Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

Abstract

We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Y^{s, η}, Z^{s, η}). In the classical (Markovian or not) literature the function $u(s,\eta ):= Y^{s,\eta }_{s}$ constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Z^{s, η})_{s, η} processes in term of a deterministic function v associated to the (above decoupled mild) solution u.

Appendix

1.1 A.1 Path-Dependent Martingale Additive Functionals

We here recall the notion of path-dependent Martingale Additive Functionals that we use in this paper. This was introduced in [6] and can be conceived as a path-dependent extension of the notion of non-homogeneous Martingale Additive Functionals of a Markov process developed in [2]. In this subsection, all results come from Section 4 in [6]. In this subsection we consider a progressive path-dependent canonical class $(\mathbbm {P}^{s,\eta })_{(s,\eta )\in \mathbbm {R}_+\times {\Omega }}$ satisfying Hypothesis 2.3 and the corresponding path-dependent system of projectors $(P_s)_{s\in \mathbbm {R}_+}$.

Definition A.1

On $({\Omega },\mathcal {F})$, a path-dependent Martingale Additive Functional, in short path-dependent MAF will be a real-valued random-field M := (M_{t, u})_0≤t≤u verifying the two following conditions.

• 1. For any 0 ≤ t ≤ u, M_{t, u} is $\mathcal {F}^o_{u}$-measurable;

• 2. for any $(s,\eta )\in \mathbbm {R}_+\times {\Omega }$, there exists a real cadlag $(\mathbbm {P}^{s,\eta },\mathbbm {F}^{s,\eta })$- martingale M^{s, η} (taken equal to zero on [0, s] by convention) such that for any η ∈Ω and s ≤ t ≤ u,

  $$ M_{t,u} = M^{s,\eta}_{u}-M^{s,\eta}_{t} \text{ } \mathbbm{P}^{s,\eta}\text{ a.s.} $$

M^{s, η} will be called the cadlag version of M under$\mathbbm {P}^{s,\eta }$.

A path-dependent MAF will be said to verify a certain property (being square integrable, having an absolutely continuous angular bracket) if under any $\mathbbm {P}^{s,\eta }$ its cadlag version verifies it.

Proposition A.2

Let$(\mathcal {D}(A), A)$bea weak generator of$(P_s)_{s\in \mathbbm {R}_+}$and$(s,\eta ) \in {\mathbbm R}_+ \times {\Omega }$. Then for every${\Phi }\in \mathcal {D}(A)$, ${\Phi }-{\int }_0^{\cdot }A({\Phi })_rdr$admitsfor all (s, η) on$[s,+\infty [$a$\mathbbm {P}^{s,\eta }$versionM[Φ]^{s, η}whichis a$(\mathbbm {P}^{s,\eta },\mathbbm {F}^{s,\eta })$-cadlagmartingale. In particular, the random field defined by$M[{\Phi }]_{t,u}(\omega ):={\Phi }_u(\omega )-{\Phi }_t(\omega )-{\int }_t^uA{\Phi }_r(\omega )dr$definesa MAF with cadlag versionM[Φ]^{s, η}under$\mathbbm {P}^{s,\eta }$.

Proposition A.3

LetM andN be two square integrable path-dependent MAFs and letM^{s, η}(respectivelyN^{s, η})be the cadlag version ofM (respectivelyN) under a fixed$\mathbbm {P}^{s,\eta }$. Assume thatN has an absolutely continuous angular bracket.

Then there exists an$\mathbbm {F}^o$-progressivelymeasurable process k such that for any$(s,\eta )\in \mathbbm {R}_+\times {\Omega }$,

$$ \langle M^{s,\eta},N^{s,\eta}\rangle = {\int}_{s}^{\cdot\vee s}k_{r}dr. $$

Notation A.4

The processk whose existence is stated in Proposition A.3 will bedenoted$\frac {d\langle M,N\rangle _t}{dt}$.

1.2 A.2 Proof of Proposition 3.10

In the sequel, we are in the framework of Section 3.

Lemma A.5

Let$\tilde {f}\in \mathcal {L}^1_{uni}$. Then$\begin {array}{rcl} (s,\eta )&\longmapsto &\mathbbm {E}^{s,\eta }[{\int }_s^{T}\tilde {f}_rdr]\\ \ [0,T]\times {\Omega }&\longrightarrow &\mathbbm {R} \end {array}$is$\mathbbm {F}^o$-progressivelymeasurable.

