Joint time-state generalized semiconcavity of the value
function of a jump diffusion optimal control problem
Ermal Feleqi
Dipartimento di Matematica
Università degli Studi di Padova
via Trieste, 63
I-35121 Padova, Italy
e-mail: [email protected]
September 1, 2014
Abstract
We prove generalized semiconcavity results, jointly in time and state variables,
for the value function of a stochastic optimal control problem where the evolution
of the state variable is described by a general stochastic differential equation (SDE)
of jump type. Assuming that terms comprising the SDE are C 1 -smooth, and that
running and terminal costs are semiconcave in generalized sense, we show that the
value function is also semiconcave in generalized sense, estimating the semiconcavity
modulus of the value function in terms of smoothness and generalized semiconcavity
moduli of data. Of course, these translate into analogous regularity results about
(viscosity) solutions of integro-differential Hamilton-Jacobi-Bellman equations due to
their controllistic interpretation. This paper may be seen as a natural continuation
of an earlier one [5], where where we dealt only with generalized semiconcavity in
state variable.
Keywords: generalized semiconcavity, value function, optimal control, partial integrodifferential equations, Hammilton-Jacobi-Bellman equations, jump diffusions
1
Introduction
In this article we continue our work initiated in [5] on establishing generalized semiconcavity results about the value function in a finite horizon optimal control problem of jumpdiffusions. While in [5] we dealt with the the problem of obtaining generalized semiconcavity estimates for the value function in the state variable, uniformly in time, here we prove
generalized semiconcavity results for the value function holding in time and state variables
jointly.
1
Under appropriate assumptions on the data-which include those made in this paper–the
value function can be interpreted as the unique viscosity solution with at most polynomial
growth of a partial integro-differential equation of Hamilton-Jacobi-Bellman type

n
∂u
1


+
inf
b(t, x, α) · ∇u + tr σ(t, x, α)σ t (t, x, α)D2 u



∂t α∈A
2

Z




+
u(·, · + H(t, x, z, α)) − u − ∇u · H(t, x, z, α) ν(dz)
(1.1)
ZE
o




u(·, · + K(t, x, z, α)) − u ν(dz) = 0 in [0, T ) × IRd ,
+



Ec


 u(T, ·) = ψ in IRd ;
where T > 0, ν is a Lévy measure on IRd (d ∈ N), IRd \ {0} = E ∪ E c where E ∪ {0} is some
open bounded set, A is some metric space–to be interpreted as the set of controls–and b, σ,
H, K, L, ψ are given maps as in (2.2) below. Before stating results we recall the following
Definition 1.1. ([4]) Given an upper semicontinuous nondecreasing function ω : IR+ →
IR+ such that ω(0+) = limρ→0+ ω(ρ) = 0 (such a function is called a semiconcavity modulus), we say that a function u : K → IR, where K is some subset of some normed space
(X, k · kX ), is an ω-semiconcave function iff
λu(x1 ) + (1 − λ)u(x2 ) − u(λx1 + (1 − λ)x2 ) ≤ λ(1 − λ)kx1 − x2 kX ω(kx1 − x2 kX )
for all x1 , x2 ∈ K such that the segment [x1 , x2 ] ⊂ K and 0 ≤ λ ≤ 1. A function u is called
ω-semiconvex iff −u is ω-semiconcave. We say that u is of class C 1,ω or C 1,ω -regular iff it
is both ω-semiconcave and ω-semiconvex1 . Finally, a vector-valued map u : K → Y , where
Y is another normed space, is said to be of class C 1,ω or C 1,ω -regular iff each “component
of u”, that is, iff < u, y ∗ > is of class C 1,ω for all2 y ∗ ∈ Y ∗ (in other words iff the inequality
above holds for the left-hand side being replaced by its own Y -norm).
Our main result goes roughly as follows. Let b, σ, H, K be, in order, of class C 1,ω1 , C 1,ω2 ,
C , C 1,ω4 , respectively, and L, ψ be ω5 and ω6 -semiconcave, jointly in time and state
variables, uniformly in control and jump variables, where all the ωi ’s are given semiconcavity moduli. Assume also that all these maps are bounded, globally Lipschitz continuous
in time and state variables, uniformly in control and jump variables, and that the Lévy
measure is finite. Then for all δ ∈]0, T ], the unique viscosity solution u of (1.1) with polynomial growth is ω-semicocave for some modulus ω that can be expressed in terms of the
given moduli ωi for i = 1, . . . , 6. (For precise results see Theorem 2.4 and its Corollaries 2.8, 2.9, 2.11.) One cannot hope to prove ω-semiconcavity of u on all of [0, T ] × IRd for
it would imply the Lipschitz continuity of u on bounded subsets of [0, T ] × IRd , which is
known to be not true in general as shown by the simple Example 3.1 in [2].
1,ω3
1
In finite-dimensional normed spaces, such a definition is justified, e.g., by [4, Theorem 3.3.7, p. 60].
Certain constants that appear in its proof are universal, that is, independent of dimension, and this fact
hints to its validity or possibility of extension to a large class of infinite-dimensional normed spaces.
2 ∗
Y stands for the topological dual of Y .
2
The results of the present paper are (to the best of our knowledge) new for two reasons: First, because the results are given for general possibly degenerate jump SDEs (in
the literature one usually either considers continuous diffusions, or either jump- diffusion
with some kind of ellipticity hypothesis); second, because the semiconcavity moduli considered are rather general (in contrast to the usual linear moduli, corresponding to classical
semiconcavity).
The case of linear moduli, that is, classical semiconcavity, has already been treated
in [6]. Same difficulty as in [6] of having to restrict
measures (ν(IRd \ {0}) <
R to finite Lévy
∞) persists here. The more general case where IRd \{0} 1 ∧ kzk2 ν(dz) < ∞ is still open.
The proof is based on interpreting the said solution of (1.1) as the value function
of a stochastic optimal control problem for jump-diffusion processes, that is, processes
which are solutions of appropriate stochastic differential equations of jump type driven by
Brownian motions and Poisson random measures independent of each other. (abbr. SDEs)
see, e.g., [10] and references therein. Further, we rely on the method of affine time changes
for Brownian motions as in [2, 3] and for Poisson random measures as in [6]. While the
corresponing change of variable formula for Weinner integrals is rather easy, for stochastic
integrals with respect to Poisson random measures, the formula is more involved and
requires a change of probability on the underlying sample space via the so called Kulik’s
transformation; see [6] for more details and references. Other tools are Burkholder type
inequalities as stated for example in [9], and of course Gronwall’s inequality.
