0 (6.3) for allω ∈ Ωt_f,v₀. LetD ⊂ C_c^∞(R^d)be a countable set which separates points, i.e. for all a 6= b ∈ R^d, there existsv0∈ Dwithv0(a) 6= v0(b). By a diagonal procedure, there exist a full measure setΩ0⊂ Ωand a sequenceεn→ 0such that properties (6.2) and (6.3) hold for alltf ∈ [0, T ] ∩ Q^,v0∈ D,n, R ∈ N^andω ∈ Ω0. Since, givenω ∈ Ω0andR ∈ N^{, there} existsR⁰_ω∈ N^{such that}B_R⊂ BR⁰_ω− σWt(ω)for allt ∈ [0, T ], we may replace (6.3) by

n→∞lim Z t_f

0

(eb^ω− eb^ω_εn) · ∇ev_ε^ω_{n 0,B}

Rds = 0 (6.4)

for alltf ∈ [0, T ] ∩ Q^,v0∈ D,R ∈ N^andω ∈ Ω0.

Step 2: C(ω, x)is a singleton for everyx ∈ R^dandω ∈ Ω0, i.e. path-by-path unique-ness holds. Givenω ∈ Ω0andy⁽ⁱ⁾∈ C(ω, x),i = 1, 2, we define the (signed) measure

ρe^ω(t) := δ_y(1)(t)−σWt(ω)− δ_y(2)(t)−σWt(ω), which satisfies

hρe^ω(tf), ϕ(tf)i = Z t_f

0

D

ρe^ω(s), eb^ω(s) · ∇ϕ(s) + ∂tϕ(s)E ds

for alltf ∈ [0, T ]andϕ ∈ C¹([0, T ]; C_c^∞(R^d)), due to Lemma 6.3. In particular, this holds forϕ =ev^ω_εn and thus, by (3.19), we get

ρe^ω(tf), v0(· + σWt_f(ω)) = Z t_f

0

D

ρe^ω(s), (eb^ω(s) − eb^ω_εn(s)) · ∇ev^ω_εn(s)E ds.

Then, ifR > 0is such that|y⁽ⁱ⁾(t)| ≤ Rfort ∈ [0, T ]andi = 1, 2, we find

= 0 is satisfied due to (6.4). This is equivalent to

ρe^ω(t_f, · − σW_tf(ω)), v₀ = 0, which impliesρe^ω(t_f, · − σW_tf(ω)) = 0since v₀ ∈ D was arbitrary andDseparates points. Consequently,y⁽¹⁾(tf) = y⁽²⁾(tf)follows. This holds true for everytf∈ [0, T ] ∩ Q, and sincet 7→ y⁽¹⁾(t)is continuous, we gety⁽¹⁾(t) = y⁽²⁾(t) for everyt ∈ [0, T ]. This finishes the proof of the theorem.

Theorem 6.4 is, in a sense, our main result on path-by-path uniqueness, although assumption (6.1) is not explicit in terms ofb. Roughly speaking, this condition is true when we have a uniform bound (in some probabilistic sense) fork∇ev^ω_εnk0,B_R. It introduces a new approach to the very difficult question of path-by-path uniqueness, which may be easily generalized, for instance, to sDEs in infinite dimensions (which will be treated in separate works). A simple consequence is:

Corollary 6.5. Let(bε)_ε∈(0,1)be a family inC_c^∞([0, T ] × R^d)which converges uniformly

wherevεis the smooth solution of the backward sPDEs (3.17) corresponding tobεandv0, withcε= 0. Then path-by-path uniqueness holds for (sDE).

In Section 2.10 we have proved (reformulated for the backward sPDE) that, for every tf ∈ [0, T ],R > 0,man even positive integer andv0∈ C_c^∞(R^d),

Therefore, by Sobolev embedding, we obtain form > d sup

which implies condition (6.5) of Corollary 6.5. Hence, we have:

Corollary 6.6. Under the conditions of Section 2.10 (Dbof class LPS ) we have path-by-path uniqueness for (sDE).

Notice that the conditions of Section 2.10 withm > dimplyb ∈ C_loc^ε (R^d, R^d)for some ε > 0. Thus, in the casem > d, this result is included in Corollary 6.8 below and already in [26]. However, also the limit casem = d ≥ 3is included in our statement. Otherwise, we may take estimate (6.6) from [41] in the case

b ∈ L^∞(0, T ; C_b^α(R^d)); (6.7) (essential boundedness in time, with values in C_b^α(R^d), is actually enough since the measure solutions to the continuity equation are only space-valued). Precisely, the following result is proved in [41]. We give here an independent proof for the sake of completeness.