Proof

We fix T₀ ∈]0, T] and we will show that on [0, T₀] × Ω, $(s,\eta )\longmapsto \mathbbm {E}^{s,\eta }[{\int }_s^T\tilde {f}_rdr]$ is $\mathcal {B}([0,T_0])\otimes \mathcal {F}^o_{T_0}$-measurable. We will start by showing that on [0, T₀] × Ω × [0, T₀], the function $k^n:(s,\eta ,t)\mapsto \mathbbm {E}^{s,\eta }[{\int }_{t}^T((-n)\vee \tilde {f}_r\wedge n)dr]$ is $\mathcal {B}([0,T_0])\otimes \mathcal {F}^o_{T_0}\otimes \mathcal {B}([0,T_0])$-measurable, where $n\in \mathbbm {N}$.

Let t ∈ [0, T₀] be fixed. Then by Remark 2.4 $(s,\eta )\mapsto \mathbbm {E}^{s,\eta }[{{\int }_{t}^{T}}((-n)\vee \tilde {f}_r\wedge n)dr]$ is $\mathcal {B}([0,T_0])\otimes \mathcal {F}^o_{T_0}$-measurable.

Let (s, η) ∈ [0, T₀] × Ω be fixed and $t_m\underset {m\rightarrow \infty }{\longrightarrow } t$ be a converging sequence in [0, T₀]. We then have

$$ {\int}_{t_{m}}^{T}((-n)\vee \tilde{f}_{r}\wedge n)dr\underset{m\rightarrow\infty}{\longrightarrow}{{\int}_{t}^{T}}((-n)\vee \tilde{f}_{r}\wedge n)dr \ \text{a.s.} $$

(A.1)

This sequence is uniformly bounded by nT, so by dominated convergence theorem, the convergence in Eq. A.1 also holds under the expectation, so that $t\mapsto \mathbbm {E}^{s,\eta }[{\int }_{t}^T((-n)\vee \tilde {f}_r\wedge n)dr]$ is continuous. By Lemma 4.51 in [1], kⁿ is therefore $\mathcal {B}([0,T_0])\otimes \mathcal {F}^o_{T_0}\otimes \mathcal {B}([0,T_0])$-measurable.

The composition of (s, η)↦(s, η, s) with the maps k_n yields that, for any n ≥ 0, $\tilde {k}^n:(s,\eta )\longmapsto \mathbbm {E}^{s,\eta }[{\int }_{s}^T((-n)\vee \tilde {f}_r\wedge n)dr]$ is (on [0, T₀] × Ω) $\mathcal {B}([0,T_0])\otimes \mathcal {F}^o_{T_0}$-measurable. $\tilde {k}^n$ therefore defines an $\mathbbm {F}^o$-progressively measurable process. Then by letting n tend to infinity, $((-n)\vee \tilde {f}\wedge n)$ tends $dt\otimes d\mathbbm {P}^{s,\eta }$ a.e. to $\tilde {f}$ and since we assumed $\mathbbm {E}^{s,\eta }[{\int }_s^T|\tilde {f}_r|dr]<\infty$, by dominated convergence, $\mathbbm {E}^{s,\eta }[{\int }_{s}^T((-n)\vee \tilde {f}_r\wedge n)dr]$ tends to $\mathbbm {E}^{s,\eta }[{\int }_s^T\tilde {f}_rdr]$. $(s,\eta )\longmapsto \mathbbm {E}^{s,\eta }[{\int }_s^T\tilde {f}(r,X_r)dr]$ is therefore an $\mathbbm {F}^o$-progressively measurable process as the pointwise limit of the $\tilde {k}^n$ which are $\mathbbm {F}^o$-progressively measurable processes. □

We recall the following immediate consequence of Fubini’s Theorem which corresponds to Lemma 5.12 in [2].

Lemma A.6

Let ℙ bea probability measure on$({\Omega },\mathcal {F})$andϕ, ψbetwo measurable processes. Ifϕandψare ℙ-modificationsof each other, then they are equal$dt\otimes d\mathbbm {P}$a.e.

The proof of Proposition 3.10 goes through a linearization lemma.

