Nonlinear Differ. Equ. Appl.

17 (2010), 715–728
c 2010 Springer Basel AG
1021-9722/10/060715-14
published online June 5, 2010 Nonlinear Differential Equations
DOI 10.1007/s00030-010-0078-x and Applications NoDEA

Lipschitz continuity and semiconcavity

properties of the value function
of a stochastic control problem
Rainer Buckdahn, Piermarco Cannarsa and Marc Quincampoix

Abstract. We investigate the Cauchy problem for a nonlinear parabolic

partial differential equation of Hamilton–Jacobi–Bellman type and prove
some regularity results, such as Lipschitz continuity and semiconcavity,
for its unique viscosity solution. Our method is based on the possibility
of representing such a solution as the value function of the associated sto-
chastic optimal control problem. The main feature of our result is the fact
that the solution is shown to be jointly regular in space and time with-
out any strong ellipticity assumption on the Hamilton–Jacobi–Bellman
equation.
Mathematics Subject Classification (2000). 93E20, 35D40, 35K55, 35K65.

1. Introduction
We consider the following partial differential equation
⎧
⎨−vt + H(t, x, v, − ∂v , − ∂ 22 v ) = 0, (t, x) ∈ [0, T ) × Rn ,
∂x ∂ x
(1.1)
⎩v(T, x) = h(x), x ∈ Rn ,

where
H(t, x, u, p, A)
 
1
= sup tr(Aσ(t, x, u)σ T (t, x, u)) + p, b(t, x, u) − f (t, x, u) (1.2)
u∈U 2
for any (t, x, p, A) ∈ [0, T ] × Rn × Rn × L(Rn , Rn ). It is well-known that,
under continuity and boundedness assumptions on b, σ, f and h, equation (1.1)
has a unique continuous viscosity solution of at most polynomial growth v :
[0, T ] × Rn → R (see, for instance, [2]). The main goal of the present article is
to prove the local Lipschitz continuity and semiconcavity of the solution. We
recall the following:
716 R. Buckdahn et al. NoDEA

Definition 1.1 ([1]) Let A ⊂ Rn be an open set and let u : A → Rn . We say

that f is semiconcave (with linear modulus) in A if there exists a constant
C ≥ 0 such that, for all λ ∈ [0, 1],
λu(x1 ) + (1 − λ)u(x0 ) − 2u(λx1 + [1 − λ]x0 ) ≤ Cλ(1 − λ)|x1 − x0 |2
for all x1 , x0 ∈ A such that the segment [x0 , x1 ] is contained in A. Any con-
stant C satisfying the above inequality is called a semiconcavity constant for
u in A.
In the literature most of the regularity results for problem (1.1) are con-
cerned with uniformly elliptic or parabolic equations [5,7]. For such equations
solutions are expected to be smooth, even without imposing the structural
assumption (1.2). In this paper, on the contrary, attention will be focussed on
degenerate parabolic equations, including the fully degenerate case of σ = 0
where our problem reduces to a first order equation. Therefore, one cannot
expect solutions to be smooth and, as is well-known, semiconcavity is the
maximal regularity that can be obtained, see. e.g., [1]. Moreover, due to the
presence of a possibly degenerate diffusion in the stochastic differential equa-
tion associated with (1.1), such a regularity fails to hold at time T , as we shall
explain later on (see Example 3.1).
The key idea of our approach is to interpret the solution of the PDE as
the value function of a suitable stochastic control problem (see e.g. [3,8]), and
then use fine properties of the associated control system to derive Lipschitz
continuity and semiconcavity. Moreover, we obtain Lispschitz and semiconcav-
ity constants for the solution that are independent of the specific stochastic
control problem that was chosen to represent the solution of (1.1).
The main novelty of this article lies in the fact that the solution is shown
to possess the same regularity with respect to both variables x and t (for
t = T ). Indeed, for our problem, the Lipschitz continuity or semiconcavity of
the solution just with respect to space variables has been obtained:
(a) in [3] and [8], using the representation of the solution of (1.1) as the value
function of the associated stochastic control problem, and
(b) in [4], by comparison arguments for viscosity solutions of Hamilton-Jacobi
equations.
On the other hand, the above results only ensure space regularity, as we have
already mentioned. In this paper, on the contrary, we are interested in joint
space–time regularity which is an essentially new property in the present con-
text because, in the stochastic case, time and space variables often play a
different role. Indeed, although our starting point is the same representation
formula for the solution of (1.1) as in (a), then we need to introduce a suit-
able change of time in the underlying Brownian motion and derive nontrivial
estimates for the resulting stochastic integrals in order to prove our result.
Let us now explain how our paper is organized. In Sect. 2, we give our
assumptions and describe the optimal stochastic control problem associated
to (1.1). Section 3 is devoted to the study of the Lipschitz regularity of the
solution while Sect. 4 investigates semiconcavity.
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 717

