Viscosity solutions of path-dependent PDEs with randomized time

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Viscosity solutions of path-dependent PDEs with randomized time

Zhenjie Ren, Mauro Rosestolato

To cite this version:

Zhenjie Ren, Mauro Rosestolato. Viscosity solutions of path-dependent PDEs with randomized time.

2019. �hal-02292995�

arXiv:1806.07654v1 [math.PR] 20 Jun 2018

Viscosity solutions of path-dependent PDEs with randomized time

Zhenjie REN ^* Mauro ROSESTOLATO ^†

Abstract

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/super-solutions.

As an application, we prove the existence of viscosity solutions using the Perron method.

Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.

Keywords: Viscosity solution; Path-dependent partial differential equations; Partial differ- ential equations in infinite dimension.

AMS 2010 subject classification:35K10; 35R15; 49L25; 60H30.

1 Introduction

This paper studies viscosity solutions of the fully nonlinear path-dependent partial differential equation

−∂tu(t,ω)−G¡

t,ω,u(t,ω),∂ωu(t,ω),∂²_ωωu(t,ω)¢

=0 on [0,T)×Ω. (1.1) Here, T>0 is a given terminal time andω∈Ωis a continuous path from [0,T] to R^m starting from the origin. The path derivatives∂t,∂ω,∂²_ωω were first introduced in the work of Dupire [6].

See also [2] for the related Itô calculus. Such equations arise naturally in many applications.

For example, the dynamic programming equation associated with a stochastic control problem of non-Markov diffusions (see [10]) and the one associated with a stochastic differential game with non-Markov dynamics (see [22]) both fall in the class of equation (1.1). The notion of nonlinear path-dependent partial differential equations was first proposed by Peng [19]. We also refer to Peng and Wang [20] for a study on classical solutions of semilinear equations.

The notion of viscosity solutions studied in this paper is a slight modification over the one introduced in Ekren et al. ([8]) in the semilinear context and further extended to the fully non- linear case in [10,11]. Following the lines of the classical Crandall and Lions notion of viscosity solutions ([5]), supersolutions and subsolutions are defined through tangent test functions. How- ever, while Crandall and Lions consider pointwise tangent functions, the tangency conditions in the path-dependent setting is in the sense of the expectation with respect to an appropriate class of probability measuresP. We refer to [27] for an overview, and to [3,7,14,16,23,25,26,28] for some of the generalizations.

*Université Paris-Dauphine, PSL Research University, CNRS, UMR [7534], Ceremade, 75016 Paris, France, ren@ceremade.dauphine.fr.

†CMAP, Ecole Polytechnique Paris, mauro.rosestolato@polytechnique.edu. Research supported by ERC 321111 Rofirm.

The authors are sincerely grateful to Nizar Touzi for valuable discussions.

Regardless of the successful development mentioned above, there were some difficulties for the theory of viscosity solutions coming from the definition adopted in [8]. First, a good stability result was missing. In [10] the authors proved a stability result in the sense that if a sequence of uniformly continuous solutionsuⁿ uniformly converges to a uniformly continuous functionu, thenuis also a solution. The assumptions of the uniform continuity and the uniform convergence are often too strong for applications, for example, for the Perron method to prove the existence of viscosity solution. Secondly, also a comparison result for semicontinuous viscosity sub-/super- solutions was missing. The aim of the present paper is to fill these two theoretical gaps. We do this by slightly modifying the definition in [8]. This modification is sufficient to let us overcome the technical difficulties for proving stability and comparison results, but does not compromise the other results till now obtained in the literature, such as existence.

In the previous definition, adopted in [8], the test function ϕ is urged to be tangent to the (sub)solutionuat a point (say 0) in the sense that

(u−ϕ)(0)=max

τ∈T sup

P∈P

E^P£

(u−ϕ)(_τ,B)¤

, (1.2)

whereT is the set of all stopping times taking values in [0,T], andP is a family of probability measures on the path spaceΩon whichBis the canonical process. In the arguments for proving the stability and the comparison, we often need to solve the optimal stopping problem on the right hand side of (1.2), which is not a simple task and requires the uniform continuity ofu (see [9]).

This turns out to be one of the main difficulties in improving the stability and the comparison results. In the present paper, in order to overcome this fundamental difficulty, we randomize the optimal stopping problem, and the new definition reads

(u−ϕ)(0)=sup

P∈fP

E^P£

(u−ϕ)(T,B)¤ ,

wherePfis a family of probability measures on the time-path product space [0,T]×Ωon which (T,B) is the canonical process. It turns out that the randomized problem can be solved for less regular functions u, and is more stable. With this change of definition, we manage to prove a general stability result for semicontinuous viscosity solutions (Theorem4.5). The recipe of our proof is composed of the classical argument for stability in [4] and the measurable selection the- orem. Moreover, by using the stability result, we are also able to prove the existence of viscosity solution through Perron’s method (Theorem5.3). Further, we prove a comparison result for semi- continuous sub-/super-solutions under some strong assumptions (Theorem6.9), by approximating semicontinuous solutions with Lipschitz continuous solutions to approximating equations.