Lemma 6.7. Letbsatisfy (6.7) and take a familybε∈ C_c^∞([0, T ] × R^d)which converges uniformly tobon compact sets. Then the flowsΦ^ε_t associated to (sDE) with coefficients bεsatisfy for everym ≥ 1

Proof. Step 1: Formula forDΦ^ε_t(x)via Itô–Tanaka trick. Let us introduce the vector field U_ε(t, x), fort ∈ [0, T ],x ∈ R^d, and with componentsU_εⁱ(t, x), fori = 1, . . . , d, satisfying the backward parabolic equation (wherebⁱ_εis thei-component ofbε)

∂tU_εⁱ+ bε· DU_εⁱ+σ²

2 ∆U_εⁱ= −bⁱ_ε+ λU_εⁱ, U_εⁱ(T, x) = 0 (6.9) for someλ > 0. As explained in [41, Section 2] based on classical results of [54] (see also a partial probabilistic proof in [39]), this equation has a unique solutionU_εⁱof class C¹([0, T ]; C_b^α(R^d)) ∩ C([0, T ]; C_b^2,α(R^d)), and there is a uniform constantC > 0such that

sup

ε∈(0,1)

sup

[0,T ]

kUεk_C2,α

b (R^d)≤ C. (6.10)

Moreover, given anyδ > 0, there existsλ > 0large enough such that

kDU_εk_∞≤ δ. (6.11)

Here and below we denote byk·k_∞theL^∞norm both in time and space. We may apply the Itô formula toU_εⁱ(t, Φ^ε_t(x))and use (6.9) to get

U_εⁱ(t, Φ^ε_t(x)) = U_εⁱ(0, x) + Z t

0

(−bⁱ+ λU_εⁱ)(s, Φ^ε_s(x))ds + σ Z t

0

∇U_εⁱ(s, Φ^ε_s(x)) · dW_s. This allows us to rewrite the equation

Φ^ε,i_t (x) = xⁱ+ Z t

0

bⁱ_ε(s, Φ^ε_s(x))ds + σW_tⁱ in the form

Φ^ε,i_t (x) = xⁱ+U_εⁱ(0, x)−U_εⁱ(t, Φ^ε_t(x))+

Z t 0

λU_εⁱ(s, Φ^ε_s(x))ds+σ Z t

0

∇U_εⁱ(s, Φ^ε_s(x))·dW_s+σW_tⁱ. Since b_ε is smooth and compactly supported, we a priori know from [57] that Φ^ε_t is differentiable; hence we may use the differentiability properties ofU_εand the result of differentiation under stochastic integral of [57] to have

∂kΦ^ε,i_t (x) = δik+ ∂kU_εⁱ(0, x) −

d

X

j=1

∂jU_εⁱ(t, Φ^ε_t(x))∂kΦ^ε,j_t (x)

+ Z t

0

λ

d

X

j=1

∂_jU_εⁱ(s, Φ^ε_s(x))∂_kΦ^ε,j_s (x)ds

+ σ Z t

0 d

X

j,l=1

∂l∂jU_εⁱ(s, Φ^ε_s(x))∂kΦ^ε,j_s (x)dW_s^l. (6.12)

Step 2: Uniform pointwise estimate forDΦ^ε_t(x). We first use the previous identity to estimateE[|∂_kΦ^ε,i_t (x)|^r]uniformly in(t, x)andε, for eachr > 1. Denoting byC_r> 0a generic constant depending only onr, we have

E[|∂kΦ^ε_t(x)|^r] ≤ Cr+ CrkDUεk^r_∞+ CrkDUεk^r_∞E[|∂kΦ^ε_t(x)|^r] + λ^rC_rkDUεk^r_∞

Z t 0

E[|∂_kΦ^ε_s(x)|^r]ds

+ σ^rCrkD²Uεk^r_∞ Z t

0

E[|∂kΦ^ε_s(x)|^r]ds,

where we have used the Burkholder–Davis–Gundy inequality in the last term. By choos-ingλso large thatC_rkDU_εk^r_∞≤ 1/2(possible by (6.11)), we find

1

2E[|∂kΦ^ε_t(x)|^r] ≤ Cr+ λ^r Z t

0

E[|∂kΦ^ε_s(x)|^r]ds + σ^rCrkD²Uεk^r_∞ Z t

0

E[|∂kΦ^ε_s(x)|^r]ds.

