J Theor Probab DOI 10.1007/s10959-017-0791-z

Viability for Stochastic Differential Equations Driven by G-Brownian Motion Peng Luo1,2 · Falei Wang3

Received: 12 October 2016 / Revised: 13 August 2017 © Springer Science+Business Media, LLC 2017

Abstract In this paper, we prove a type of Nagumo theorem on viability properties for stochastic differential equations driven by G-Brownian motion (G-SDEs). In particular, an equivalent criterion is formulated through stochastic contingent and tangent sets. Moreover, by the approach of direct and inverse images for stochastic tangent sets we present checkable conditions which keep the solution of a given G-SDE evolving in some particular sets. Keywords Stochastic viability · Stochastic differential equation · Stochastic tangent set · G-Brownian motion Mathematics Subject Classification (2010) 60H30 · 60H10

Peng Luo: Research partially supported by National Science Foundation of China, “Research Fund for International Young Scientists” (No. 11550110184) and National Natural Science Foundation of China (No. 11671257). Falei Wang: Research partially supported by the National Natural Science Foundation of China (No. 11601282), the Natural Science Foundation of Shandong Province (No. ZR2016AQ10) and the China Scholarship Council (No. 201606225002). Luo and Wang’s research was partially supported by the Tian Yuan Projection of the National Natural Sciences Foundation of China (Nos. 11526205 and 11626247) and the 111 Project (No. B12023).

B

Falei Wang [email protected] Peng Luo [email protected]

1

Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, China

2

Department of Mathematics, ETH Zurich, Zurich, Switzerland

3

Zhongtai Securities Institute for Financial Studies and Institute for Advanced Research, Shandong University, Jinan, China

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1 Introduction Motivated by uncertainty problems, risk measures and the superhedging in finance, Peng has systemically established a time-consistent fully nonlinear expectation theory ( see [24–26]). As a typical and important case, Peng has introduced the G-expectation theory in 2006 (see [27] and the references therein). In the G-expectation framework, the notion of G-Brownian motion and the corresponding stochastic calculus of Itô’s type have been also established. On this basis, Gao [14] and Peng [27] had studied the existence and uniqueness of the solution to G-SDEs under standard Lipschitz condition. Moreover, Bai and Lin [5] had obtained the existence and uniqueness of the solution of G-SDEs under some general conditions. We refer the reader to Lin [17] and Li et al. [19] for a closest related approach on this field. The viability property for deterministic differential equations has been studied for the first time by Nagumo [22] in 1942. In this framework, he has established a criterion of the viability in terms of contingent sets. To the best of our knowledge, the viability problem for stochastic differential equations and inclusions has been firstly studied by Aubin and Da Prato [1–3]. The key point of this approach is based on the formulation of “stochastic tangent cone,” which generalizes the classical cone corresponding to deterministic systems. Afterward, Milian [21] had extended Aubin and Da Prato’s result to arbitrary subsets which could be time-dependent and random. Recently, in spirit of Aubin and Da Prato’s argument, Ciotir and R˘as¸canu [9] had discussed the viability property for stochastic differential equation driven by fractional Brownian motion, based on which Nie and R˘as¸canu [23] had considered direct and inverse images for fractional stochastic tangent. From a different point of view, Buckdahn et al. [6] had applied a tricky viscosity solution approach to tackle this question (see also [7,8,28] and the references therein for more research). The main idea of this work is to prove that the viability property for SDE (respectively, BSDE) holds true if and only if the square of the distance to the constraint sets is a viscosity supersolution (respectively, subsolution) of the related Hamilton–Jacobi–Bellman equations. We also refer to Da Prato and Frankowska [10,11] for some other arguments. The aim of the present paper is to investigate the viability property for G-SDEs by the approach of stochastic contingent and tangent sets. The constraint set considered in the sequel could be time-dependent and this generalizes the one in [20] which is independent of time argument. Note that the G-expectation space is a proper subset of classical Winener space and the measurable selection argument cannot be applied directly to deal with G-stochastic analysis questions, which is crucial for the stochastic tangent set argument. To overcome this difficulty, we shall consider the universally augmented filtration instead of the natural filtration of the canonical process considered in [27]. Thanks to the works of Peng [27] and Li and Peng [18], we could establish the theory of G-Itô stochastic calculus in this new setting. Moreover, the G-SDE has a unique solution in a new space M 2 (0, T ) (see Sect. 1) under the standard Lipschitz conditions. In this extended G-expectation space, we introduce the notions of viability, contingent set and tangent set by the language of “quasi-surely” (see [13]). Then, the equivalent criteria of viability for G-SDEs is obtained via combining the argument of stochastic tangent sets and nonlinear stochastic analysis technique. In contrast to

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the ones in [1], we construct more complicated approximate viable solutions to adapt the argument of Aubin and Da Prato to the nonlinear expectation framework (see, for example, Remak 3.11). Moreover, the direct and inverse images for stochastic tangent sets are formulated to characterize the viability for G-SDEs. In particular, some deterministic sufficient and necessary conditions are also given to guarantee the solution of a given G-SDE evolves in some particular sets, which provides a potential tool for the research of viability theory in the nonlinear expectation framework. The rest of this paper is organized as follow: In Sect. 2, we recall some notations and results which will be used in the sequel. Section 3 is devoted to the study of stochastic viability for G-SDE. In Sect. 4, we give the deterministic characterization of viability conditions for some particular constraint sets by using direct and inverse images for stochastic tangent sets.