The paper is organized as follows. Main results (Theorem 2.4 and its corollaries) are
stated in the next section. The proof of technical lemmas is postponed to the Appendix
(Sect. 3) in order to ensure a better readability of the paper.
Notation. Usually, we denote by Cu , Lu , ωu a bound on the sup-norm, a Lipschitz
constant, and a semiconcavity modulus, respectively, of a map u.
Acknowledgment. I wish to thank Piermarco Cannarsa and Martino Bardi for useful
suggestions and improvements.paper.
2
The optimal control of jump diffusions
Let T > 0 be a fixed time horizon, A be a metric space–to be interpreted as the set of
controls–and ν a finite Lévy measure on Z = IRd \ {0}, Z = E ∪ E c , where E ∪ {0} is
some open bounded subset of IRd . For all s ∈ [0, T ] we denote by Rs the collection of the
following entities
R = Ω, F, G = (Ft )s≤t≤T , P, W (·), N = N (dtdz) ,
(2.1)
that satisfy the following conditions:
•
Ω, F, F = (Ft )s≤t≤T , P
is a complete filtered probability space such that the filtration F satisfies the usual
hypotheses (that is, F is right continuous and each sub-σ-algebra Ft , for 0 ≤ t ≤ T ,
is complete with respect to the probability measure P;
3
• W = W (·) is a standard m-dimensional (m ∈ N) F-adapted Brownian motion on
(Ω, F, P);
• N = N (dtdz) is a F-adapted Poisson random measure on IR+ × Z and on probability
space (Ω, F, P) with intensity measure ν on Z, and with associated compensator
Ñ = Ñ (dtdz) = N (dtdz) − dtν(dz);
• W and N are independent of each-other and moreover have increments that are
independent of the filtration F, that is, W (t2 )−W (t1 ), N (t2 )−N (t1 ) are independent
of Ft1 for all s ≤ t1 ≤ t2 ≤ T .
S
We set also R = 0≤s≤T Rs .
Let

b : [0, T ] × Rd × A → IRd ,
σ : [0, T ] × Rd × A → IRd×m ,


(2.2)
H : [0, T ] × Rd × E × A → IRd ,
K : [0, T ] × Rd × E c × A → IRd ,


L : [0, T ] × IRd × A → IR,
ψ : IRd → IR
be measurable maps that satisfy the following assumptions. Let p ≥ 2. Then for some
constants Ci ≥ 0, Li ≥ 0, and some regularity moduli ωi for i = 1, . . . , 6
(A1) (Boundedness)
kb(r, x, α)k ≤ C1 ,
kσ(r, x, α)k ≤ C2 ,
kH(r, x, z, α)k ≤ C3 ,
c
kK(r, x, z , α)k ≤ C4 ,
|L(r, x, α)| ≤ C5 ,
|ψ(x)| ≤ C6
(A2) (Lipschitz continuity)
kb(r1 , x1 , α) − b(r2 , x2 , α)k ≤ L1 (|r1 − r2 | + kx1 − x2 k),
kσ(r, x1 , α) − σ(r, x2 , α)k ≤ L2 kx1 − x2 k,
Z
ZE
Ec
kH(r, x1 , z, α) − H(r, x2 , z, α)kp ν(dz) ≤ C p Lp3 (|r1 − r2 | + kx1 − x2 k)p ,
kK(r, x1 , z, α) − K(r, x2 , z, α)kp ν(dz) ≤ C p Lp6 (|r1 − r2 | + kx1 − x2 k)p ,
|L(r1 , x1 , α) − L(r2 , x2 , α)| ≤ L5 (|r1 − r2 | + kx1 − x2 k),
|ψ(x1 ) − ψ(x2 )| ≤ L6 kx1 − x2 k,
(A3) (C 1,ω -regular dynamics)
kλb(r1 , x1 , α) + (1 − λ)b(r2 , x2 , α) − b rλ , xλ , α k
≤ λ(1 − λ)(|r1 − r2 | + kx1 − x2 k)ω1 (|r1 − r2 | + kx1 − x2 k),
kλσ(r1 , x1 , α) + (1 − λ)σ(r2 , x2 , α) − σ rλ , xλ , α k
≤ λ(1 − λ)(|r1 − r2 | + kx1 − x2 )kω2 (|r1 − r2 | + kx1 − x2 k),
4
Z
kλH(r1 , x1 , z, α) + (1 − λ)H(r2 , x1 , z, α) − H rλ , xλ , z, α kp ν(dz)
E
≤ (λ(1 − λ)(|r1 − r2 | + kx1 − x2 k)ω3 (|r1 − r2 | + kx1 − x2 k))p ,
kλK(r1 , x1 , z c , α) + (1 − λ)K(r2 , x1 , z c , α) − K rλ , xλ , z c , α kp ν(dz c )
Z
Ec
≤ (λ(1 − λ)(|r1 − r2 | + kx1 − x2 k)ω4 (|r1 − r2 | + kx1 − x2 k))p
(A4) (ω-semiconcave costs)
λL(r1 , x1 , α) + (1 − λ)L(r, x2 , α) − L (rλ , xλ , α)
≤ λ(1 − λ)(|r1 − r2 | + kx1 − x2 k)ω5 (|r1 − r2 | + kx1 − x2 k),
λψ(x1 ) + (1 − λ)ψ(x2 ) − ψ(xλ ) ≤ λ(1 − λ)kx1 − x2 kω6 (kx1 − x2 k)
for all r, r1 , r2 ∈ [0, T ], x, x1 , x2 ∈ IRd , 0 ≤ λ ≤ 1, α ∈ A, z ∈ E, z c ∈ E, where
rλ = λr1 + (1 − λ)r2 , xλ = λx1 + (1 − λ)x2 . Since p ≥ 2 and ν(Z) < ∞, then it follows
that estimates for H and K hold also for p = 2.
The larger the p the more restrictive these assumptions become, so we aim at proving
results for p ≥ 2 as small as possible.
For any s ∈ [0, T ], R ∈ Rs as in (2.1), we consider the following optimal control
problem:
(admissible controls) we take as set of admissible controls AR (s, T ) the set of Rpredictable3 processes α(·) : [0, T ] → A;
(controlled system) for any x0 ∈ IRd , α(·) ∈ AR (s, T ) we consider the stochastic differential equation of jump type
Z t
Z t
0
σ r, x(r−), α(r) dW (r)
b r, x(r−), α(r) dr +
x(t) = x +
s
(2.3)
Z tZ
Z tZ s
H r, x(r−), z, α(r) Ñ (drdz) +
K r, x(r−), z, α(r) N (drdz) ;
+
s
s
E
Ec
(cost functionals) for any x0 ∈ IRd , α(·) ∈ AR (s, T ), if x(·) is the solution4 to (2.3)
Z T
0
JR s, x , α(·) = E
L t, x(t), α(t) dt + ψ x(T ) ;
(2.4)
s
(value function) the value function VR is given by
VR (s, x0 ) =
inf
α(·)∈AR (s,T )
JR s, x0 , α(·) j
(2.5)
we consider also
V (s, x0 ) = inf VR (s, x0 ).