Another major contribution of this paper is to connect the path-dependent partial differential equation (1.1) with the partial differential equation

−u_t− 〈A x,D_xu〉 −G(t,x,u,D_x0u,D²_x

0x0u)=0 on (0,T)×H, (1.3) whereHis a Hilbert space into which the path spaceΩcan embed,Ais an unbounded operator, and D_x0,D²_x

0x0 are the first- and second-order differentials with respect to a finite dimensional subspace of H. Both equations (1.1) and (1.3) can be used to characterize the value function of non-Markov stochastic control problem. In this paper, we prove that a viscosity solution to the path-dependent equation (1.1), under some regularity assumption, is also a viscosity solution to the corresponding equation (1.3) (Theorem 7.6). The theory of viscosity solutions for partial differential equations on Hilbert spaces (we mainly refer to [12]) is designed of a large class of equations not rescrited to those of delay type, as (1.3). We notice that, till now, when applied to PDEs of the form (1.3), such a theory can deliver a comparison result under more general assump- tion on the nonlinearity functionG, but only for more regular sub-/super-solutions. Moreover, the

theory of path-dependent equations as here developed can treat solutions (semi)continuous in the L^∞-norm, which cannot be settled in a Hilbert space framework.

The rest of the paper is organized as follows. Section2introduces the main notations. Sec- tion3 presents the modified definition of the viscosity solutions to the path-dependent partial differential equations. In Section4 we prove the stability result (Theorem 4.5), and using it in Section5 we prove the existence of viscosity solution with Perron’s method (Theorem 5.3). In Section6we show the comparison result for semicontinuous solutions (Theorem6.9). In Section 7we clarify the connection between the path-dependent equation and the correponding equation on the Hilbert space (Theorem7.6). Finally, we complete some proofs in Appendix.

2 Notations

Canonical Space. Letm>0 be a natural number,T>0 be a real number. Define Ω:={ω∈C([0,T],R^m) :ω(0)=0}.

We denote by| · |the Euclidean norm inR^mand by| · |∞the uniform norm onΩ. In this paper, we study equations set on the spacetime space:

Θ:=[0,T]×Ω.

For technical reasons, we also often work on the enlarged canonical space:

e

Θ:=Θ×Ω×Ω×C₀([0,T],S^m)

whereS^mis the space of symmetricm×mreal matrices endowed with the supremum norm and C₀([0,T],S^m) :={f ∈C([0,T],S^m) : f(0)=0}.

Hereafter, we will reserve the letter ω for a generic element of Ω, the letter θ for the generic couple_θ=(t,_ω)∈Θ, and the letter_ϑfor the generic element_ϑ=(_θ,a,_µ,q)∈Θe. We introduce onΘ the pseudo-metricd∞defined by

d∞(θ,θ^′) := |t−t^′| + |ωt∧·−ωt^′∧·|∞.

Remark 2.1. In this paper, without being otherwise stated, the (semi-)continuity on the canon- ical spaces is under| · |∞. The (semi-)continuity related to other pseudo-metrics will be explicitly expressed.

Remark 2.2. Note the following facts:

• Let F:={Ft}_t∈[0,T] denote the canonical filtration on Ω. All d∞-continuous functions are F-progressively measurable.

• lim_n→∞|θⁿ−θ|∞=0 implies that lim_n→∞d∞(_θⁿ,_θ)=0, and thus all d∞-continuous func- tions are continuous.

LetF^S:={F_t^S}_t∈[0,T]denote the canonical filtration onC₀([0,T],S^m). We introduce onΘe the filtrationG:={Gt}_t∈[0,T]defined by

Gt:=(σ{[0,r], r≤t}⊗Ft)⊗Ft⊗Ft⊗F_t^S ∀t∈[0,T].

Shifted functions . Forω,ω^′∈Ω,t∈[0,T], we define (ω⊗_tω^′)_s:=

(ωs ifs∈[0,t]

ωt+ω^′_s−t ifs∈(t,T].

For any functionξ:Ω→Rtaking values in some setR, and for anyθ=(t,ω)∈Θ, we denote ξ^θ(ω^′) :=ξ(ω⊗_tω^′) ∀ω^′∈Ω.

Similarly, given a functionu:Θ→R, we denote, forθ∈Θ, u^θ(t^′,ω^′) :=u¡

(t+t^′)∧T,ω⊗_tω^′¢

∀θ^′∈Θ.

Clearly, ifξisFT-measurable thenξ^θ isFT−t-measurable, and ifX F-adapted then so isX^θ. Similarly, we can also shift functions defined on the enlarged canonical spaceΘe. Fort^′∈[0,T], θ∈Θ,ϑ∈Θe, we denote

θt^′∧·:=(t^′∧t,ωt^′∧·), ϑt^′∧·=(θt^′∧·,a_t^′∧·,µt^′∧·,q_t^′∧·).

Given a functionv:Θe→R, we define forϑ∈Θe ands∈[0,T] v^s,ϑ(ϑ^′) :=v^ϑs∧·(ϑ^′)=v³¡

t∧s+t^′¢

∧T,ω⊗_t∧sω^′,a⊗_t∧sa^′,µ⊗_t∧sµ^′,q⊗_t∧sq^′´

∀ϑ^′∈Θe. Further, for aG-stopping time_τ, we define

v^τ,ϑ:=v^τ(ϑ),ϑ. In particular, note thatv^τ,ϑ=v^t∧τ,ϑ.