2 Preliminaries The main purpose of this section is to recall some preliminary results in G-framework which are needed in the sequel. More details can be found in Peng [27], Li and Peng [18] and Denis et al. [12]. Let Ω = C0 (R+ ) be the space of all R-valued continuous paths (ωt )t≥0 starting from origin, equipped with the distance

ρ(ω1 , ω2 ) :=

∞  i=1

2−i



  max |ωt1 − ωt2 | ∧ 1 .

t∈[0,i]

For each t ∈ [0, ∞), we denote • Bt (ω) := ωt for each ω ∈ Ω; • B(Ω): the Borel σ -algebra of Ω, Ωt := {ω·∧t : ω ∈ Ω}, Ft0 := B(Ωt ); • L i p (Ω) := {ϕ(Bt1 , . . . , Btk ) : k ∈ N, t1 , . . . , tk ∈ [0, ∞), ϕ ∈ Cb,Li p (Rk )}, where Cb,Li p (Rk ) is the space of all bounded and Lipschitz functions on Rk . Let the canonical process Bt be the G-Brownian motion under the G-expectation 2 2 ˆ ˆ ˆ 2 (Eˆ t [·])t≥0 ), where σ 2 := −E[−B space (Ω, L i p (Ω), E[·], 1 ] ≤ E[B1 ] =: σ . For ˆ each p ≥ 1, the completion of L i p (Ω) under the norm ||X || L p := (E[|X | p ])1/ p is p

p

denoted by L G (Ω). Similarly, we can define L G (Ωt ).

G

Lemma 2.1 ([12,15]) There exists a weakly compact set P of probability measures on (Ω, B(Ω)) such that ˆ ] = sup E P [ξ ] for all ξ ∈ L 1 (Ω). E[ξ G P∈P

ˆ P is called a set that represents E.

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ˆ Then, it is natural to introduce Let P be a weakly compact set that represents E. the following capacity: c(A) := sup P(A), A ∈ B(Ω). P∈P

A set A ⊂ B(Ω) is polar if c(A) = 0. A property holds “quasi-surely” (q.s.) if it holds outside a polar set. In the following, we do not distinguish between two random variables X and Y if X = Y q.s. Denote by FtP the P-completion of Ft0 and Ft := ∩ P∈P FtP for each t. We set L p (Ft ) := {X ∈ Ft : sup E P [|X | p ] < ∞} for p ≥ 1. P∈P

It is important to note that L G (Ωt ) ⊆ L p (Ft ). Then, the G-expectation Eˆ could be ˆ i.e., for each X ∈ L 1 (Ft ), extended to the space of L p (Ft ) and we still denote it by E, p

ˆ ] = sup E P [X ]. E[X P∈P

ˆ · | p ])1/ p . For p ≥ 1, L p (Ft ) is a Banach space under the norm (E[| From Theorem 8.2.13 in [4], we have the following result, which is crucial for the main results in this paper. Lemma 2.2 (Projection map) Assume that a set-valued map K : Ω → Rn with closed images is Ft -measurable. Let X : Ω → Rn be an Ft -measurable random variable. Then, the projection map ω ∈ Ω →  K (ω) (X (ω)) is given by  K (ω) (X (ω)) := {x ∈ K (ω)|d(x, X (ω)) = d(X (ω), K (ω))}, where d is the usual distance on Rn . If  K (ω) (X (ω)) = ∅ for every ω ∈ Ω, then there exists a Ft -measurable selection g(ω) ∈ K (ω) such that d(X (ω), g(ω)) = d(X (ω), K (ω)). Remark 2.3 In general we cannot obtain that g given in Lemma 2.2 belongs to Gexpectation space. Now, we set M p,0 (0, T ) ={η := ηt (ω) =

N −1 

ξ j (ω)I[t j ,t j+1 ) (t).

j=0

∀ N > 0, 0 = t0 < · · · < t N = T, ξ j (ω) ∈ L p (Ft j ), j = 0, 1, 2, . . . , N − 1}.

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For each p ≥ 1, denote by M p (0, T ) the completion of M p,0 (0, T ) under the norm  ||η|| M p (0,T ) =

T

ˆ t | p ]dt E[|η

1/ p .

0

T From Li and Peng [18], for each η ∈ M 2 (0, T ), the Itô’s integral 0 ηs dBs is well defined. Moreover, by Proposition 2.10 in [18] and classical Burkholder–Davis–Gundy inequality, the following properties hold. Lemma 2.4 For each η, θ ∈ M p (0, T ), p ≥ 2, ξ ∈ L ∞ (Ft ), we have  ηs dBs = 0, 0   t p

 p/2

T   p 2  ˆ ˆ σ cpE |ηs | ds ≤ E sup  ηs dBs  0 t∈[0,T ] 0

  p/2 T ≤ σ¯ p C p Eˆ |ηs |2 ds ,

ˆ E



T

0



T



(ξ ηs + θs )dBs = ξ

t

T

 ηs dBs +

t

T

θs dBs ,

t

where 0 < c p < C p < ∞ are some constants. Then, the quadratic variation process of B can be formulated in the following way:  Bt := Bt2 − 2

t

Bs dBs .

0

Moreover, Itô’s integral with respect to B can also be defined in a similar fashion as in Peng [27]. In particular, Lemma 2.5 Let p ≥ 1, η ∈ M p (0, T ). Then, it holds that ˆ E



t

sup | 0≤t≤T

ηs dBs |

0

p

 ≤σ

2p

T

p−1

T

ˆ s | p ]ds. E[|η

0

3 Stochastic Viability for G-Stochastic Differential Equation Let us first recall some notations. • C n (Rd ; Rk ): the space of all functions of class C n from Rd into Rk , n d k n d k • Cb,Li p (R ; R ): the space of all bounded functions of class C (R ; R ) whose partial derivatives of order less than or equal to n are bounded Lipschitz continuous functions.

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Let ·, · and | · | denote the inner product and norm in Rd , respectively. Given a family of closed subsets K := (Kt )t∈[0,T ] of Rd . In the rest of this paper we shall make use of the following assumption. (H) b, h, σ are given Rd -valued functions on Rd . Moreover, b, h, σ ∈ Cb,Li p (Rd ; Rd ). Consider the following SDE driven by a 1-dimensional G-Brownian motion: dX t = f (X t )dt + g(X t )dBt + h(X t )dBt .