R∈Rs
3
4
That is, predictable with respect to the filtration F of R.
Under assumptions made precise below, (2.3) is indeed uniquely solvable.
5
(2.6)
Under these assumptions VR (s, ·) = V (s, ·) for all s ∈ [0, T ], R ∈ Rs , and V is actually
the unique viscosity solutions of (1.1) with polynomial growth [10], [11]. Actually, as in [6]
it can be proved that V is Lipschitz continuous on [0, T − δ] × IRd for any δ ∈]0, T ] (under
Assumptions (A1), (A2) for p = 2).
Let δ ∈]0, T ]. In order to prove generalized semiconcavity estimates we take s1 , s2 ∈
[0, T − δ], x01 , x02 ∈ IRd , λ ∈ [0, 1], and set sλ = λs1 + (1 − λ)s2 , x0λ = λx01 + (1 − λ)x02 . Let
R ∈ Rsλ , α(·) ∈ AR (sλ , T ), and denote by τ1 , τ2 the affine “time changes” that transform,
[s1 , T ], respectively [s2 , T ], into [sλ , T ], that is,
τi : [si , T ] → [sλ , T ],
τi (t) = sλ +
T − sλ
(t − si ) ∀t ∈ [si , T ], i = 1, 2,
T − si
(2.7)
(which have derivatives τ̇i = (T − sλ )/(T − si )). We take
Ri = (Ω, F, Qi , Fi , τi (N ), τi (N )).
(2.8)
as in the proof of Lemma 2.1 bellow, and it is easy to see that Ri ∈ Rsi , α1 (·) = α(·) ◦ τ1 ∈
AR1 (s1 , T ), α1 (·) = α(·) ◦ τi ∈ AR1 (s1 , T )
Denoting by xi (·) the solutions of equation (2.3) for R = Ri , α(·) = αi (·) and initial
conditions s = si , x0 = x0i , for i = 1, 2, respectively; and by xλ (·) the solution of (2.3)
for the previously fixed R ∈ Rsσ , α(·) ∈ AR (sλ , T ), initial conditions s = sλ , x0 = x0λ ,
setting x̃1 (·) = x1 (·) ◦ τ1−1 , x̃2 (·) = x2 (·) ◦ τ2−1 x̃λ (·) = λx̃1 (·) + (1 − λ)x̃2 (·), we obtain, by
Burkholder inequalities and change of variable formulas stochastic integrals with respect
to affine time changes–see the detailed proof in the Appendix–the following estimates:
Lemma 2.1.
kx̃1 (t) − x̃2 (t) − x01 + x02 kpC1R ≤
4 Z t
X
p1
φi (τ̇1 )fi τ1−1 (r), x̃1 (r), α(r) − φi (τ̇2 )fi τ2−1 (r), x̃2 (r), α(r) 0 dr
i=1
kx̃λ (t) −
xλ (t)kpB2R
(2.9)
Ci,R
sλ
3 Z t
X
≤
λφi (τ̇1 )fi τ1−1 (r), x̃1 (r), α(r)
i=1
sσ
p2
+ (1 − λ)φi (τ̇2 )fi τ2−1 (r), x̃2 (r), α(r) − f r, xλ (r), α(r) 0 dr
Bi,R
Z t
p2
−1
−1
+
λφ
(
τ̇
)f
τ
(r),
x̃
(r),
α(r)
+
(1
−
λ)λφ
(
τ̇
)f
τ
(r),
x̃
(r),
α(r)
0 dr5
4 1 4 1
1
4 1 4 2
2
B4,R
sσ
(2.10)
√
for p1 = p and p2 = 2, φ1 (τ̇ ) = 1/τ̇ , φ2 (τ̇ ) = 1/ τ̇ , φ3 (τ̇ ) = 1, φ4 (τ̇ ) = 1 − 1/τ̇ for
all τ̇ ∈ IR, f1 = c b̄, f2 = c σ̄, f3 = c H̄, f4 = c K̄ for some c ≥ 0 that depends only
0
on T, d, p, ν(Z), BR = L2 (Ω, F, P; IRd ), CR = Lp (Ω, F, P; IRd ), Bi,R
= L2 (Ω, F, P; Yi ),
6
for i = 1, 2, 3, 4, with Y1 , Y2 , Y3 , Y4 equal, in order, to IRd , IRd×m , L2 (E, E, ν; IRd ),
0
L2 (E c , E c , ν; IRd ), Ci,R
= Lp (Ω, F, P; Yi ) for i = 1, 2, 3, 4, with Y1 , Y2 , Y3 , Y4 equal, in
order, to IRd , IRd×m , Lp (E, E, ν; IRd ), Lp (E c , E c , ν; IRd ) with E, E c the Borel σ-algebras on
E and E c , respectively, and where
b̄ : [0, T ] × L2 (Ω, F, P; IRd ) × AR → L2 (Ω, F, P; IRd ),
σ̄ : [0, T ] × L2 (Ω, F, P; IRd ) × AR → L2 (Ω, F, P; IRd×m ),
H̄ : [0, T ] × [0, T ] × L2 (Ω, F, P; IRd ) × AR → L2 (Ω × E, F ⊗ E, P ⊗ ν(dz); IRd ),
(2.11)
K̄ : [0, T ] × L2 (Ω, F, P; IRd ) × AR → L2 (Ω × E c , F ⊗ E c , P ⊗ ν(dz); IRd ),
are defined by setting
b̄(t, x, α)(ω) = b(t, x(ω), α(ω)),
H̄(t, x, α)(ω, z) = H(t, x(ω), z, α(ω)),
σ̄(t, x, α)(ω) = σ(t, x(ω), α(ω)),
K̄(t, x, α)(ω, z c ) = K(t, x(ω), z c , α(ω)),
for all 0 ≤ t ≤ T , ω ∈ Ω, x ∈ L2 (Ω, F, P; IRd ), z ∈ E, z c ∈ E c , α ∈ AR , where AR is the
set of A-valued random variables on (Ω, F, P).
For a better readability of the paper, the proof of the lemmas stated in this section is
postponed to the Appendix.