Probability Space. We denote byP the set of probability measures on (eΘ,GT). Unless other- wise specified, the setP is always endowed with the topology of the weak convergence. We recall thatΘe is a Polish space with respect to the product topology, henceP is a Polish space too. We denote byT,B,A,M,Qthe first, second, third, fourth, and fifth projection ofΘe, respectively

T:Θe→R, ϑ7→t B:Θe→Ω, ϑ7→ω A:Θe→Ω, ϑ7→a M:Θe→Ω, ϑ7→µ Q:Θe→C₀([0,T],S^m), ϑ7→q.

We stress the fact thatTis aG-stopping time andB,A,M,Qare allG-adapted processes. In order to simplify the notations, we also denote the quintuple (T,B,A,M,Q) byX,

X:=(T,B,A,M,Q).

Remark 2.3. Using the notations defined previously, we note that T^τ,ϑ(_ϑ^′)=¡

t∧τ(_ϑ)+t^′¢

∧T, B^τ,ϑ(_ϑ^′)=ω⊗_t∧τ(ϑ)ω^′, ∀ϑ,_ϑ^′∈Θe, where_τis aG-stopping time. Further, given a functionu:Θ→R, we have

u¡

T^τ,ϑ(_ϑ^′),B^τ,ϑ(_ϑ^′)¢

=u³¡

t∧τ(_ϑ)+t^′¢

∧T,_ω⊗_t∧τ(ϑ)ω^′

´

=u^{t∧τ(ϑ),ω}(_θ^′).

In this paper, we will use the following probability family on the enlarged canonical spaceΘe to define the viscosity solutions to path-dependent PDEs.

Definition 2.4. For L>0, we define the subset ofΘe e

Θ_L:=©

ϑ∈Θe:t∈[0,T], ω=a+µ, a,q∈AC([0,T],R^m), |a|˙_∞≤L, |q|˙_∞≤Lª

, (2.1)

and the set of probabilities PL:=©

P∈P:P(eΘ_L)=1,M is a square-integrableP-martingale, 〈M〉 =Q ^P-a.s.ª . Using the canonical processes, we have for allP∈PLthat

T∈[0,T], B=A+M P-a.s. (2.2)

M is aP-martingale (2.3)

〈M〉 =Q ^P-a.s. (2.4)

A∈AC([0,T],R^m) and|A˙|∞≤LP-a.s. (2.5) Q∈AC([0,T],R^m×m) and|Q˙|_∞≤LP-a.s. (2.6) Recall thatTis aG-stopping time, soB_T∧·,A_T∧·,M_T∧·areG-adapted andM_T∧· is aP-martingale.

We introduce the sublinear and superlinear expectation operators associated withPL: EL:= sup

P∈PL

E^P E_L:= inf

P∈PL

E^P. (2.7)

3 Definition of _P

-viscosity solution

In this paper, we consider the fully nonlinear parabolic path-dependent PDE (PPDE):

−∂tu−G(θ,u,∂_ωu,∂²_ωωu)=0. (3.1) In [27, 26], it is showed that one can define viscosity solutions for PPDEs via jets. In this manuscript, we start directly from the definition via jets. For_α∈R, _β∈R^m, _γ∈S^m, let

ϕ^α,β,γ(θ) :=αt+ 〈β,ωt〉 +1

2〈γωt,ωt〉 ∀θ∈Θ, (3.2) where〈·,·〉denotes the standard scalar product onR^m.

A further tool for the type of localization that we will implement in the definition of viscosity solution if the functionH_δ. For anyδ∈(0,T], define the functionH_δ:Θe→[0,T] by

H_δ(ϑ) :=inf{s>0 : d_∞¡

(s,ω), 0¢

=s+ |ωs∧·|_∞≥δ}. (3.3) It is not difficult to show thatH_δis continuous.

Let_θ∈Θ, witht<T. Foru:Θ→Rupper semicontinuous, locally bounded from above, the subjet ofuinθis defined by

JLu(ϑ) :=n

(α,β,γ)∈R×R^m×S^m: u(θ)=EL

h

(u^θ−ϕ^α,β,γ)(T∧H_δ,B)i

for someδ∈(0,T−t]o , In a symmetric way, for a lower semicontinuous functionu: Θ→R, locally bounded from below, the superjet ofuinθis defined by

J_Lu(θ) :=

n

(α,β,γ)∈R×R^m×S^m:u(θ)=E_L h

(u^θ−ϕ^α,β,γ)(T∧H_δ,B) i

for someδ∈(0,T−t]

o . We denoteR:=R∪{−∞,+∞}.

Definition 3.1 (PL-viscosity sub-/supersolution). Let G:Θ×R×R^m×S^m→Rbe a function. A d∞-upper semicontinuous (resp. d∞-lower semicontinuous) function u:Θ→R, locally bounded from above (resp.locally bounded from below) is aPL-viscosity subsolution (resp.PL-viscosity supersolution) of (3.1)if

−α−G(θ,u(θ),β,γ)≤0 ∀θ∈Θ, (α,β,γ)∈J

Lu(θ), (resp. −α−G(θ,u(θ),β,γ)≥0 ∀θ∈Θ, (α,β,γ)∈J_Lu(θ)).