(1)

Similar to Peng [27], we have the following result by a fixed point argument. Theorem 3.1 Under the assumption (H ), for any given initial condition X 0 ∈ Rd , there exists a unique solution X ∈ M 2 (0, T ) to the G-SDE (1). In order to focus on the main ideas, in this paper we content ourselves with the simple G-SDE (1) under somewhat strong technical conditions, and leave more general cases for future studies. In particular, by slightly more involved estimates, we can extend our results to multi-dimensional G-Brownian motion. Now we introduce the definition of the viability property for G-SDE. Definition 3.2 K is said to be viable for G-SDE (1) if starting at any time t ∈ [0, T ] t,ξ and from any random variable ξ ∈ L 2 (Ft ) in Kt , the solution (X s )t≤s≤T to (1) satisfies for each s ∈ [t, T ], X st,ξ ∈ Ks ,

q.s.

Then, we shall state the concept of contingent set and tangent set, which is inspired from [1,2]. Consider a random variable ξ ∈ L 2 (Ft ) in Kt for some time point t ∈ [0, T ]. Definition 3.3 The stochastic contingent set CK (t, ξ ) to K at ξ is the set of all bounded triples (u, v, w) of Ft -measurable random variables, such that for any  > 0, there exists δ  > 0 such that for each δ ∈ (0, δ  ], we can find three Ft+δ -measurable random variables a δ , bδ and cδ so that, ⎧ ˆ δ |2 ] ≤  2 (i) E[|a ⎪ ⎪ ⎪ ⎪ ⎪ ˆ δ |2 ] ≤  2 ⎪ (ii) E[|b ⎪ ⎨ ˆ δ |2 ] ≤  2 (iii) E[|c ⎪ ⎪ ⎪ ⎪ ˆ δ] = 0 (iv) E[c ⎪ ⎪ ⎪ ⎩ ˆ (v) E[η, cδ ] = 0, ∀ Ft -measurable random variable η and satisfy ξ + w(Bt+δ − Bt ) + uδ + v(Bt+δ − Bt ) + δa δ + δbδ √ + δcδ ∈ Kt+δ , q.s.

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Definition 3.4 The stochastic tangent set TK (t, ξ ) to K at ξ is the set of all bounded triples (u, v, w) of Ft -measurable random variables, such that there exist three bounded adapted stochastic processes as , bs , cs converging to 0 when s → t so that for some δ  > 0, 



t+δ

∀δ ∈ [0, δ ], ξ + 

t



t

t+δ

+

t+δ

+

(u + as )ds (v + bs )dBs (w + cs )dBs ∈ Kt+δ , q.s.,

t

where the stochastic process d = a, b, c satisfies: for any p > 0 there exists some constant C p depending on p and T such that,     p Eˆ |ds1 − ds2 | p ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ t, t + δ  . Example 3.5 Let K be a vector subspace independent of t. If for each x ∈ Rd , f (x) ∈ K, h(x) ∈ K, g(x) ∈ K, taking a = b = c = 0 indicates that  ξ+ t

s



s

f (ξ )ds +



s

h(ξ )dBs +

t

g(ξ )dBu ∈ K.

t

Thus, ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ). By the above definitions, we have the following relationship between contingent set and tangent set. Lemma 3.6 For each ξ ∈ L 2 (Ft ) in Kt , TK (t, ξ ) ⊂ CK (t, ξ ). Proof Assume that (u, v, w) ∈ TK (t, ξ ). Then, there exist three bounded adapted stochastic processes as , bs , cs converging to 0 when s → t such that for some δ  > 0, ∀δ ∈ [0, δ  ], ξ +



t+δ



t



t

t+δ

+

t+δ

+

(u + as )ds (v + bs )dBs (w + cs )dBs ∈ Kt+δ .

t

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 t+δ Set a δ := 1δ t as ds, bδ := Hölder’s inequality yields that

 1 t+δ bs dBs δ t

and cδ :=

√1 δ

 t+δ t

cs dBs . Applying

   t+δ    t+δ    1 1 1 t+δ ˆ  2  δ 2 2 2 ˆ ˆ ˆ E |a | = E 2 | as ds| ≤ E |as | ds ≤ E |as | ds. δ δ t δ t t ˆ s |2 ] ≤ C2 |s − t| for some constant C2 , we Recalling that for each s ∈ [t, t + δ  ], E[|a get   ˆ |a δ |2 ≤ C2 δ. E 2 In a similar way, we can find some constant K so that,   Eˆ |bδ |2 + |cδ |2 ≤ K δ. In view of the property of G-stochastic integral, we obtain for every Ft -measurable random variable η,    t+δ   1 δ ˆ ˆ η, cs dBs = 0, E η, c  = E √ δ t  

which completes the proof. Now we are in a position to give the main result of this section.

Theorem 3.7 Let K be a family of closed subsets of Rd , then the following conditions are equivalent: (1) K is viable for G-SDE (1). (2) For each t ∈ [0, T ] and any ξ ∈ L 2 (Ft ) in Kt , ( f (ξ ), h(ξ ), g(ξ )) ∈ CK (t, ξ ). (3) For each t ∈ [0, T ] and any ξ ∈ L 2 (Ft ) in Kt , ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ). Remark 3.8 Note that the viscosity solution approach can be also applied to discuss the viability property for G-SDEs (see [20]). However, due to the structure of the viscosity solution this method requires that the square of the distance to the constraint sets is lower semicontinuous in time variable (see [7]). It is obvious that (3) ⇒ (2). Before proceeding to the proof, we need the following lemmas.

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Lemma 3.9 If the family K is viable for G-SDE (1), then for every ξ ∈ L 2 (Ft ) in Kt , ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ). Proof Consider the following stochastic process 

s

Xs = ξ + t



s

f (X u )du + t



s

h(X u )dBu +

g(X u )dBu , s ∈ [t, T ].