Lemma 2.2 (Lipschitz estimates in terms of initial conditions). Assume that
kfi (r, x, α)kCi0 ≤ Cfi
(2.12)
for all R ∈ R, r ∈ [0, T ], x ∈ CR , α ∈ A, i = 1, . . . , 4; and
0
kfi (r1 , x1 , α) − fi (r2 , x2 , α)kCi,R
≤ Lfi ,K (|r1 − r2 | + kx1 − x2 kC )
for all R ∈ R, x1 , x2 ∈ CR , r1 , r2 ∈ [0, T ], α ∈ AR , i = 1, . . . , 4.
Then
kx̃1 (t) − x̃2 (t)kC ≤ LΦ |s1 − s2 | + kx01 − x02 k .
(2.13)
(2.14)
for some constant LΦ ≥ 0 that depends only on T, p1 , p2 , δ, Cfi , Lfi , for i = 1, . . . , 4.
Lemma 2.3 (C 1,ω -estimates in terms of initial conditions). Let maps fi for i = 1, . . . , 4,
in addition to (2.12), (2.13), satisfy also
0
kfi (r1 , x1 , α) − f (r2 , x2 , α)kBi,R
≤ Lfi (|r1 − r2 | + kx1 − x2 kBR )
(2.15)
for all R ∈ R, r1 , r2 ∈ [0, T ], x1 , x2 ∈ BR , α ∈ AR , i = 1, 2, 3, 4, for given constants
Cfi ≥ 0, Lfi ≥ 0. Assume that for some given moduli ωfi 6 ,
λfi (r1 , x1 ) + (1 − λ)f (r2 , x2 ) − f (rλ , xλ ) 0
B
i,R
6
Independent of R ∈ R.
7
≤ λ(1 − λ)(|r1 − r2 | + kx1 − x2 kCR )ωfi (|r1 − r2 | + kx1 − x2 kCR ), (2.16)
for all r1 , r2 ∈ [0, T ], R ∈ R, x1 , x2 ∈ CR, 0 ≤ λ ≤ 1,, α ∈ AR , i = 1, . . . , 4, where
(rλ , xλ ) = λr1 + (1 − λ)r2 , λx1 + (1 − λ)x2 .
Then
kx̃λ (t) − xλ (t)kB ≤ λ(1 − λ)(|s1 − s2 | + kx01 − x02 kC )ωΦ (|s1 − s2 | + kx01 − x02 kC ),
(2.17)
with x̃λ (·) = λx̃1 (·) + (1 − λ)x̃2 (·), where
ωΦ (ρ) =
4
X
c0i ωfi (ci ρ) + c5 ρ,
ρ ≥ 0,
(2.18)
i=1
with constants ci , c0i ≥ 0 for i = 1, 2, 3, 4, c5 ≥ 0 depending only on p1 , p2 , T, δ, Cfi , Lfi ,.
We notice that the cost JR (s, x0 , α(·)) in (2.4) can be written as
Z T
0
JR s, x , α(·) =
L t, x(t), α(t) dt + ψ x(T ) 7 ,
(2.19)
s
where
L : [s, T ] × CR × AR → IR,
ψ : CR → IR
(2.20)
are defined by setting
L(t, x, α) = E[L(t, x, α)],
ψ(x) = E[ψ(x)]
(2.21)
(2.22)
for all x ∈ CR , α ∈ AR .
Theorem 2.4 (ω-semiconcave value function). Let maps fi for i = 1, 2, 3, 4 satisfy (2.12),
(2.13), (2.15), (2.16). Assume that maps L, ψ are Lipschitz continuous in time and state
variables jointly, uniformly in control variables, that is,
|L(r1 , x1 , α) − L(r2 , x2 , α)| ≤ LL (|r1 − r2 | + kx1 − x2 kCR ),
(2.23)
|ψ(x1 ) − ψ(x2 )| ≤ Lψ (|r1 − r2 | + kx1 − x2 kCR )
(2.24)
for all R ∈ R, r1 , r2 ∈ [0, T ], x1 , x2 ∈ CR , α ∈ AR ; and semiconcave in generalized
sense in time and state variables jointly, uniformly in control variables, that is, for some
semiconcavity moduli ωL , ωψ
λL(r1 , x1 , α) + (1 − λ)L(r2 , x2 , α) − L rλ , xλ , α
(2.25)
≤ λ(1 − λ)(|r1 − r2 | + kx1 − x2 kCR )ωL (|r1 − r2 | + kx1 − x2 kCR )
λψ(x1 ) + (1 − λ)ψ(x2 ) − ψ xλ ≤ λ(1 − λ)kx1 − x2 kCR ωψ (kx1 − x2 kCR )
(2.26)
for all 0 ≤ r1 , r2 ≤ T , R ∈ R, x1 , x2 ∈ CR , α ∈ AR , where rλ = λr1 + (1 − λ)r2 ,
xλ = λx1 + (1 − λ)x2 .
8
Then, for all δ ∈]0, T ], V is ω-semiconcave on [0, T − δ] × IRd , where
ω(ρ) =
4
X
c0i ωfi (ci ρ) + c05 ωL (c5 ρ) + c06 ωψ (c6 ρ) + c7 ρ
∀ρ ≥ 0
(2.27)
i=1
for suitable constants ci , c0i ≥ 0, for i = 1, . . . , 6, c7 ≥ 0 that depend only on T , δ, p1 , p2 ,
Cfi , Lfi , LL , Lψ .
We need the following simple technical lemma which can be checked by straightforward
computation, see e.g. [2], hence its proof is omitted.
Lemma 2.5. For any 0 < δ ≤ T there exists Cδ > 0 such that
1
1 1
1 −1
−1
|τ1 (r) − τ2 (r)| + − + √ − √ ≤ Cδ |s1 − s2 |,
τ̇1 τ̇2
τ̇1
τ̇2
1 1 1
λ 1 − √ + (1 − λ) 1 − √ ≤ λ(1 − λ)|s1 − s2 |,
2δ
τ̇1
τ̇
2
1
1 λ 1 − √1
≤ 2 λ(1 − λ)|s1 − s2 |
+ (1 − λ) 1 − √
2δ
τ̇1
τ̇2
(2.28)
(2.29)
(2.30)
for all 0 ≤ si ≤ T − δ, i = 1, 2, 0 ≤ λ ≤ 1, s ≤ r ≤ T , where sλ = λs1 + (1 − λ)s2 .
Moreover,
λτ1−1 (r) + (1 − λ)τ2−1 (r) = r,
1
1
= −(1 − λ) 1 −
= λ(1 − λ)(s1 − s2 ).