Now it is sufficient to apply Gronwall’s lemma and the uniform estimate (6.10) to arrive at

sup

ε∈(0,1)

sup

t∈[0,T ]

sup

x∈R^d

E[|DΦ^ε_t(x)|^r] < ∞. (6.13)

Step 3: Conclusion via Kolmogorov’s regularity criterion. To get the supremum inx inside the expectation, we want to apply the Kolmogorov regularity criterion. Given x, y ∈ R^d,r > 1, we derive from (6.12) (using suitable vector notations)

E[|∂_kΦ^ε_t(x) − ∂_kΦ^ε_t(y)|^r] ≤ C_r(I₁+ I₂₁+ I₂₂+ λ^rI₃₁+ λ^rI₃₂+ σ^rI₄₁+ σ^rI₄₂) with the following abbreviations

I1:= |∂kUε(0, x) − ∂kUε(0, y)|^r

I21:= E[|DUε(t, Φε,t(x))|^r|∂kΦ^ε_t(x) − ∂kΦ^ε_t(y)|^r] I22:= E[|DUε(t, Φε,t(x)) − DUε(t, Φε,t(y))|^r|∂kΦ^ε_t(y)|^r] I₃₁:=

Z t 0

E[|DU_ε(s, Φ^ε_t(x))|^r|∂kΦ^ε_s(x) − ∂_kΦ^ε_s(y)|^r]ds

I32:=

Z t 0

E[|DUε(s, Φ^ε_s(x)) − DUε(s, Φ^ε_s(y))|^r|∂kΦ^ε_s(y)|^r]ds

I41:=

Z t 0

E[|D²Uε(s, Φ^ε_t(x))|^r|∂kΦ^ε_s(x) − ∂kΦ^ε_s(y)|^r]ds I₄₂:=

Z t 0

E|D²U_ε(s, Φ^ε_s(x)) − D²U_ε(s, Φ^ε_s(y))|^r|∂kΦ^ε_s(y)|^rds.

For the last term we have used again the Burkholder–Davis–Gundy inequality. Let us denote byCU > 0(resp.Cr,Φ > 0) a constant independent of ε ∈ (0, 1), based on the uniform estimate (6.10) (resp. on (6.13)) and let us writeδ > 0for the constant in (6.11).

We have

I₁≤ kDUεk^r_∞|x − y|^r≤ C_U^r|x − y|^r

I21≤ kDUεk^r_∞E[|∂kΦ^ε_t(x) − ∂kΦ^ε_t(y)|^r] ≤ δ^rE[|∂kΦ^ε_t(x) − ∂kΦ^ε_t(y)|^r] I22≤ kD²Uεk^r_∞EhZ 1

0

|DΦ^ε_t(θx + (1 − θ)y)|^rdθ|∂kΦ^ε_t(y)|^ri

|x − y|^r

≤ kD²Uεk^r_∞Eh

|∂kΦ^ε_t(y)|^2ri1/2Z 1 0

E|DΦ^ε_t(ux + (1 − u)y)|^2rdu1/2

|x − y|^r

≤ C_U^rC_2r,Φ^1/2C_2r,Φ^1/2|x − y|^r= C_U^rC_2r,Φ|x − y|^r. Similarly, we get

I₃₁≤ C_U^r Z t

0

E[|∂_kΦ^ε_s(x) − ∂_kΦ^ε_s(y)|^r]ds I32≤ T C_U^rC2r,Φ|x − y|^r

and finally

I41≤ C_U^r Z t

0

E|∂kΦ^ε_s(x) − ∂kΦ^ε_s(y)|^rds

I42≤ sup

[0,T ]

kD²Uεk^r_Cα

Z t 0

EhZ 1 0

|DΦ^ε_s(ux + (1 − u)y)|^αrdu|∂kΦ^ε_s(y)|^ri

ds|x − y|^αr

≤ T C_U^rC_2αr,Φ^1/2 C_2r,Φ^1/2|x − y|^αr.

Takingδsufficiently small (and thusλlarge enough), from Gronwall’s lemma we deduce E|∂kΦ^ε_t(x) − ∂kΦ^ε_t(y)|^r ≤ Cr|x − y|^αr.

Sincer > 1is arbitrary, we may apply Kolmogorov’s regularity criterion (see for instance the quantitative version of [57] for the bound on the moments of supremum norm inx) and entail (6.8), which finishes the proof of the lemma.

As a straight-forward consequence of Theorem 6.4 in combination with Lemma 6.7, it follows:

Corollary 6.8. Under condition (6.7) we have path-by-path uniqueness for (sDE).

Proof. In view of the formula vε(t, x) = v0(Φ^ε_t(x)) and (6.8), the estimate (6.6) holds, which in turn implies (6.1). Hence, the path-by-path uniqueness follows immediately from Theorem 6.4.

In document Propuesta de procedimiento para enmienda de AOC para un concesionario de servicios aéreos en la República Mexicana (página 55-69)