(3)

t

We can rewrite it in the following form: for each s ∈ [t, T ],  X s =ξ + f (ξ )(s − t) + h(ξ )(Bs − Bt ) + g(ξ )(Bs − Bt ) +  s  s bu dBu + cu dBu , + t

s

au du t

t

where au = f (X u ) − f (ξ ), bu = g(X u ) − g(ξ ) and cu = h(X u ) − h(ξ ). Then, for each p > 0 and d = a, b, c, there exists some constant C p depending on p and T such that: p

p ˆ ˆ s1 − ds2 | p ] ≤ C p E[|X E[|d s1 − X s2 | ] ≤ C p |s1 − s2 | 2 ,

∀ [s1 , s2 ] ⊂ [t, T ].

Since X s ∈ Ks , we can obtain ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ), which is the desired result.   Next we will construct approximate viable solutions to the G-stochastic differential equation (1). Set: c := max(| f |, |g|, |h|) < +∞. Lemma 3.10 Suppose that the condition (2) in Theorem 3.7 holds. Then, for any  > 0, the set S (t, ξ ) of stochastic processes Y on [t, T ] satisfying Yt = ξ and ˆ 2 (Ys , Ks )] ≤  2 , (i) ∀s ∈ [t, T ], E[d ˆ s1 − Ys2 |2 ] ≤ e|s1 −s2 | (5 + 3σ 2 + 5σ 4 )c2 |s1 − s2 |, (ii) ∀s1 , s2 ∈ [t, T ], E[|Y    ˆ s − ξ − s f (Yu )du − s h(Yu )dBu − s g(Yu )dBu |2 ] ≤  2 (iii) ∀s ∈ [t, T ], E[|Y t t t is not empty. Remark 3.11 Under condition (ii), one can check that f (Ys ), g(Ys ), h(Ys ) ∈ M 2 (0, T ) and the G-stochastic integral is well defined. This is different from the classical case, in which condition (ii) is not necessary (see, for example, [2]). Proof Without loss of generality, assume [t, T ] = [0, 1] and the Lipschitz coefficient with respect to f, g, h is a constant C > 1. Let us fix  > 0 and choose η ∈ (0, ] such that 2 . η2 ≤ 3C 2 (1 + σ 2 + σ 4 ) Denote by A (ξ ) the set of pairs (TY , Y ) where TY ∈ [0, 1] and Y is a stochastic process satisfying Y0 = ξ and

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ˆ 2 (Yt , Kt )] ≤ η2 TY , (i) ∀t ∈ [0, TY ], E[d ˆ t − Ys |2 ] ≤ e|t−s| (5 + 3σ 2 + 5σ 4 )c2 |t − s|, (ii) ∀t, s ∈ [0, TY ], E[|Y (iii) ∀t ∈ [0, TY ],    ˆ E Yt − ξ −

t



t

f (Ys )ds −

0



t

g(Ys )dBs −

0

0

2

 g(Ys )dBs  ≤  2 .

(4)

The set A (ξ ) is not empty: take TY = 0 and Y0 = ξ . It is an inductive set for the order relation (TY 1 , Y 1 )  (TY 2 , Y 2 ) if and only if TY 1 ≤ TY 2 and Y 2 |[0,TY 1 ] = Y 1 . Zorn’s Lemma implies that there exists a maximal element (TY , Y ) ∈ A (ξ ). We will prove by contradiction that TY = 1. Assume that TY < 1. Since KTY and X TY are FTY -measurable, the projection map KTY (YTY ) is also FTY -measurable. ˆ ∈ KTY (YTY (ω)), which is Then, there exists an FTY -measurable selection ρ(ω) said to be a projection of the random variable YTY onto the set KTY . For simplicity, set θ = YTY and thus choose a projection θˆ ∈ KTY (θ ). √ η 1 − TY ρ := >0 4

We take

and denote



(1 − TY )η2 α := min η, η2 + (5 + 3σ 2 + 5σ 4 )c2

 > 0,

ˆ h(θˆ )) belongs to the stochastic which is positive whenever TY < 1. Since ( f (θˆ ), g(θ), contingent set CK (TY , θˆ ), there exists a constant h Y ∈ (0, α] such that for each δ ∈ (0, h Y ], we can find three FTY +δ -measurable random variables a δ , bδ and cδ so that ⎧ 2 ˆ δ2 ⎪ ⎪ (i) E[|a | ] ≤ ρ ⎪ ⎪ ⎪ ˆ δ |2 ] ≤ ρ 2 ⎪ (ii) E[|b ⎪ ⎨ ˆ δ |2 ] ≤ ρ 2 (iii) E[|c ⎪ ⎪ ⎪ ⎪ ˆ δ] = 0 (iv) E[c ⎪ ⎪ ⎪ ⎩ ˆ (v) E[η, cδ ] = 0, ∀ FTY -measurable random variable η and satisfy ˆ TY +δ − Bt ) + f (θ)δ ˆ + h(θ)(B ˆ θˆ + g(θ)(B TY +δ − BTY ) √ δ δ δ + δa + δb + δc ∈ KTY +δ .

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Then, set SY = TY + h Y > TY and define the stochastic process Yˆ on the interval [TY , SY ] by Yˆt := θ + (t − TY ) f (θˆ ) ˆ + (Bt − BTY )g(θ), ˆ + (Bt − BTY )h(θ) and Yˆ = Y on the interval [0, TY ]. Therefore, for each δ ∈ (0, h Y ], √ ˆTY +δ ) − d 2 (YˆTY ) ≤ |θ − θˆ − δa δ − δbδ − δcδ |2 − |θ − θ| ˆ2 ( Y K TY Y +δ √ √ = |δa δ + δbδ + δcδ |2 − 2θ − θˆ , δa δ  − 2θ − θˆ , δbδ  − 2θ − θˆ , δcδ .