λ 1−
τ̇1
τ̇2
(2.31)
(2.32)
Proof of Theorem 2.4. We have
λJR1 s1 , x01 , α1 (·) + (1 − λ)JR2 s2 , x02 , α2 (·) − JR sλ , x0λ , α(·) =
Z T
Z T
Z T
λ
L t, x1 (t), α1 (t) dt + (1 − λ)
L t, x2 (t), α2 (t) dt −
L t, xλ (t), α(t) dt
s1
s2
sλ
+ λψ x1 (T ) + (1 − λ)ψ x2 (T ) − ψ xλ (T ) .
In the first two integrals we apply the change of variables τ1 , τ2 , respectively (defined by
(2.7)), notice that αi (·) ◦ τi−1 = α(·) for i = 1, 2; we have
λJR1 s1 , x01 , α1 (·) + (1 − λ)JR2 s2 , x02 , α2 (·) − JR sλ , x0λ , α(·)
Z T
1−λ
λ
=
L τ1−1 (r), x̃1 (r), α(r) +
L τ2−1 (r), x̃2 (r), α(r) − L t, xλ (t), α(t) dt
τ̇1
τ̇2
sλ
+ λψ x1 (T ) + (1 − λ)ψ x2 (T ) − ψ xλ (T )
Z T
1−λ
λ
=
L τ1−1 (r), x̃1 (r), α(r) +
L τ2−1 (r), x̃2 (r), α(r) − L t, x̃λ (t), α(t) dt
τ̇1
τ̇2
sλ
9
Z
T
L t, x̃λ (t), α(t) − L t, xλ (t), α(t) dt
sλ
+ λψ x1 (T ) + (1 − λ)ψ x2 (T ) − ψ x̃λ (T ) + ψ x̃λ (T ) − ψ xλ (T ) ,
+
where x̃λ (·) = λx1 (·) + (1 − λ)x2 (·). Further, noting that λ/τ̇1 + (1 − λ)/τ̇2 = 1 as follows
by differentiating (2.31), we can write
λJR1 s1 , x01 , α1 (·) + (1 − λ)JR2 s2 , x02 , α2 (·) − JR sλ , x0λ , α(·)
Z T
λL τ1−1 (r), x̃1 (r), α(r) + (1 − λ)L τ2−1 (r), x̃2 (r), α(r) − L t, x̃λ (t), α(t) dt
=
sλ
Z
T
+
sλ
1
L τ1−1 (r), x̃1 (r), α(r) − (1 − λ)L τ2−1 (r), x̃2 (r), α(r) dt
(1 − λ) 1 −
τ̇2
Z T
+
L t, x̃λ (t), α(t) − L t, xλ (t), α(t) dt
sλ
+ λψ x1 (T ) + (1 − λ)ψ x2 (T ) − ψ x̃λ (T ) + ψ x̃λ (T ) − ψ xλ (T ) ,
By the ω-semiconcavity of L, ψ (that is, (2.25), (2.26)), and by the Lipschitz continuity
of L, ψ (that is, (2.23), (2.24)), we obtain
λJR1 s1 , x01 , α1 (·) + (1 − λ)JR2 s2 , x02 , α2 (·) − JR sλ , x0λ , α(·)
Z T
≤ λ(1 − λ)
|τ1−1 (t) − τ2−1 (t)| + kx1 (t) − x2 (t)kC
sλ
× ωL,K̄ |τ1−1 (t) − τ2−1 (t)| + kx1 (t) − x2 (t)kC dt
Z T
1 −1
−1
|τ1 (t) − τ2 (t)| + kx1 (t) − x2 (t)kC + LL
kx̃λ (t) − xλ (t)kC dt
+ LL (1 − λ) 1 −
τ̇2
sλ
+ λ(1 − λ)kx1 (T ) − x2 (T )kC ωψ,K̄ (kx1 (T ) − x2 (T )kC ) + Lψ kx̃λ (T ) − xλ (T )kC .
Now, using estimates (2.14), (2.17)) and (2.28), (2.32), we obtain
λJR1 s1 , x01 , α1 (·) + (1 − λ)JR2 s2 , x02 , α2 (·) − JR sλ , x0λ , α(·)
≤ λ(1 − λ)(|s1 − s2 | + kx01 − x02 kC0 )ω(|s1 − s2 | + kx01 − x02 kC0 )
for an ω as in (2.27). The fact that R ∈ Rsλ and α(·) ∈ AR (sλ , T ) are arbitrary implies
the claimed result about V .
Using the following lemmas we can obtain several corollaries from Theorem 2.4.
Lemma 2.6. Let X, Y be normed spaces, (Ω, F, P) a probability space, C, B 0 normed
spaces of X-valued and Y -valued random variables, respectively, and
f : [0, T ] × C → B 0
g : [0, T ] × X → Y,
10
maps such that
f (r, x)(ω) = g(r, x(ω))
for all r ∈ [0, T ], x ∈ C, ω ∈ Ω. Let g : X → Y be of class C 1,ωg for some modulus ωg .
Then f : C → B 0 is of class C 1,ωf with ωf = ωg if either one of the following conditions
hold:
• (power moduli) ωg (ρ) = k ρα for some k ≥ 0, 0 < α(≤ 1), and Lp (Ω; Y ) ,→ B 0 ,
C ,→ Lp(1+α) (Ω; X) for some 1 ≤ p ≤ ∞;
q
• (moduli with concavity properties) γg (ρ) = ρβ ωg2 (ρ) , where 0 ≤ β ≤ 2, 1 ≤ q ≤ ∞,
r−1 + q −1 = 1, is concave, and L2 (Ω; Y ) ,→ B 0 , C ,→ L(2−β)r (Ω; X), (2 − β)r ≥ 18 .
Lemma 2.7. Let X be a normed space, (Ω, F, P) a probability space, C a normed space of
X-valued random variables, and
L : [0, T ] × X → IR,
L : [0, T ] × C → IR
maps such that
L(r, x)(ω) = g(r, x(ω))
for all r ∈ [0, T ], x ∈ C, ω ∈ Ω L() is ωL -semiconcave with ωL = ωL if either one of the
following happens:
• (power moduli) ωL (ρ) = k ρα for k ≥ 0, 0 < α(≤ 1) and C ,→ L1+α (Ω; X);
q
• (general moduli) γL (ρ) = ρβ ωL (ρ) , where 0 ≤ β ≤ 1, 1 ≤ q, r ≤ ∞, q −1 + r−1 = 1,
is concave and C ,→ L(1−β)r (Ω; X), (1 − β)r ≥ 1.