A locally bounded continuous function u is a PL-viscosity solution of (3.1) if it is both a PL- viscosity sub- and supersolution of (3.1).

Remark 3.2. Let us recall the previous definition of viscosity solution of path-dependent PDEs.

Define the probability familyP_L^′ on the spaceΘe^′:=Ω×Ω×Ω×C₀¡

[0,T],S^m¢

(we still use the notations of canonical processesB,A,M,Q):

P_L^′ :=n

P: B=A+M, M is aP-martingale, 〈M〉 =Q, A∈AC([0,T],R^m),

Q∈AC([0,T],R^m,m), |A|˙ ∞≤L, |Q|˙ ∞≤L, P-a.s.o Note that the space Θe^′ misses the time dimension, compared to the canonical space Θe in the present paper. Define the nonlinear expectationE^′_L:=supP∈P_L^′E^Pand the subjet

J^′

Lu(_θ) :=

½

(_α,_β,_θ)∈R×R^m×S^m: u(_θ)=max

τ∈T E^′_L h

(u^θ−ϕ^α,β,θ)(_τ∧H_δ,B)i

for some_δ∈(0,T−t]

¾ , whereT is the set of allF^B-stopping times. Similarly we can define the superjetJ^′_L. Then we defineP_L^′-viscosity solutions as in Definition 3.1, by replacing the jetsJ_L,J

L byJ^′_L,J^′

L. As we see, in our new definition of subjet the function (u−ϕ)^θ_H

δ∧· reaches its maximum at 0 in the sense ofELinstead of max_τ∈T E^′_L. By doing so, the maximization over the stopping times_τ∈T is replaced by the maximization of the laws applied on the canonical variableT. Indeed, the new nonlinear expectationELis a randomized optimal stopping operator. In general, one need fewer assumptions to ensure the existence of

P^∗∈arg max

P∈PL

E^P[f] than that of _τ^∗∈arg max

τ∈T E^′_L[f], for f:Θ→R (3.4) In particular, the optimal probability P^∗ for the first optimization exists if f is d_∞-u.s.c. and bounded from above (see SectionA), while in [9] the authors proved for the second optimization the optimal stopping time τ^∗ exists if f is bounded d∞-uniformly continuous. This change al- lows the viscosity solution under the new definition to have better properties. For example, it allows us to prove a stronger stability of solutions (see Section 4) and a comparison result for semicontinuous solutions (see Section6).

Finally, the change of definition does not threat the already proven results in the path- dependant PDE literature, namely, comparison [11, 26, 28], convergence of numerical schemes [25], etc. In fact, the arguments for these results will stay in the same lines, while necessary modifications need to be made concerning the optimization problems in (3.4).

4 Stability

For allG-stopping timeτ≤T, define the family:

PL(τ,ϑ) :=

n

P∈PL:T≤T−τ(ϑ), P-a.s.

o .

We list the following important properties concerning the families of probabilitiesPLandPL(τ,ϑ).

The proofs are postponed to Appendix.

Proposition 4.1. The setPLis compact.

Lemma 4.2. Letτ:Θe→[0,T]be a continuousG-stopping time. Then the graph of the function e

Θ→P, ϑ7→PL(τ,ϑ)

is closed, i.e., given{(ϑⁿ,Pⁿ)}_n∈N, (ϑ^∗,P^∗)such thatPⁿ∈PL(τ,ϑⁿ) for each n,|ϑⁿ−ϑ^∗|∞→0 and Pⁿ→P^∗, we haveP^∗∈PL(τ,ϑ^∗).

Define the nonlinear (conditional) expectations:

E^τ_L[·](ϑ) := sup

P∈PL(τ,ϑ)

E^P[·] E^τ_L[·](ϑ) := inf

P∈PL(τ,ϑ)

E^P[·].

Proposition 4.3. Let f:Θ→Rbe d∞-u.s.c. bounded from above, andτbe a continuousG-stopping time. We have

EL

h

f1_{T<τ}+E^τ_L£ f^θ¤

|θ=(τ,B)1_{τ≤T}

i

=EL

£f¤

. (4.1)

In particular we have

EL

h E^T_L£

f^θ¤

|θ=(T,B)

i

=EL

£f¤

. (4.2)

Moreover, letP^∗∈PLbe such thatEL[f]=E^P∗[f]. Then we have f =E^T_L£

f^T,B¤

P^∗-a.s.

As explained in Remark3.2, in this section we will exploit the advantage of the new definition of the jets in order to prove a better stability result. As we will see in the rest of the paper, one may apply many pseudo-metrics onΘother thand∞.

In what follows, we denote byda pseudo-metric onΘwhich isd_∞-continuous and such that d(θ,θ^′)=0 implies t=t^′ andωt∧·=ω^′_t∧·. In order to recall that d has these properties, we will often writed≪d∞.

Proposition 4.4. Let d be a pseudo-metric defined onΘsuch that d≪d∞. Let u_nbe a sequence of d_∞-u.s.c. functions uniformly bounded from above. Define the function u:Θ→Rby

u(θ) :=lim sup

d( ˆθ,θ)→0 n→∞

u_n( ˆθ).