Lemma A.7

Let$\tilde {f}\in \mathcal {L}^2_{uni}$. Let, for every (s, η) ∈ [0, T] × Ω, $(\tilde Y^{s,\eta },\tilde M^{s,\eta })$bethe unique solution of

$$ \tilde Y^{s,\eta}_{t} = \xi + {{\int}_{t}^{T}} \tilde f_{r}dr -(\tilde M^{s,\eta}_{T} - \tilde M^{s,\eta}_{t}),\quad t\in[s,T], $$

(A.2)

in$\left ({\Omega },\mathcal {F}^{s,\eta },\mathbbm {F}^{s,\eta },\mathbbm {P}^{s,\eta }\right )$. Then there exists a process$\tilde Y\in \mathcal {L}^2_{uni}$, a square integrable path-dependent MAF$(\tilde M_{t,u})_{0\leq t\leq u}$and$\tilde Z^1,\cdots ,\tilde Z^d\in \mathcal {L}^2_{uni}$, such that for all (s, η) ∈ [0, T] × Ω the following holds.

• 1. $\tilde Y^{s,\eta }$is on [s, T] a$\mathbbm {P}^{s,\eta }$-modification of$\tilde Y$;

• 2. $\tilde M^{s,\eta }$is the cadlag version of$\tilde M$under$\mathbbm {P}^{s,\eta }$.

• 3. For each integer 1 ≤ i ≤ d, $\tilde Z^i=\frac {d\langle \tilde M^{s,\eta },N^{i,s,\eta }\rangle _t}{dt}$$dt\otimes d\mathbbm {P}^{s,\eta }$ a.e.

Remark A.8

The existence, for any (s, η), of a unique solution $(\tilde Y^{s,\eta },\tilde M^{s,\eta })$ of Eq. A.2 holds because ξ and $(t,\omega ,y,z)\mapsto \tilde {f}_t(\omega )$ trivially verify the hypothesis of Theorem 3.7.

Proof

We set $\tilde Y:(s,\eta )\mapsto \mathbbm {E}^{s,\eta }\left [\xi + {\int }_s^T \tilde {f}_rdr\right ]$ which is $\mathbbm {F}^o$-progressively measurable by Remark 2.4 and Lemma A.5. Therefore, for a fixed t ∈ [s, T] we have $\mathbbm {P}^{s,\eta }$-a.s.

$$ \begin{array}{rcl} \tilde Y_{t}(\omega) &=& \mathbbm{E}^{t,\omega}\left[\xi + {{\int}_{t}^{T}} \tilde{f}_{r}dr\right]\\ &=& \mathbbm{E}^{s,\eta}\left[\xi + {{\int}_{t}^{T}} \tilde{f}_{r}dr\middle|\mathcal{F}_{t}\right](\omega)\\ &=& \mathbbm{E}^{s,\eta}\left[\tilde Y^{s,\eta}_{t}+(\tilde M^{s,\eta}_{T}-\tilde M^{s,\eta}_{t})|\mathcal{F}_{t}\right](\omega)\\ &=& \tilde Y^{s,\eta}_{t}(\omega). \end{array} $$

The second equality follows by Remark 2.4 and the third one uses (A.2). For every 0 ≤ t ≤ u and ω ∈Ω we set

$$ \tilde M_{t,u}(\omega):=\left\{\begin{array}{l} \tilde Y_{u\wedge T}(\omega)-\tilde Y_{t\wedge T}(\omega)-{\int}_{t\wedge T}^{u\wedge T}\tilde{f}_{r}(\omega)dr \text{ if }{\int}_{t\wedge T}^{u\wedge T}|\tilde{f}_{r}(\omega)|dr<+\infty,\\ 0\text{ otherwise}. \end{array} \right. $$

(A.3)