Corollary 2.8 (power moduli). Let assumptions (A1)-(A4) be in force and let ωi (ρ) =
ki ραi for all ρ ≥ 0, i = 1, . . . , 6, for given 0 < αi (≤ 1), ki ≥ 0. Assume that p ≥
2(1 + max{α1 , α2 , α3 , α4 }), p ≥ 1 + max{α5 , α6 }. Let also 0 < δ ≤ T . Then, the value
function V is ω-semiconcave on [0, T − δ] × IRd for some modulus ω of the form
ω(ρ) =
6
X
ci ρ α i + c7 ρ
∀ρ ≥ 0,
(2.33)
i=1
for constants ci ≥ 0 for i = 1, . . . , 7 that depend only on d, T, δ, ν(Z), p, Ci , Li for i =
1, . . . , 6.
Proof. Clearly maps fi for i = 1, 2, 3, 4 defined in Lemma 2.1 satisfy (2.12), (2.13),
(2.15) for constants Cfi ≥ 0, Lfi ≥ 0 that depend only on Ci , Li , T , d, p, ν(Z). Similarly,
we can easily check that maps L, ψ satisfy (2.23), (2.24) for constants LL ≥ 0, Lψ ≥ 0
depending only on Ci , Li for i = 5, 6, T , d, p. By the first bullet in Lemma 2.6 and
Lemma 2.7, we deduce that fi satisfy (2.16) with ωfi = c oi for i = 1, 2, 3, 4, where c ≥ 0
is as in Lemma 2.1, and L, ψ satisfy (2.25), (2.26) with ωoL = ω5 , ωψ = ω8 . Therefore
Theorem 2.4 applies and we obtain the conclusion.
8
This implies C ,→ L1 (Ω; X)
11
Corollary 2.9 (moduli with concavity properties).
Let assumptions (A1)-(A4) be in
q
force, and assume that maps γi (ρ) = ρβi ωi2 (ρ) i for i = 1, . . . , 4, and some 0 ≤ βi ≤ 2,
q
γi (ρ) = ρβi ωi (ρ) i for i = 5, 6, and some 0 ≤ βi ≤ 2, where 1 ≤ qi ≤ ∞, ri−1 + qi−1 = 1
for all i = 1, . . . 6, are concave. Assume also that 1 ≤ (2 − βi )ri ≤ p for i = 1, . . . , 4,
1 ≤ (1 − βi )ri ≤ p for i = 5, 6. Then, for all δ ∈]0, T ], the value function V is ωsemiconcave on [0, T − δ] × IRd for some modulus ω of the form
ω(ρ) =
6
X
c0i ωi (ci ρ) + c7 ρ
∀ρ ≥ 0
(2.34)
i=1
for constants ci , c0i ≥ 0 for i = 1, . . . , 6, c7 ≥ 0 that depend only on d, T, δ, ν(Z), p, Ci , Li
for i = 1, . . . , 6.
It should be now rather straightforward to state results under the assumption that some
of the moduli ωi are of power type while the others satisfy suitable concavity properties
(as stated in Lemma 2.6 and Lemma 2.7).
It is always possible to choose the moduli ωi concave, and by growth assumptions
contained in (A1)-(A4) it is also possible to these moduli ωi bounded too. This remark
can be used to derive ω-semiconcavity results by mean of the following lemma.
Lemma 2.10 (bounded concave moduli). Fix q, r ∈ [1, ∞] such that 1/q + 1/r = 1.
• Let maps f , g be as in Lemma 2.6, and assume that ωgq is concave for some q > 0,
and ωg is bounded by some constant k ≥ 0. Then f is of class C 1,ωf with ωf =
q/(2q)
k 1−q/(2q) ωg
if L2 (Ω; Y ) ,→ B 0 , C ,→ L2r (Ω; X).
• Let functions L and L be as in Lemma 2.7, and assume that ωLq is concave for some
q > 0, and ωL is bounded by some constant k ≥ 0. Then L is ωL -semiconcave with
q/q
ωL = k 1−q/q ωg if C ,→ Lr (Ω; X).
Then Theorem 2.4 combined with this lemma has the following corollary.
Corollary 2.11 (concave bounded moduli with concavity). Let assumptions (A1)-(A4)
be in force, and assume that, for suitable 1 < qi , ri < ∞, q i > 0 such that 1/qi + 1/ri = 1,
q
for i = 1, . . . , 6, 2ri ≤ p for i = 1, . . . , 4, r5 , r6 ≤ p, maps ωi i are concave and bounded.
Then, for all δ ∈]0, T ], the value function V is ω-semiconcave on [0, T − δ] × IRd for some
modulus ω of the form
ω(ρ) =
4
X
i=1
q /(2q )
c0i ωi i i (ci ρ)
+
6
X
q /(qi )
c0i ωi i
(ci ρ) + c7 ρ
∀ρ ≥ 0
i=5
for constants ci , c0i ≥ 0 for i = 1, . . . , 6, c7 ≥ 0 that depend only on d, T, δ, ν(Z), p, Ci , Li , qi , qi
and upper bounds of ωi for i = 1, . . . , 6.
Further corollaries of the previous result can be obtained by noticing that in assumptions (A1)-(A4) the smoothness and semiconcavity moduli ωi , for i = 1, . . . , 6, can always
be chosen to be concave and bounded.
12
3
Appendix
Proof of Lemma 2.1.
Fact 1. (Burkholder-Davis-Gundy inequalities [9]) For any 2 ≤ p < ∞ there exist c0p , cp > 0
such that
" Z
p Z t
p/2 #
t
0
E kσ(r)k2 dr
,
(3.1)
σ(r)dW (r) ≤ cp E
s
s
and
p Z t Z
Z t Z
p
E H(r, z)Ñ (drdz) ≤ cp E
kH(r, z)k drν(dz)
s
F
s
F
"Z Z
p/2 #
t
+ c00p E
kH(r, z)k2 drν(dz)
(3.2)
s
F
p
d×m
for all predictable
processes
σ
∈
L
[s,
T
]
×
Ω,
dt
⊗
P;
I
R
, H ∈ Lp [s, T ] × F × Ω, dt ⊗
d
ν ⊗ P; IR , where F is any measurable subset of Z; see e.g., [1, Theroem 4.4.22, p.263
and Theorem 4.4.23, p.265 ], or [9, Section 2.5]. Actually, for p = 2, by the L2 -isometry of
stochastic integrals, we can take cp = c0p = 1 and c00p = 0 above, and these inequalities are
in fact equalities.