Then, for any(α,β,γ)∈J

Lu(θ), there exist_θˆ_n∈Θand(αn,βn,γ)∈J

Lu_n( ˆθn)such that d( ˆθn,θ)→0, ¡

u_n( ˆθn),αn,βn

¢→¡

u(θ),α,β¢ . Proof. Step 1.Since (_α,_β,_γ)∈J

Lu(_θ), we also have (_α+ε,_β,_γ)∈J

Lu(_θ) for any_ε>0. Let_θn be a sequence such that

d(θn,θ)→0 and u(θ)= lim

n→∞u_n(θn).

For the simplicity of notation, we denote U(θ^′) :=¡

u^θ−ϕ^α+ε,β,γ¢¡

t^′∧H_δ(θ^′),ω^′¢

, U_n(θ^′) :=¡

u^θ_nⁿ−ϕ^α+ε,β,γ¢¡

t^′∧H_δ(θ^′),ω^′¢

, for allθ^′∈Θ.

SincePLis compact (Proposition4.1) and the mapsP →R,P7→E^P[U_n] are upper semicontinuous, there existsP_n∈PLfor eachn∈Nsuch that

E^Pn£ U_n¤

=EL

£U_n¤

. (4.3)

By considering a subsequence if necessary, again denoted byP_n, we can findP^∗∈PL such that P_n→P^∗. By Skorohod’s representation there exists a probability space (Ωb,Gb,bP) on which

∃ r.v.’s Xⁿ

¯¯

¯_b

P

=dX

¯¯

¯P_n, X^∗

¯¯

¯_b

P

=dX

¯¯

¯P^∗ such that ¯¯Xⁿ−X^∗¯¯

∞→0 Pb-a.s., (4.4) where=^d denotes equality in distribution. Together with (4.3) we obtain

u(_θ)= lim

n→∞u_n(_θn)≤lim sup

n→∞ EL

£U_n¤

=lim sup

n→∞

E^Pn£ U_n¤

=lim sup

n→∞

E^bP£

U_n(Xⁿ)¤

≤E^bP

· lim sup

n→∞

U_n(Xⁿ)

¸

≤E^Pb£ U(X^∗)¤

=E^P∗£ U¤

≤EL

£U¤

=u(θ),

(4.5)

The first inequality in the last line is due to Fatou’s lemma, the second one is due to the definition ofu, whereas the last equality is due to the assumption (α,β,γ)∈J

Lu(θ) and by choosing anyδ sufficiently small. Therefore

E^P∗£ U¤

=EL

£U¤ , which, together with (α,β,γ)∈J_Lu(θ), provides

P^∗[T=0]=1. (4.6)

SinceH_δis continuous, from (4.6) and by denotingT^∗=T(X^∗), Tⁿ=T(Xⁿ), we have 1=bP£

T^∗=0¤

=Pb£

T^∗<H_δ(X^∗)¤

=Pb£

n→∞lim(Tⁿ−H_δ(Xⁿ))<0¤

≤Pb£ lim inf

n→∞

©Tⁿ<H_δ(Xⁿ)ª¤

. (4.7) Further, denotingB^∗=B(X^∗) andBⁿ=B(Xⁿ), by (4.4),(4.6) and byd_∞-continuity ofd, we have

1=Pb h

d¡

θ, (t+T^∗,ω⊗_tB^∗)¢

<δi

=Pb h

n→∞lim d¡

θn, (t_n+Tⁿ,ωn⊗_tnBⁿ)¢

<δi

≤Pb h

lim inf

n→∞

n d¡

θn, (t_n+Tⁿ,_ωn⊗_tnBⁿ)¢

<δ oi

.

(4.8)

Step 2.It follows from (4.3) and Proposition4.3that U_n=E^T_L£

U_n^T,B¤

, P_n-a.s.

Therefore, for alln∈N, P_n£

Ξⁿ¤

=1, where Ξⁿ:=n

ϑ^′∈Θe:U_n(_θ^′)=E^t

′

L

£U_n^θ′¤ (_ϑ^′)o

. (4.9)

Note that

Ξⁿ∩{T<H_δ}⊂n

ϑ^′∈Θe:¡

α+ε,β+2γω^′_t′,γ¢

∈J

Lu^θ_nⁿ(θ^′)o , and by (4.7), (4.8) and (4.9) we have

b P h

lim inf

n→∞

n¡α+ε,_β+2_γBⁿ

Tⁿ,_γ¢

∈J

Lu^θ_nⁿ(Tⁿ,Bⁿ), d¡

θn, (t_n+Tⁿ,_ωn⊗_tnBⁿ)¢

<δ oi

=1. (4.10) Step 3.It follows from (4.6) that

b P£

n→∞lim Tⁿ=0¤

=1.

Together with (4.5) we obtain u(θ)=E^bPh

lim sup

n→∞

U_n(Xⁿ)i

=E^Pbh

lim sup

n→∞

u^θ_nⁿ(Tⁿ,Bⁿ)i .

By the definition ofuwe know that lim sup_n→∞u^θ_nⁿ(Tⁿ,Bⁿ)≤u(_θ)Pb-a.s. Therefore, b

P h

lim sup

n→∞

u^θ_nⁿ(Tⁿ,Bⁿ)=u(θ) i

=1.