2 dK T

Now we shall take expectation to both sides of the above inequality and estimate each term of the second line. First, we can get          √ ˆ |bδ |2 + E ˆ |cδ |2 . Eˆ |δa δ + δbδ + δcδ |2 ≤ 3δ δ Eˆ |a δ |2 + δ E Next, applying Hölder’s inequality yields  1  1   ˆ θ − θˆ , a δ  ≤ E ˆ |θ − θˆ |2 2 E ˆ |a δ |2 2 . E   1 δ ˆ ˆ E θ − θ, √ c  = 0. δ

Note that

Then, by the very choice of ρ and (4) (i), we obtain   ˆ d 2 (YˆT +δ , KT +δ ) E Y Y 1  1 1  1     2 2 ˆ 2 2 Eˆ |bδ |2 2 ≤ Eˆ d 2 (YˆTY , KTY ) + 2δ Eˆ |θ − θˆ |2 Eˆ |a δ |2 + 2δ Eˆ |θ − θ|        + 3δ δ Eˆ |a δ |2 + h Y Eˆ |bδ |2 + Eˆ |cδ |2            2 ˆ d 2 (YˆT , KT ) + δ Eˆ |θ − θ| ˆ |a δ |2 + 5Eˆ |bδ |2 + 3Eˆ |cδ |2 ˆ ≤ E + 5 E Y Y ≤ η2 TY + δ(η2 TY + 13ρ 2 ) ≤ η2 TY + δη2 = η2 (TY + δ). Hence Yˆ satisfies (4) (i) for SY . For each t, s ∈ [TY , SY ], by a similar analysis we conclude that     ˆ |(t − s) f (θˆ ) + (Bt − Bs )h(θ) ˆ + (Bt − Bs )g(θ)| ˆ 2 Eˆ |Yˆt − Yˆs |2 ≤E   ˆ 2 + |(Bt − Bs )g(θ)| ˆ 2 ≤ 3Eˆ |(t − s) f (θˆ )|2 + |(Bt − Bs )h(θ)| ≤ 3(1 + σ 2 + σ 4 )c2 |t − s|,

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and for each t ∈ [TY , SY ], s ∈ [0, TY ],     ˆ |θ − Ys + (t − TY ) f (θ) ˆ |Yˆt − Yˆs |2 ≤E ˆ + (Bt − BTY )h(θˆ ) + (Bt − BTY )g(θ)| ˆ 2 E  ˆ |θ − Ys |2 + 2θ − Ys , (t − TY ) f (θ) ˆ + (Bt − BTY )h(θˆ ) + (Bt − BTY )g(θ) ˆ ≤E  ˆ + (Bt − BTY )h(θ) ˆ + (Bt − BTY )g(θ)| ˆ 2 + |(t − TY ) f (θ)   ˆ |θ − Ys |2 | ≤ (5 + 3σ 2 + 5σ 4 )c2 |t − TY | + (1 + t − TY )E ≤ (5 + 3σ 2 + 5σ 4 )c2 |t − TY | + (1 + t − TY )(5 + 3σ 2 + 5σ 4 )c2 |TY − s|e|TY −s| ≤ (5 + 3σ 2 + 5σ 4 )c2 e|t−s| |t − s|.

Thus, Yˆ satisfies (4) (ii) for SY . For any t ∈ [TY , SY ], we have ˆ + (Bt − BTY )h(θ) ˆ + (Bt − BTY )g(θ)| ˆ 2 |Yˆt − θˆ |2 = |θ − θˆ + (t − TY ) f (θ) 2 ˆ (t − TY ) f (θ) ˆ + (Bt − BTY )h(θ) ˆ = dK (θ ) + 2θ − θ, T Y

ˆ + (Bt − BTY )g(θ) ˆ + (Bt − BTY )g(θ)| ˆ 2. + |(t − TY ) f (θˆ ) + (Bt − BTY )h(θ) Taking expectation to both sides of this equality, we derive that       2 ˆTY ) ≤ (t − TY )Eˆ d 2 (YˆTY ) ( Y Eˆ |Yˆt − θˆ |2 − Eˆ dK K TY T   Y 2 ˆ + (2(t − TY ) + 3(t − TY ) )E | f (θˆ )|2   ˆ 2 + (2(t − TY ) + 3(t − TY )2 )σ 4 Eˆ |h(θ)|   ˆ |g(θ)| ˆ 2 . + 3σ 2 (t − TY )E Since c2 = max(| f |2 , |g|2 , |h|2 ) < +∞, we deduce that   ˆ |Yˆt − θ| ˆ 2 ≤ η2 TY + (t − TY )(η2 TY + (5 + 3σ 2 + 5σ 4 )c2 ) ≤ η2 TY + α(η2 + (5 + 3σ 2 + 5σ 4 )c2 ). E

By the choice of α, it is easy to check that α(η2 + (5 + 3σ 2 + 5σ 4 )c2 ) ≤ (1 − TY )η2 . Therefore,   Eˆ |Yˆt − θˆ |2 ≤ η2 , ∀t ∈ [TY , SY ]. (6) Recalling Lemmas 2.4 and 2.5, we have that  2

 t  t  t    ˆ ˆ ˆ ˆ ˆ E Yt − θ − f (Ys )ds − g(Ys )dBs − g(Ys )dBs  TY TY TY  2

 t  t t    ˆ ˆ ˆ ˆ ˆ ˆ ˆ = E  ( f (θ ) − f (Ys ))ds + (h(θ ) − h(Ys ))dBs + (g(θ ) − g(Ys ))dBs  TY

123

TY

TY

J Theor Probab

 ≤ 3

t

   Eˆ | f (θˆ ) − f (Yˆs )|2 ds + σ 4

TY

 +σ2

t

   ˆ |g(θˆ ) − g(Yˆs )|2 ds E

TY



≤ 3(1 + σ 2 + σ 4 )C 2

t

t

  ˆ − h(Yˆs )|2 ds Eˆ |h(θ)