By first compensating, that is, using N = Ñ + dtν(dz), and then inequality (3.2) with
F = E c , and Hölder’s inequality, we obtain
" Z Z
p #
Z t Z
t
p
kK(r, z)k drν(dz)
(3.3)
E K(r, z)N (drdz)
≤ cE
c
s
kzk>δ
s
E
for come c > 0 that depends only p, T, ν(E c ); recall that ν(E c ) < ∞ which is essential
above. Clearly (3.1) implies
p Z t
Z t
p
kσ(r)k dr ,
(3.4)
E σ(r)dW (r) ≤ c E
s
s
and since ν is a finite measure, (3.2) implies
p Z t Z
Z t Z
p
E H(r, z)Ñ (drdz)
kH(r, z)k drν(dz)
≤ cE
s
F
s
(3.5)
F
for some constant c ≥ 0 depending only on p, T , ν(Z), d.
Fact 2. If, for i = 1, 2, we define
F ◦ τi−1 = {Fτi−1 (t0 ) }sλ ≤t0 ≤T
1
τi (W )(t) = √ W (τi (t)) − W (sλ ) ,
τi
13
(3.6)
si ≤ t ≤ T,
then τi (W ) is a (IRm -valued) Brownian motion on Ω, F, F ◦ τi−1 , P . Moreover, we have
Z
τi−1 (t)
Z
t
σ(r)τi (W )(d r) =
si
sλ
1
√ σ(τi−1 (r0 ))W (d r0 )
τ̇i
(3.7)
for all predictable processes σ ∈ L2 si , τi−1 (t) × Ω, dr ⊗ P; IRm×d ), t ∈ [si , T ].
Next, we use a transformation of a Poisson random measure with respect to affine time
changes which is called Kulik’s transformation. The reader interested for more information
on this transformation is referred to papers [8, 7], or even [6] for a quick and very readable
introduction. We define
τi (N )([si , t] × ∆) = N ([sλ , τi (t)] × ∆),
si ≤ t ≤ T, ∆ ∈ Z.
Fact 3. For each i = 1, 2, τi (N ) is a Poisson random measure on the filtered probability
space (Ω, F, F ◦ τi−1 , Qi ) where Qi is another probability on (Ω, F), which, is absolutely
continuous with respect to P and has Radon-Nikodym density
T − sλ
dQi
= exp − ln
τi (N )([si , T ] × Z) + (si − sλ )ν(Z) ,
dP
T − si
while the time changed filtration F◦τi−1 is defined by (3.6). Moreover, we have the following
change of variables formulas
Z
τi−1 (t)
si
Z
Qi
H (r, z) τ^
i (N )(dr dz)
E
Z t Z
Z tZ
1
−1 0
0
P
e (dr dz) + 1 −
H τi (r ), z N
=
H τi−1 (r0 ), z dr0 ν(dz), (3.8)
τ̇i
sλ E
sλ E
τi−1 (t)
Z
Z tZ
Z
Qi
K (r, z) τi (N )(dr dz)
si
=
Ec
sλ
Ec
K τi−1 (r0 ), z N (dr0 dz)P
(3.9)
for all predictable
processes H ∈ L2 si , τi−1 (t) × E × Ω, dr ⊗ ν ⊗ Qi ; IRd , and K ∈
L2 si , τi−1 (t) × E c × Ω, dt ⊗ ν ⊗ P; IRd , t ∈ [si , T ]. We have put probability measures
Qi or P on some of stochastic integrals above in order to emphasize that the probability
measure in which the integral is being carried out.
We have, by (3.7), (3.8), (3.9)
Z
t
x̃i (t) =
sλ
Z t
0
1
1
−1 0
0
0
√ σ τi−1 (r0 ), x̃i (r0 ), α(r0 ) W (dr)P
b τi (r ), x̃i (r ), α(r ) dr +
τ̇i
τ̇i
sλ
Z tZ
0
P
+
H τi−1 (r0 ), x̃i (r0 ), z, α(r0 ) τ^
i (N )(dr dz)
s
E
λ Z t Z
1
+ 1−
H τi−1 (r0 ), x̃i (r0 ), z, α(r0 ) dr0 ν(dz)
τ̇i
sλ E
14
Z tZ
+
sλ
Ec
K τi−1 (r0 ), x̃i (r0 ), z, α(r0 ) τi (N )(dr0 dz)P . (3.10)
We have also used the fact that τi (W ) is also a Brownian motion with respect to probability
Qi and
Z
τi−1 (t)
σ
Q
τi−1 (r), x̃i (r) i
Z
τi−1 (t)
τi (W )(d r) =
si
P
σ τi−1 (r), x̃i (r) τi (W )(d r)
si
because Qi is absolutely continuous with respect to P and dQi /dP is bounded almost surely
with respect to P (and hence Qi ). We have also used the following simple change of variable
formula for ordinary (deterministic) integrals:
Z
τi−1 (t)
Z
t
b(r, x(r)) dr =
sλ
si
1
b τi−1 (r0 ), x̃i (r0 ) dr0 .
τ̇i
Subtracting the two identities in (3.10) for i = 1, 2, taking norms in CR = Lp (Ω, F, P; IRd ),
using moment inequalities (3.4), (3.5), we obtain
Z
t
p i
h 1
1
≤c
E b (r, x̃1 (r), α(r)) − b (r, x̃2 (r), α(r)) kx̃1 (t) − x̃2 (t) − +
τ̇1
τ̇2
sλ
Z t h
i
1
1
E √ σ(r, x̃1 (r), α(r)) − √ σ(r, x̃2 (r), α(r)) dr
+c
τ̇
τ̇
1
2
sλ
p
i
h Z 1
1
H (r, x̃1 (r), z, α(r)) − 1 −
H (r, x̃2 (r), z, α(r)) ν(dz) dr
+ cE
1−
τ̇1
τ̇2
E
i
hZ
kK(r, x̃1 (r), z, α(r)) − K(r, x̃2 (r), z, α(r))kp ν(dz) dr
+ cE
x01
x02 kpCR
Ec
for some c ≥ 0 that depends only on d, T , p and ν(Z). Hence, recalling the definition of
0
maps b̄, σ̄, H̄, K̄, maps fi , φi , and normed spaces Ci,R
, for i = 1, . . . , 4, the right-hand
side of inequality above equals we have proved compatibility estimate (2.9). The other
compatibly estimate (2.10) is proved similarly.
0
0
, Ci,R
Now for brevity we write simply B, C, Bi0 , Ci0 instead of, respectively, BR , CR , Bi,r
for the normed spaces introduced in Lemma 2.1.