Together with (4.10), we obtain b

P hn

lim sup

n→∞

u^θ_nⁿ(Tⁿ,Bⁿ)=u(θ) o

∩lim inf

n→∞

n¡α+ε,β+2γBⁿ

Tⁿ,γ¢

∈J

Lu^θ_nⁿ(Tⁿ,Bⁿ), d¡

θn, (t_n+Tⁿ,ωn⊗_tnBⁿ)¢

<δoi

=1.

(4.11) Denote ˆθn:=¡

t_n+Tⁿ(ϑ^′),ωn⊗_tnBⁿ(ϑ^′)¢

andβn:=_β+2γBⁿ

Tⁿ(ϑ^′) for allϑ^′∈Θe,n∈N. It follows from (4.11) that there existϑ^′ andnsuch that

d( ˆθ_n,θ)<d(θ_n,θ)+δ, ¯¯u_n( ˆθ_n)−u(θ)¯¯<δ, |β_n−β| <2γδ, and (α+ε,β_n,γ)∈J

Lu_n( ˆθ_n).

Finally, since the constantsδ,εcan be chosen arbitrarily small, the desired result follows.

As direct consequence of Proposition4.4we have the following

Theorem 4.5. Let d,u_n,u be as in Proposition4.4. Let G_n:Θ×R×R^m×S^m→Rbe a sequence of functions. Define G:Θ×R×R^m×S^m→Rby

G(θ,r,β,γ) := lim sup

n→∞,d(θ_n,θ)→0 rn→r,βn→β

G_n(θn,r_n,βn,γ).

If u_nis aPL-viscosity subsolution of

−∂tu_n−G_n(_θ,u_n,_∂ωu_n,_∂²_ωωu_n)=0, then u is aPL-viscosity subsolution of

−∂_tu−G(θ,u,∂_ωu,∂²_ωωu)=0.

5 Existence of _P

-viscosity solution: Perron’s method

As we see in the classical literature on the viscosity solutions, one can apply the stability and the comparison results for semi-continuous solutions to prove the existence of viscosity solution via the so-called Perron’s method. In Section4, we have established a quite general stability result (Theorem4.5), and we will leave the discussion on the comparison result to the next sections. In this section, we adapt the Perron’s method to the context ofPL-viscosity solution, assuming some comparison result holds true.

Assumptions 5.1. Letdbe a pseudo-metric onΘsuch thatd≪d∞.

(i) G:Θ×R×R^m×S^m→Risd+|·|+|·|∞+|·|∞-uniformly continuous ond+|·|+|·|∞+|·|∞-bounded sets.

(ii) For every (θ,β,γ)∈Θ×R^m×S^m, the functionr7→G(θ,r,β,γ) is non-decreasing.

(iii) For every (θ,r,β)∈Θ×R×R^m,

G(θ,r,β,γ)≤G(θ,r,β,γ^′) ∀γ,γ^′∈S^m, γ≤γ^′. (iv) For all (_θ,r)∈Θ×R,_β,_β^′∈R^m,_γ,_γ^′∈S^m,

|G(θ,r,β+β^′,γ+γ^′)−G(θ,r,β,γ)| ≤L¡

|β^′| + |γ^′|¢ .

Assumption 5.2. Let d be a pseudo-metric on Θ such that d ≪d_∞. Assume that there is a bounded d-u.s.c. viscosity subsolution u and a bounded d-l.s.c. viscosity supersolution v of PPDE (3.1) such that

i) u≤vonΘ; ifu_∗denote thed-l.s.c. envelop ofu, thenu_∗(T,·)=v(T,·)=:_ξ.

ii) For any d-u.s.c. viscosity subsolution u andd-l.s.c. viscosity supersolution vof PPDE (3.1) taking values betweenu_∗andv, we haveu≤vonΘ.

.

Theorem 5.3. Let Assumptions5.1and5.2hold true. Denote D:=©

φ:Θ→R:φis a bounded d-u.s.c.PL-viscosity subsolution of (3.1)and u≤φ≤vª . Then u(θ) :=sup©

φ(θ) :φ∈Dª

is a d-continuousPL-viscosity solution of (3.1), and satisfies the boundary condition u(T,·)=ξ.

Before proving Theorem 5.3, we show some useful lemmas. The following proposition is a direct corollary of the stability result in Proposition4.4.

Proposition 5.4. Let u be d∞-u.s.c., θ∈Θ and (α,β,γ)∈J

Lu(θ). Suppose also that u_n is a sequence of d∞-u.s.c. functions uniformly bounded from above such that

((1) there existθn∈Θsuch that d(θn,θ)→0, u_n(θn)→u(θ), (2) ifθ^′_n∈Θand d(θ^′_n,θ^′)→0, then lim sup_n→∞u_n(θ^′_n)≤u(θ^′).

Then there exist_θˆ_n∈Θ,(αn,βn,γ)∈J

Lu_n( ˆθn)such that d( ˆθn,θ)→0, ¡

u_n( ˆθn),αn,βn

¢→¡

u(θ),α,β¢ . Proof. Recall the definition ofuin Proposition4.4. We clearly have

u(θ)=u(θ) and u(θ^′)≥u(θ^′) for all θ^′∈Θ. It the follows that (α,β,γ)∈J

Lu(θ). Then the desired result follows from Proposition4.4.