For fixed (s, η), Eq. A.2 implies $d\tilde Y^{s,\eta }_r=-\tilde {f}_rdr+d\tilde M^{s,\eta }_r$. On the other hand ${\int }_s^T|\tilde {f}|_rdr<+\infty$$\mathbbm {P}^{s,\eta }$ a.s.; so for any s ≤ t ≤ u we have $\tilde M^{s,\eta }_u-\tilde M^{s,\eta }_t=\tilde M_{t,u}$$\mathbbm {P}^{s,\eta }$- a.s. Taking into account that $\tilde M^{s,\eta }$ is square integrable and the fact that previous equality holds for any (s, η) and t ≤ u, then $(\tilde M_{t,u})_{0\leq t\leq u}$ indeed defines a square integrable path-dependent MAF. Y belongs to $\mathcal {L}^2_{uni}$ because the validity of the two following arguments hold for all (s, η). First Y is a $\mathbbm {P}^{s,\eta }$-modification of Y^{s, η} on [s, T], so by Lemma A.6 Y = Y^{s, η}$dt\otimes d\mathbbm {P}^{s,\eta }$ a.e.; second $Y^{s,\eta }\in \mathcal {L}^2(dt\otimes d\mathbbm {P}^{s,\eta })$. The existence of Z follows setting for all i, $Z^i=\frac {d\langle \tilde M,N^i\rangle _t}{dt}$, see Notation A.4 and Proposition A.3. □

Notation A.9

For every fixed (s, η) ∈ [0, T] × Ω, we will denote by$(Y^{k,s,\eta },M^{k,s,\eta })_{k\in \mathbbm {N}}$thePicard iterations associated toBSDE^{s, η}(f, ξ) asdefined in Notation A.13 in [4] andZ^{k, s, η} := (Z^{1, k, s, η},⋯Z^{d, k, s, η}) willdenote$\frac {\langle M^{k,s,\eta },N^{s,\eta }\rangle _t}{dt}$.

This means that for all (s, η) ∈ [0, T] × Ω,(Y^{0, s, η}, M^{0, s, η}) ≡ (0, 0) andfor allk ≥ 1, we have on [s, T]

$$ Y^{k,s,\eta}=\xi+{\int}_{\cdot}^{T}f(r,\cdot, Y^{k-1,s,\eta}_{r},Z^{k-1,s,\eta}_{r})dr-(M^{k,s,\eta}_{T}-M^{k,s,\eta}_{\cdot}), $$

(A.4)

in the sense of$\mathbbm {P}^{s,\eta }$-indistinguishability, and that for all (s, η) ∈ [0, T] × Ω, k ≥ 0, Y^{k, s, η}, Z^{1, k, s, η},⋯Z^{d, k, s, η}belong to$\mathcal {L}^2(dt\otimes d\mathbbm {P}^{s,\eta })$, see Notation A.13 and Lemma A.2 in [4].

A direct consequence of Proposition A.15 in [4] and the lines above it, is the following.

Proposition A.10

For every (s, η) ∈ [0, T] × Ω, each component of (Y^{k, s, η}, Z^{1, k, s, η},⋯ , Z^{d, k, s, η}) tendsto each component of (Y^{s, η}, Z^{1, s, η},⋯ , Z^{d, s, η}) in$\mathcal {L}^2(dt\otimes d\mathbbm {P}^{s,\eta })$and$dt\otimes d\mathbbm {P}^{s,\eta }$-a.e.when k tends to infinity.

Proposition A.11

For each$k\in \mathbbm {N}$, there existprocesses$Y^k\in \mathcal {L}^2_{uni}, Z^{k,1},\cdots , Z^{k,d}\in \mathcal {L}^2_{uni}$, a square integrablepath-dependent MAF$(M^k_{t,u})_{0\leq t\leq u}$suchthat for all (s, η) ∈ [0, T] × Ω, we have the following.

• 1. Y^{k, s, η}ison [s, T] a$\mathbbm {P}^{s,\eta }$-modificationofY^k;

• 2. M^{k, s, η}isthe cadlag version ofM^kunder$\mathbbm {P}^{s,\eta }$.

• 3. For all (s, η) ∈ [0, T] × Ω andi ∈ ⟦1; d⟧, $Z^{k,i}=\frac {d\langle M^{k,s,\eta },N^{i,s,\eta }\rangle _t}{dt}$$dt\otimes d\mathbbm {P}^{s,\eta }$a.e.

Proof

We prove the statement by induction on k ≥ 0. It is clear that Y⁰ ≡ 0 and M⁰ ≡ 0 verify the assertion for k = 0.

Suppose the existence, for k ≥ 1, of a square integrable path-dependent MAF M^k− 1 and processes Y^k− 1$Z^{k-1,1},\cdots , Z^{k-1,d}\in \mathcal {L}^2_{uni}$ such that the statements 1. 2. 3. above hold replacing k with k − 1.