TY

  Eˆ |θˆ − Yˆs |2 ds

TY 2 2

≤ 3(1 + σ + σ )C η ≤  2 , 2

4

where we have used Eq. (6) in the last inequality. Thus, Yˆ satisfies (4) (iii). Therefore, the maximal solution (TY , Y ) has been extended to the interval [0, SY ] and we obtain the desired contradiction. The proof is complete.   Now we are ready to state the proof of Theorem 3.7. The proof of Theorem 3.7 By Lemma 3.9, (1) ⇒ (3) ⇒ (2). Then, it suffices to prove that (2) ⇒ (1). Recalling Lemma 3.10, we choose a sequence of approximate solutions {X  }>0 , which can be written in the form of  s  s  s f (X u )du + h(X u )dBu + g(X u )dBu + ζs , ∀ s ∈ [t, T ], X s = ξ + t

t

t

ˆ s |2 ] ≤  2 . Then, for any  > 0, we get for some constant C indewhere sup E[|ζ s∈[t,T ]

pendent of ,   ˆ |X s − X s |2 E  2

 s  s  s     ˆ  ( f (X u ) − f (X u ))du + ≤E (h(X ) − h(X ))dB + (g(X ) − g(X )dB + ζ u u u u u u s  t



s

≤C t

t

t

  ˆ |X u − X u |2 du + C 2 . E

Applying Gronwall’s lemma, we can find some constant C T such that   Eˆ |X s − X s |2 ≤ C T  2 . Furthermore, inequalities (4) (i) and (ii) imply that for each s ∈ [t, T ]   Eˆ dKs (X s ) = 0. Consequently, K is viable for Eq. (1), which completes the proof. Suppose that ϕ ∈ Cb,Li p

(Rd ).



 

Then, consider the following fully nonlinear PDE:

∂t u(t, x) − LG u(t, x) = 0, u(0, x) = ϕ(x).

(7)

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J Theor Probab

Here, for ψ ∈ C 1,2 ([0, T ] × Rd ; R) the operator LG is given by LG ψ(t, x) = G(∂x2x ϕ(t, x)g(x), g(x) + 2∂x ψ(t, x), h(x)) +∂x ψ(t, x), f (x), (t, x) ∈ [0, T ] × Rd . 0,x ˆ From Peng [27], u(t, x) := E[ϕ(X t )] is the unique viscosity solution to Eq. (7). Then, the following result is a direct consequence of Theorem 3.7.

Proposition 3.12 Suppose ϕ(K t ) is convex subset of R for each t. If for all t ∈ [0, T ] and ξ ∈ L 2 (Ft ) in Kt ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ), then u(t, x) ∈ ϕ(Kt ) for any 0 ≤ t ≤ T .

4 Stochastic Tangents to Direct and Inverse Images This section is devoted to the study of direct and inverse images for stochastic tangent set, which is introduced by Aubin and Da Prato [1] in the classical case. As an application, we also state the deterministic sufficient and necessary conditions for viability in some particular cases. Theorem 4.1 Let K := (Kt )0≤t≤T be a family of closed subsets of Rd and ϕ ∈ 2 d m Cb,Li p (R ; R ). If ( f (ξ ), h(ξ ), g(ξ )) ∈ TK (t, ξ ), then 1 (ϕ  (ξ ) f (ξ ), ϕ  (ξ )h(ξ ) +  ϕ  (ξ )g(ξ ), g(ξ ), ϕ  (ξ )g(ξ )) ∈ Tϕ(K) (t, ϕ(ξ )). 2 Proof Without loss of generality, assume t = 0 and ξ = x ∈ Rd . Since ( f (x), h(x), g(x)) ∈ TK (0, x), then there exist three bounded adapted stochastic processes as , bs , cs converging to 0 when s → 0 such that for some δ  > 0, ∀h ∈ [0, δ  ], X h = x +

h

( f (x) + as )ds

0



h

+ 0

123





h

(h(x) + bs )dBs + 0

(g(x) + cs )dBs ∈ Kh ,

J Theor Probab

where the stochastic process d = a, b, c satisfies: for any p > 0 there exists some constant C p depending on T and p such that,   p ˆ |ds1 − ds2 | p ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [0, h]. E Note that ϕ(X h ) ∈ ϕ(Kh ). Therefore, applying G-Itô formula (see, for example, [18]), we deduce that 

h

ϕ(X h ) = ϕ(x) +

ϕ  (X s )( f (x) + as )ds +

0



h

[ϕ  (X s )(h(x) + bs )

0

1 +  ϕ  (X s )(g(x) + cs ), g(x) + cs ]dBs 2  h + ϕ  (X s )(g(x) + cs )dBs 0   1 = ϕ(x) + ϕ  (x) f (x)h + ϕ  (x)h(x) +  ϕ  (x)g(x), g(x) Bh 2  + ϕ (x)g(x)Bh  h    (ϕ (X s ) − ϕ  (x)) f (x) + ϕ  (X s )as ds +  +

0 h

   (ϕ (X s ) − ϕ  (x))h(x) + ϕ  (X s )bs dBs

0

 1 h   (ϕ (X s ) − ϕ  (x))g(x), g(x) + 2 0  +ϕ  (X s )cs , cs  + 2ϕ  (X s )cs , g(x) dBs  h    (ϕ (X s ) − ϕ  (x))g(x) + ϕ  (X s )cs dBs ∈ ϕ(Kh ). + 0

Set a¯ s := (ϕ  (X s ) − ϕ  (x)) f (x) + ϕ  (X s )as , b¯s := (ϕ  (X s ) − ϕ  (x))h(x) + ϕ  (X s )bs + 21 (ϕ  (X s ) − ϕ  (x))g(x), g(x) + ϕ  (X s )cs , cs  + 2ϕ  (X s )cs , g(x), c¯s := (ϕ  (X s ) − ϕ  (x))g(x) + ϕ  (X s )cs , one can easily check that the stochastic pro¯ c¯ satisfies: for any p > 0 there exists some constant C p depending on cess d = a, ¯ b, T and p such that, p

ˆ s1 − ds2 | p ] ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [0, h], E[|d which is the desired result.