Forγ∈S^m, we define

(γ)⁺:= sup

x∈R^m,|x|≤1

〈γx,x〉. (5.1)

Lemma 5.5. Let G as in Assumption5.1. Then, for allθ∈Θ,r∈R,β∈R^m,γ,γ^′∈S^m, G(θ,r,β,γ)−G(θ,r,β,γ^′)≤(γ−γ^′)⁺.

Proof. By definition of (·)⁺ in (5.1), we haveγ−γ^′≤L(γ−γ^′)⁺. By the ellipticity condition As- sumption5.1(iii) and the Lipschitz continuity condition in Assumption5.1(iv) we then have

G(θ,r,β,γ)=G(θ,r,β,γ−γ^′+γ^′)≤G(θ,r,β, (γ−γ^′)⁺+γ^′)≤L(γ−γ^′)+G(θ,r,β,γ^′),

which concludes the proof.

Lemma 5.6. Recall the definition of the nonlinear expectationE^′_Lin Remark3.2. Letξ:Ω→Rbe bounded,| · |_∞-u.s.c. Define

φ(θ) :=E^′_L h

ξ^θ i

+ϕ^α,β,γ(θ).

Thenφis a locally bounded d∞-u.s.c. function and is aPL-viscosity subsolution to

−∂tφ+α+L|β+γωt−∂_ωφ| +L(γ−∂²_ωωφ)⁺=0. (5.2) Proof. It is clear thatφis a locally boundedd_∞-u.s.c. function. Define

ψ(θ) :=φ(θ)−ϕ^α,β,γ(θ)=E^′_L h

ξ^θ i

.

By the dynamic programming result (see e.g. Theorem 2.3 in [18]), we have ψ(θ)=E^′_L

h

ψ^θ(η,B) i

for allF-stopping timeη≥0.

Then it is easy to verifyψis aPL-viscosity subsolution to

−∂tψ+L|∂_ωψ| +L(−∂²_ωωψ)⁺=0. (5.3) and, using (5.3), it is not difficult too see that_φis aPL-viscosity subsolution to (5.2).

Lemma 5.7. Let_ξ:Ω→Rbe bounded and d_∞-l.s.c. Define φ(θ) :=E^′_L

h ξ^θ

i . Then_φis a bounded d_∞-l.s.c. function.

Proof. Define the function

f(θ,P) :=E^P[ξ^θ], for allθ∈ΘandP∈P_L^′.

Letθn∈ΘandP_n∈P_L^′ such thatd∞(θn,θ)→0 andP_n→P. By Skorohod’s representation there exists a probability space (Ωb,Gb,bP) on which

∃ r.v.’s X^n¯¯_¯

b P

=dX^¯¯_¯

P_n, X^∗¯¯_¯

b P

=dX^¯¯_¯

P such that ¯¯Xⁿ−X^∗¯¯

∞→0, Pb-a.s.

So we have

lim inf

n→∞ f(θn,P_n)=lim inf

n→∞ E^bP h

ξ¡

ωn⊗_tnB^n¢i≥E^bP h

ξ¡

ω⊗_tB^∗¢i=f(θ,P),

where the inequality is due to Fatou’s lemma and thed_∞-l.s.c. ofξ. Therefore,f is l.s.c. onΘ×P_L^′. Further, note that

φ(θ)= sup

P∈P_L^′

f(θ,P)

andP_L^′ is compact (see [17]). By Proposition 7.32 of [1], we obtain that_φisd_∞-l.s.c.

Lemma 5.8. Let Assumption5.1 hold true. Let u be a bounded PL-viscosity subsolution (resp.

supersolution) and u^∗ (resp. u∗) be its d-u.s.c. (resp. l.s.c.) envelop. Then u^∗ (resp. u∗) is also a PL-viscosity subsolution (resp. supersolution).

Proof. We only show the result for subsolution. The one for the supersolution follows from the same argument. First, notice thatd≪d∞implies thatu^∗isd∞-u.s.c. Now letθ∈Θand (α,β,γ)∈ Ju^∗(θ). By the very definition of u^∗, it is clear that the assumptions of Proposition 5.4 are fulfilled (with the functions u_n,u appearing in the statement replaced by u,u^∗, respectively).

Hence there exist ˆθn∈Θ, (αn,βn,γ)∈Ju( ˆθn) such that d( ˆθn,θ)→0, ¡

u( ˆθn),αn,βn

¢→¡

u^∗(θ),α,β¢ . Sinceuis aPL-viscosity subsolution to (3.1), we have

−αn−G( ˆθn,u( ˆθn),βn,γ)≤0.

By lettingn→ ∞, we have−α−G(θ,u^∗(θ),β,γ)≤0.

Lemma 5.9. Forθ∈Θ, with t<T, define D(θ) :=©

φ: [0,T−t]×Ω→R: u^θ≤φ≤v^θ and

φis a bounded d-u.s.c. viscosity subsolution of (3.1)on[0,T−t]ª , andu(e _θ^′) :=sup{_φ(_θ^′) :_φ∈D(_θ)}. Then we have u(_θ)=u(0).e

Proof. First it is obvious that, for anyφ∈D, we haveφ^θ∈D(θ). Sou(θ)≤u(0).e

On the other hand, suppose u(_θ)<eu(0). Then there is_φ∈D(_θ) such that_φ(0)>u(_θ). Now take anyφe∈D, and define, forθ^′∈Θ,

Φ(_θ^′) :=

(φe(θ^′)∨φ( ˆθ) if ˆt+t=t^′≥tandω^′=ω⊗_tωˆ, φe(θ^′) elsewhere.