We fix (s, η) ∈ [0, T] × Ω. By Lemma A.6, (Y^{k− 1, s, η}, Z^{k− 1, s, η}) = (Y^k− 1, Z^k− 1) $dt\otimes d\mathbbm {P}^{s,\eta }$ a.e. Therefore by Eq. A.4

$$ Y_{t}^{k,s,\eta} = \xi + {{\int}_{t}^{T}} f\left( r,\cdot,Y^{k-1}_{r},Z^{k-1}_{r}\right)dr -(M^{k,s,\eta}_{T} - M^{k,s,\eta}_{t}), t \in [s,T].$$

According to Notation 3.8, the Eq. A.4 can be seen as a BSDE of the type $BSDE^{s,\eta }(\tilde f,\xi )$ where $\tilde f:(t,\omega )\longmapsto f(t,\omega ,Y^{k-1}_t(\omega ),Z^{k-1}_t(\omega ))$. We now verify that $\tilde f$ verifies the conditions under which Lemma A.7 applies.

$\tilde f$ is $\mathbbm {F}^o$-progressively measurable since Y^k− 1, Z^k− 1 are $\mathbbm {F}^o$-progressively measurable and f is $\mathcal {P}ro^o\otimes \mathcal {B}(\mathbbm {R})\otimes \mathcal {B}(\mathbbm {R}^d)$-measurable. Since

$$|\tilde f(t,\omega)|=|f(t,\omega,Y^{k-1}_{t}(\omega),Z^{k-1}_{t}(\omega))|\leq |f(t,\omega,0,0)|+K(|Y^{k-1}_{t}(\omega)|+\|Z^{k-1}_{t}(\omega)\|),$$

for all t, ω, with $f(\cdot ,\cdot ,0,0),Y^{k-1},Z^{k-1,1},\cdots ,Z^{k-1,d}\in \mathcal {L}^2_{uni}$ by recurrence hypothesis, it is clear that $\tilde f\in \mathcal {L}^2_{uni}$. Since (Y^{k, s, η}, M^{k, s, η}) is a solution of $BSDE^{s,\eta }(\tilde f,\xi )$, Lemma A.7 shows the existence of suitable Y^k, M^k, Z^k,1,⋯ , Z^{k, d} verifying the statement for the integer k. □

Proof of Proposition 3.10

We define $\bar {Y}$ and $\bar Z^i, 1 \le i \le d$ by ${\bar Y}_s(\eta ):= \underset {k\in \mathbbm {N}}{\text {limsup }}Y^k_s(\eta )$ and $\bar Z^i_s(\eta ):=\underset {k\in \mathbbm {N}}{\text {limsup }}Z^{k,i}_s(\eta ),$ for every (s, η) ∈ [0, T] × Ω. $\bar {Y}$ and $\bar Z:=(\bar Z^1,\cdots ,\bar Z^d)$ are $\mathbbm {F}^o$-progressively measurable. Combining Propositions A.11, A.10 and Lemma A.6 it follows that, for every (s, η) ∈ [0, T] × Ω,

$$ \left\{\begin{array}{rcl} Y^{k}&\underset{k\rightarrow\infty}{\longrightarrow}& Y^{s,\eta} \quad dt\otimes d\mathbbm{P}^{s,\eta}\\ Z^{k,i}&\underset{k\rightarrow\infty}{\longrightarrow}& Z^{i,s,\eta} \quad dt\otimes d\mathbbm{P}^{s,\eta}, \text{ for all } 1 \le i \le d. \end{array}\right. $$

(A.5)

Let us fix 1 ≤ i ≤ d and (s, η) ∈ [0, T] × Ω. There is a set A^{s, η} of full $dt\otimes d\mathbbm {P}^{s,\eta }$ measure such that for all (t, ω) ∈ A^{s, η} we have

$$ \left\{\begin{array}{rcccccl} \bar Y_{t}(\omega)&=&\underset{k\in\mathbbm{N}}{\text{limsup }}{Y^{k}_{t}}(\omega)&=&\underset{k\in\mathbbm{N}}{\text{lim }}{Y^{k}_{t}}(\omega) &=& Y^{s,\eta}_{t}(\omega) \\ \bar Z_{t}(\omega)&=&\left( \underset{k\in\mathbbm{N}}{\text{limsup }} Z^{k,i}_{t}(\omega)\right)_{i\leq d}&=&\left( \underset{k\in\mathbbm{N}}{\text{lim }} Z^{k,i}_{t}(\omega)\right)_{i\leq d} &=& Z^{s,\eta}_{t}(\omega). \end{array}\right. $$