Proof of Lemma 2.2. Let L = max{Lfi : i = 1, 2, 3, 4} ≥ 0 be a Lipschitz constant
for all fi for i = 1, 2, 2, 4 on [0, T ] × C. Since
φi (τ̇1 )fi (τ1−1 (r), x̃1 (r)) − φi (τ̇2 )fi (τ2−1 (r), x̃2 (r)) 0
C
i −1
−1
≤ |φi (τ̇1 ) − φi (τ̇2 )| fi (τ1 (r), x̃1 (r)) C 0 + φi (τ̇2 ) fi (τ1 (r), x̃1 (r)) − fi (τ2−1 (r), x̃1 (r))C 0
i
i
for i = 1, 2, 3, 4, where maps φi are defined in Lemma 2.1, then by Lemma 2.5, (2.12),
(2.15), we deduce
15
φi (τ̇1 )fi (τ1−1 (r), x̃1 (r)) − φi (τ̇2 )fi (τ1−1 (r), x̃1 (r)) 0 ≤ Cδ (|s1 − s2 | + kx̃1 (r) − x̃2 (r)kC )
C
i
for some constant Cδ ≥ 0 that depends on δ, p1 , T, L, which in turn, by (2.9), (2.12) yields
Z t
p1
p1
0
0 p1
p1
kx̃1 (t) − x̃2 (t)kC dr .
kx̃1 (t) − x̃2 (t)kC ≤ Cδ |s1 − s2 | + kx1 − x2 kC +
σλ
for another constant Cδ ≥ 0 that depends on δ, p1 , T, Cfi , Lfi . This last estimate yields the
claimed estimate (2.14) via Gronwall’s inequality.
Proof of Lemma 2.3. We can write
λφi (τ̇1 )fi (τ1−1 (r), x̃1 (r)) + (1 − λ)φi (τ̇2 )fi (τ2−1 (r), x̃2 (r)) − f (r, xλ (r))
=λfi (τ1−1 (r), x̃1 (r)) + (1 − λ)fi (τ2−1 (r), x̃2 (r)) − fi (r, x̃λ (r))
+ (λφi (τ̇1 ) + (1 − λ)φi (τ̇2 ) − 1) fi (τ1−1 (r), x̃1 (r))
fi (τ1−1 (r), x̃1 (r))
+ (1 − λ) (1 − φi (τ̇2 ))
+ fi (r, x̃λ (r)) − fi (r, xλ (r)),
−
(3.11)
fi (τ2−1 (r), x̃2 (r))
i = 1, . . . , 4, where maps φi are defined as in Lemma 2.1. By Lemma 2.5 we deduce that
|λφi (τ̇1 ) + (1 − λ)φi (τ̇2 ) − 1| ≤
1
λ(1 − λ)|s1 − s2 |2
2
δ
(3.12)
for i = 1, . . . , 4. Indeed, for i = 3 the left-hand side of (3.12) obviously vanishes; by
differentiating (2.31) one discovers that it also vanishes for i = 1; for i = 4 it vanishes too
as follows from the case for i = 1, and the case i = 2 is covered by (2.30).
Using identity (3.11) and assumptions (2.16), (2.12), the fact that IRd ,→ CR , C ,→ B,
Ci0 ,→ Bi0 with embedding constants equal to 1, and estimates (2.14), (3.12), we deduce
λφi (τ̇1 )fi (τ1−1 (r), x̃1 (r)) + (1 − λ)φi (τ̇2 )fi (τ2−1 (r), x̃2 (r)) − f (r, xλ (r)) 0
Bi
≤ λ(1 − λ) (|τ1−1 (r) − τ2−1 (r)| + kx̃1 (r) − x̃2 (r)kC )
× ωfi (|τ1−1 (r) − τ2−1 (r)| + kx̃1 (r) − x̃2 (r)kC )
+ C|s1 − s2 |(|s1 − s2 | + kx̃1 (r) − x̃2 (r)kC ) + Ckx̃λ (r) − xλ (r)kB
≤ Cλ(1 − λ) (|s1 − s2 | + kx01 − x02 kC )ωfi (|s1 − s2 | + kx01 − x02 kC )
+ |s1 − s2 |(|s1 − s2 | + kx01 − x02 kC ) + Ckx̃λ (r) − xλ (r)kB .
for constants C ≥ 0 that depend on δ, p1 , T, Cfi , Lfi .
Thus, by the -compatibility estimate (2.10),
kx̃λ (t) − xλ (t)kpB2 ≤ Cλp2 (1 − λ)p2
4 X
(|s1 − s2 | + kx01 − x02 kC )ωfi (|s1 − s2 | + kx01 − x02 kC )
i=1
16
+ |s1 − s2 |(|s1 − s2 | +
kx01
−
p 2
x02 kC )
Z
t
+C
kx̃λ (r) − xλ (r)kpB2 dr,
sλ
for some constant C ≥ 0 depending only on T, δ, p1 , p2 , Cfi , Lfi , and therefore, by Gronwall’s
inequality we derive estimate (2.17) with ωΦ as in (2.18).
Proof of Lemma 2.6. The proof of the result under the first set of assumptions is
trivial. As to the second, by Hölder’s and Jensen’s inequalities and assumptions, we can
estimate as follows
kλf (r1 ,x1 ) + (1 − λ)f (x2 ) − f (rλ , xλ )k2B 0
≤ E[kλg(r1 , x1 ) + (1 − λ)g(r2 , x2 ) − g(rλ , xλ )k2Y ]
≤ λ2 (1 − λ)2 E[(|r1 − r2 | + kx1 − x2 k2X )ωg2 (|r1 − r2 | + kx1 − x2 kX )]
1
1
≤ λ2 (1 − λ)2 E[(|r1 − r2 | + kx1 − x2 kX )r(2−β) ] r (E[γ(|r1 − r2 | + kx1 − x2 kX )]) q
2−β
1
(2−β)r
r(2−β)
2
2
]
≤ λ (1 − λ) |r1 − r2 | + E[kx1 − x2 kX
1
× (γ (|r1 − r2 | + E[kx1 − x2 kX ])) q
1
≤ λ2 (1 − λ)2 (|r1 − r2 | + kx1 − x2 kC )2−β (γ(|r1 − r2 | + kx1 − x2 kC )) q
= λ2 (1 − λ)2 (|r1 − r2 | + kx1 − x2 kC )2 ωg2 (|r1 − r2 | + kx1 − x2 kC ).
for all r1 , r2 ∈ [0, T ], x1 , x2 ∈ C, 0 ≤ λ ≤ 1, where rλ = λr1 + (1 − λ)r2 , xλ = λx1 + (1 − λ)x2
Thus, the proof is over.
Lemma 2.7 and Lemma 2.10 are proved similarly.
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