 

Theorem 4.2 Let K := (Kt )0≤t≤T be a family of closed subsets of Rd and ϕ ∈ 2 d m   + Cb,Li p (R ; R ). If the matrix ϕ (x) has a right inverse denoted by ϕ (x) , which is a bounded Lipschitz function, then ( f (ξ ), h(ξ ), g(ξ )) ∈ Tϕ −1 (ϕ(K)) (t, ξ ).

123

J Theor Probab

if and only if 1 (ϕ  (ξ ) f (ξ ), ϕ  (ξ )h(ξ ) +  ϕ  (ξ )g(ξ ), g(ξ ), ϕ  (ξ )g(ξ )) ∈ Tϕ(K) (t, ϕ(ξ )). 2 Proof Without loss of generality, assume t = 0 and ξ = x ∈ Rd . We only have to prove that from (ϕ  (x) f (x), ϕ  (x)h(x)+ 21 ϕ  (x)g(x), g(x), ϕ  (x)g(x)) ∈ Tϕ(K) (0, ϕ(x)), we infer that ( f (x), h(x), g(x)) ∈ Tϕ −1 (ϕ(K)) (0, x). Since (ϕ  (x) f (x), ϕ  (x)h(x) +  21 ϕ  (x)g(x), g(x), ϕ  (x)g(x)) ∈ Tϕ(K) (t, ϕ(x)), there exist three bounded adapted stochastic processes a¯ s , b¯s , c¯s converging to 0 when s → 0 such that for some δ  > 0 and each h ∈ [0, δ  ], 

h

ϕ(x) +

(ϕ  (x) f (x) + a¯ s )ds +

0

+ b¯s )dBs +



h 0



h

1 (ϕ  (x)h(x) +  ϕ  (x)g(x), g(x) 2

(ϕ  (x)g(x) + c¯s )dBs ∈ ϕ(Kh ),

0

¯ c¯ satisfies: for any p > 0 there exists some where the stochastic process d = a, ¯ b, constant C p depending on T and p such that,   p Eˆ |ds1 − ds2 | p ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [0, h]. Denote a(s, y) : = ϕ  (y)+ (a¯ s − (ϕ  (y) − ϕ  (x)) f (x)),  1  + ¯ b(s, y) : = ϕ (y) bs − (ϕ  (y) − ϕ  (x))h(x) − (ϕ  (y) − ϕ  (x))g(x), g(x) 2  1 −ϕ  (y)cs , cs  −  ϕ  (y)c(s, y), g(x) , 2 c(s, y) = ϕ  (y)+ (c¯s − (ϕ  (y) − ϕ  (x))g(x)), where ϕ  (y)+ is the right inverse of ϕ  (y). Then, the following G-SDE  Xt = x + 0

t

 ( f (x) + a(s, X s ))ds +

+c(s, X s ))dBs

t



t

(h(x) + b(s, X s ))dBs +

0

(g(x)

0

has a unique solution X ∈ M 2 (0, T ). Moreover, for any p > 0 there exists some constant C p depending on T and p such that,   p ˆ |X s1 − X s2 | p ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [0, h]. E

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J Theor Probab

Then, according to G-Itô formula, we have 

 h   ϕ (X s )(h(x) + bs ) ϕ  (X s )( f (x) + as )ds + 0 0 1 +  ϕ  (X s )(g(x) + cs ), g(x) + cs  dBs 2  h + ϕ  (X s )(g(x) + cs )dBs

ϕ(X h ) = ϕ(x) +

h

0

1 = ϕ(x) + ϕ  (x) f (x)h + (ϕ  (x)h(x) +  ϕ  (x)g(x), g(x))Bh 2 + ϕ  (x)g(x)Bh  h  h  h + a¯ s ds + c¯s dBs ∈ ϕ(Kh ). b¯s dBs + 0

0

0

Consequently, it holds that ( f (x), h(x), g(x)) ∈ Tϕ −1 (ϕ(K)) (0, x),  

which completes the proof.

From Theorem 4.2, the deterministic sufficient and necessary conditions for viability can be established when K takes some particular forms. Assume K is independent of t, i.e., Kt = K for each t ∈ [0, T ]. Lemma 4.3 Let K be the unit ball. Then, for |x|2 = 1, ( f (x), h(x), g(x)) ∈ TK (t, x) if and only if x, g(x) = 0, x, f (x) + G(2x, h(x) + g(x), g(x)) ≤ 0. Proof We take ϕ(x) = |x|2 , then it suffices to prove 1 (x, f (x), x, h(x) + g(x), g(x), x, g(x)) ∈ Tϕ(K) (t, 1) 2 ⇔ (x, g(x) = 0, x, f (x) + G(2x, h(x) + g(x), g(x)) ≤ 0). Sufficient If (x, g(x) = 0, x, f (x) + G(2x, h(x) + g(x), g(x)) ≤ 0) and |x| = 1, we can take a = b = c = 0, then 

 s 1 (x, h(x) + g(x), g(x))dBu + x, g(x)dBu 2 t t t  s  s 1 ≤ 1+ (x, h(x)+ g(x), g(x))dBu − G(2x, h(x)+g(x), g(x))du 2 t t ≤ 1.