It is easy to verify thatΦisd∞-u.s.c. and that, by the viscosity subsolution property ofφeandφ, Φis aPL-viscosity subsolution to (3.1). Then, it follows from Lemma5.8that thed-u.s.c. envelop Φ^∗is also aPL-viscosity subsolution to (3.1). Finally note that

Φ^∗(θ)≥Φ(θ)≥φ(0)>u(θ),

which is a contradiction to the definition ofu. Therefore, we haveu(θ)=eu(0).

Proof of Theorem5.3. Thanks to Lemma 5.9, we only need to check the property of viscosity solution at the pointθ=(0, 0).

Step 1.We first prove that u isd-u.s.c. and that it is a PL-viscosity subsolution. Let (α,β,γ)∈ Ju^∗(0, 0), where u^∗ is the d-u.s.c. envelop of u. By the definition of u and u^∗, there exists a sequence ofu_n∈Dandθn∈Θsuch that

d(θn, (0, 0))→0, u_n(θn)→u^∗(0, 0). (5.4) Moreover,

ifθ_n^′ ∈Θandd(θ^′_n,θ^′)→0, then lim sup

n→∞

u_n(θ_n^′)≤u^∗(θ^′). (5.5)

It follows from Proposition5.4that there exist ˆθ_n∈Θand (α_n,β_n,γ)∈Ju_n( ˆθ_n) such that d( ˆθn, (0, 0))→0, ¡

u_n( ˆθn),αn,βn

¢→¡

u^∗(0, 0),α,β¢ . Sinceu_nare allPL-viscosity subsolution to (3.1), we have

−αn−G( ˆ_θn,u_n( ˆ_θn),_βn,_γ)≤0.

If we let n→ ∞, we obtain−α−G¡

(0, 0),u^∗(0, 0),β,γ¢

≤0. Therefore, u^∗ is aPL-viscosity subso- lution. By definition of u, we have u^∗≤u. On the other hand, by definition ofu^∗, we also have u^∗≥u. So finally we conclude that

u=u^∗ isd-u.s.c. and aPL-viscosity subsolution.

Step 2. Let ue_∗ be the d_∞-l.s.c. envelop of the function u. We now prove ue_∗ is a PL-viscosity supersolution atθ=0. Suppose it is not the case and there is (α,β,γ)∈Jue∗(0, 0) such that

−α−G¡

(0, 0),ue_∗(0, 0),β,γ¢

= −3ε<0.

Therefore, forδsmall enough we have e

u∗(0, 0)=E_L h

(ue∗−ϕ^α,β,γ)(T∧H_δ,B) i

, δ≤ ^ε

L|γ| and −α−G¡

θ,ue_∗(θ),β+γω_t,γ¢

< −2ε for allt≤H_δ(ω). (5.6) For the simplicity of notation, we denote for allθ∈Θ

ξ(θ) :=u¡

H_δ(ω),ω¢

−ϕ^{α−ε,β,γ}¡

H_δ(ω),ω¢

and ξ(θ) :=ue∗

¡H_δ(ω),ω¢

−ϕ^{α−ε,β,γ}¡

H_δ(ω),ω¢ . Recall the nonlinear expectationE^′_Ldefined in Remark3.2. Define

φ(θ) :=E^′_L£ ξ^θ¤

+ϕ^{α−ε,β,γ}(θ), φ(θ) :=E^′_L£ ξ^θ¤

+ϕ^{α−ε,β,γ}(θ).

By recalling the definitions ofE^′_LandE_L, we have φ(0, 0)=E^′_L£

ξ¤

≥E_L£ ξ¤

=E_L£

(ue∗−ϕ^{α−ε,β,γ})¡

H_δ,B^¢¤>eu∗(0, 0).

By Lemma5.7,_φisd_∞-l.s.c. ByStep 1we know thatuisd- henced_∞-u.s.c. Then, by Lemma5.6, it follows thatφis a locally boundedd∞-u.s.c. function and aPL-viscosity subsolution to

−∂_tφ+α−ε+L|β+γω_t−∂_ωφ| +L(γ−∂²_ωωφ)⁺=0. (5.7) It follows from (5.6) that α>2ε−G¡

θ,ue∗(θ),β+γωt,γ¢ on ©

θ: t<H_δ(ω)ª

. By the Lipschitz con- tinuity assumptions on G and Lemma5.5, we then have that _φ is a PL-viscosity subsolution to

−∂tφ−G(·,ue∗,∂ωφ,∂²_ωωφ)=0, on ©

θ:t<H_δ(ω)ª .

Further, since eu∗ ≤u≤u∨φ and G is non-decreasing in r, the function φ is a PL-viscosity subsolution to

−∂tφ−G(·,u∨φ,∂ωφ,∂²_ωωφ)=0, on ©

θ: t<H_δ(ω)ª

. (5.8)

Now define

U:=(u∨φ)1_{t<Hδ(ω)}+u1_{{t≥Hδ(ω)}}.