(A.6)

This implies

$$ \begin{array}{@{}rcl@{}} \bar Y_{t}(\omega)&=& Y^{s,\eta} \ dt\otimes d\mathbbm{P}^{s,\eta} \text{a.e.} \\ \bar Z_{t}(\omega)&=& Z^{s,\eta} \ dt\otimes d\mathbbm{P}^{s,\eta} \text{a.e.} \end{array} $$

(A.7)

By Eqs. A.7 and ??, under every $\mathbbm {P}^{s,\eta }$, we actually have

$$ Y^{s,\eta} = \xi + {\int}_{\cdot}^{T} f\left( r,\cdot,\bar{Y}_{r},\bar Z_{r}\right)dr -(M^{s,\eta}_{T} - M^{s,\eta}_{\cdot}), $$

(A.8)

in the sense of $\mathbbm {P}^{s,\eta }$-indistinguishability, on the interval [s, T]. At this stage, in spite of Eq. A.7, $\bar Y$ is not necessarily a modification of Y^{s, η}. We will construct processes Y, Z fulfilling indeed the statement of Proposition 3.10. In particular Y fulfills item 1. that is a bit stronger than (A.7).

We set now $\tilde f:(t,\omega )\mapsto f(t,\omega ,\bar {Y}_t(\omega ),\bar Z_t(\omega ))$; Eq. A.8 is now of the form (A.2) and we show that $\tilde f$ so defined verifies the conditions under which Lemma A.7 applies. $\tilde f$ is $\mathbbm {F}^o$-progressively measurable since f is $\mathcal {P}ro^o\otimes \mathcal {B}(\mathbbm {R})\otimes \mathcal {B}(\mathbbm {R}^d)$-measurable and $\bar Y,\bar Z$ are $\mathbbm {F}^o$-progressively measurable.

Moreover, for any (s, η) ∈ [0, T] × Ω, Y^{s, η} and Z^{1, s, η},⋯ , Z^{d, s, η} belong to $\mathcal {L}^2(dt\otimes d\mathbbm {P}^{s,\eta })$; therefore by Eq. A.6, so do $\bar {Y}$ and $\bar Z^1,\cdots ,\bar Z^d$.

Since this holds for all (s, η), then $\bar {Y}$ and $\bar Z^1,\cdots ,\bar Z^d$ belong to $\mathcal {L}^2_{uni}$.

Finally, since $|\tilde f(t,\omega )|=|f(t,\omega ,\bar Y_t(\omega ),\bar Z_t(\omega ))|\leq |f(t,\omega ,0,0)|+K(|\bar Y_t(\omega )|+\|\bar Z_t(\omega )\|)$ for all t, ω, with $f(\cdot ,\cdot ,0,0),\bar Y,\bar Z^1,\cdots ,\bar Z^{d}\in \mathcal {L}^2_{uni}$, it is clear that $\tilde f\in \mathcal {L}^2_{uni}$. Now Eq. A.8 can be considered as a BSDE where the driver does not depend on y and z of the form Eq. A.2. We can therefore apply Lemma A.7 to $\tilde {f}$ and conclude on the existence of (Y, M, Z¹,⋯ , Z^d) verifying the three items of the proposition.

It remains to prove now the last assertion of Proposition 3.10. We fix some (s, η). The first item implies that $Y_s= Y^{s,\eta }_s$$\mathbbm {P}^{s,\eta }$ a.s. But since Y_s is $\mathcal {F}^o_s$-measurable and $\mathbbm {P}^{s,\eta }(\omega ^s=\eta ^s)= 1$, it also yields that Y_s is $\mathbbm {P}^{s,\eta }$ a.s. equal to the deterministic value Y_s(η) hence $Y^{s,\eta }_s$ is $\mathbbm {P}^{s,\eta }$ a.s. equal to the deterministic value Y_s(η). This also proves that Y is unique because it is given by $Y:(s,\eta )\longmapsto Y^{s,\eta }_s$. The uniqueness of Z up to zero potential sets is immediate by the third item of the proposition and Definition 3.2. □