|x|2 +

s



x, f (x)du +

s

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J Theor Probab

Thus, 1 (x, f (x), x, h(x) + g(x), g(x), x, g(x)) ∈ Tϕ(K) (t, 1). 2 Necessary If (x, f (x), x, h(x) + 21 g(x), g(x), x, g(x)) ∈ Tϕ(K) (t, 1), then there exist three bounded adapted stochastic processes a¯ s , b¯s , c¯s converging to 0 such that for some δ  > 0 and each h ∈ [0, δ  ], 

t+h



t+h

(x, f (x) + a¯ s )ds +

t

+b¯s )dBs +



t t+h

1 (x, h(x) + g(x), g(x) 2

(x, g(x) + c¯s )dBs ≤ 0,

t

¯ c¯ satisfies: for any p > 0 there exists some where the stochastic process d = a, ¯ b, constant C p depending on T and p such that, p

ˆ s1 − ds2 | p ] ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [t, t + h]. E[|d Thus, for each h, 1 x, f (x)h + (x, h(x) + g(x), g(x))(Bt+h − Bt ) + x, g(x)(Bt+h − Bt ) 2   t+h  t+h  t+h (8) ≤− a¯ s ds + c¯s dBs . b¯s dBs + t

t

t

By the standard argument, we conclude for some constant C, which is allowed to change from line to line,   Eˆ 

t

t+h

2

  c¯s dBs  ≤ C

t+h

  ˆ |c¯s |2 ds ≤ Ch 2 , E

t

which implies that   Eˆ 

t+h t

2

 c¯s √ dBs  ≤ Ch. h

Thus, we have  lim

h↓0 t

123

t+h

c¯s √ dBs = 0, q.s. h

J Theor Probab

By a similar analysis, we obtain 1 lim √ h↓0 h

 t

t+h

1 a¯ s ds = 0, lim √ t↓0 h



t+h

b¯s dBs = 0, q.s.

t

Recalling the property of G-Brownian motion, we have that 

   Bt+h − Bt Bt+h − Bt c lim sup = +∞ = 1, c lim inf = −∞ = 1. √ √ h↓0 h h h↓0 Then, there exists Ω0 ⊂ Ω with c(Ω0 ) = 1, such that for every ω ∈ Ω0 , we can find −B B a sequence (rn := rn (ω)) so that lim t+r√nr t = +∞. By Eq. (8), we derive that rn ↓0

x, g(x) lim

rn ↓0

n

Bt+rn − Bt ≤ 0. √ rn

Consequently, x, g(x) ≤ 0. Similarly we can prove x, g(x) ≥ 0, thus x, g(x) = 0. Finally, taking conditional expectation to both sides of Eq. (8) yields that x, f (x) + G(2x, h(x) + g(x), g(x))   t+h  t+h 1ˆ ≤ lim E a¯ s ds + b¯s dBs = 0, h→0 h t t  

which completes the proof.

Lemma 4.4 Let K be the unit sphere. If σ 2 > σ 2 , then for x ∈ K, ( f (x), h(x), g(x)) ∈ TK (t, x) if and only if 1 x, f (x) = 0, x, h(x) + g(x), g(x) = 0, x, g(x) = 0. 2 Proof We take ϕ(x) = |x|2 , then it suffices to prove 1 (x, f (x), x, h(x) + g(x), g(x), x, g(x)) ∈ T1 (t, 1) 2 1 ⇔ x, f (x) = 0, x, h(x) + g(x), g(x) = 0, x, g(x) = 0. 2

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Sufficient If (x, f (x) = 0, x, h(x) + 21 g(x), g(x) = 0, x, g(x) = 0) and |x| = 1, we can take a = b = c = 0, then for each s ∈ [t, T ],  s  s 1 x, f (x)du + (x, h(x) + g(x), g(x))dBu |x|2 + 2 t  st + x, g(x)dBu = 1. t

Thus, 1 (x, f (x), x, h(x) + g(x), g(x), x, g(x)) ∈ T1 (t, 1). 2 Necessary If ((x, f (x), x, h(x) + 21 g(x), g(x), x, g(x)) ∈ T1 (t, 1)), then there exist three bounded adapted stochastic process a¯ s , b¯s , c¯s converging to 0 such that for some δ  > 0 and each h ∈ [0, δ  ], 

t+h

|x|2 +



t+h

(x, f (x) + a¯ s )ds +

t



+ b¯s dBs +



t t+h

 1 x, h(x) + g(x), g(x) 2

(x, g(x) + c¯s )dBs = 1,

t

¯ c¯ satisfies: for any p > 0 there exists some where the stochastic processes d = a, ¯ b, constant C p such that, p

ˆ s1 − ds2 | p ] ≤ C p |s2 − s1 | 2 , ∀ [s1 , s2 ] ⊂ [t, t + h]. E[|d Since |x|2 = 1, by a similar argument as in Lemma 4.3, we can obtain x, g(x) = 0, x, f (x) + G(2x, h(x) + g(x), g(x)) = 0, −x, f (x) + G(−2x, h(x) − g(x), g(x)) = 0. Since for each a ∈ R, it holds that G(a) + G(−a) =

σ 2 −σ 2 2 |a|,

then we get

1 x, h(x) + g(x), g(x) = 0. 2 Consequently, x, f (x) = 0, which completes the proof.

 

The following result is a direct consequence of Theorems 3.7, 4.2 and Lemma 4.4.

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Corollary 4.5 Let K be the unit sphere, then K is viable for G-SDE (1) if and only if 1 x, f (x) = 0, x, h(x) + g(x), g(x) = 0, x, g(x) = 0, ∀x ∈ Rd . 2 Remark 4.6 In [20], we have got a necessary and sufficient condition for comparison of G-SDEs based on the viscosity solution argument (see also [16] and [28]). With the help of stochastic tangent approach, [23] had also got a similar result for 1-dimensional stochastic differential equation driven by fractional Brownian motion. However, we cannot use the stochastic tangent approach to obtain the similar result for G-SDEs since the G-Itô integral is not a pathwise integral, which is crucial for the one in [23]. Acknowledgements The authors would like to thank the editor and the anonymous referee for their helpful discussions and suggestions.

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