2. Stochastic optimal control problem

Let T > 0 be a fixed time horizon, let U be a metric space— the control space—
and let (Ω, F, P) be a complete probability space. For any (s, y) ∈ [0, T ) × Rn ,
consider the state equation

dx(t) = b(t, x(t), u(t))dt + σ(t, x(t), u(t))dW (t), t ∈ [s, T ],
(2.3)
x(s) = y
governed by an m-dimensional Brownian motion W and a control process u.
For any s ∈ [0, T ), we define U w [s, T ] to be the set of all pairs ν =
(W (·), u(·)) satisfying the following:
(i) {W (t)}t≥s is an m-dimensional standard Brownian motion on (Ω, F, P)
over [s, T ] (with W (s) = 0 almost surely) for which we denote by Fts =
σ{W (r) : s ≤ r ≤ t} the associated filtration, augmented by all the
P−null sets of F.
(ii) u ∈ L0{F s }t≥s (0, T ; U ) is an {Fts }t≥s − adapted process on (Ω, F, P).
t

Sometimes, we simply write u(·) ∈ U w [s, T ] instead of (W (·), u(·)) ∈ U w [s, T ].

Under standard assumptions, that will be explicitly recalled below, for
any y ∈ Rn Eq. (2.3) admits a unique solution x(·). Then, for any (s, y) ∈
[0, T ] × Rn and u ∈ U w [s, T ] one can compute the cost

T
J(s, y; u) = E f (t, x(t), u(t))dt + h(x(T )) (2.4)
s

and the corresponding value function

V (s, y) = inf J(s, y; u) ∀(s, y) ∈ [0, T ] × Rn . (2.5)
u(·)∈U w [s,T ]

We underline that the mathematical expectation E in (2.4) is taken with

respect to probability measure P.
Let us now introduce some assumptions.
(S1) U is a complete separable metric space.
(S2) Maps b : [0, T ] × Rn × U → Rn , σ : [0, T ] × Rn × U → Rn×m , f :
[0, T ] × Rn × U → R and h : Rn → R are uniformly continuous and
there exists a positive constant L such that the function
φ(t, x, u) = (b(t, x, u), σ(t, x, u), f (t, x, u), h(x))
satisfies
|φ(t, x, u) − φ(t, x̂, u)| ≤ L|x − x̂| ∀t ∈ [0, T ], x, x̂ ∈ Rn , u ∈ U,
|φ(t, x, u)| ≤ L ∀(t, x) ∈ [0, T ] × Rn , u ∈ U.
Under assumptions (S1) and (S2), for any (s, y) ∈ [0, T ] × Rn and u(·) ∈
U w [s, T ], (2.3) admits a unique solution x(·) and the cost functional (2.4) is
well-defined. We also recall that V is the unique continuous viscosity solution
of (1.1) with at most polynomial growth (see [2,3,8]).
718 R. Buckdahn et al. NoDEA

Remark 2.1. In fact, (2.3) is well-posed even when functions b, σ and f have
linear growth with respect to x—instead of being bounded as assumed in (S2).
The results of our paper remain valid under linear growth conditions but, for
simplicity, here we prefer to restrict the analysis to the bounded case.

3. Lipschitz continuity
The fact that, under assumption (S1) and (S2), the solution to (1.1) ought
to be jointly Lipschitz continuous in (t, x) is somewhat expected in the litera-
ture, see, e.g., [4,6]. On the other hand, some care is needed to give a precise
Lipschitz regularity result for V , as our next example shows.
Example 3.1. We consider the following one dimensional example m = n = 1
without control where b = 0, σ = 1, f = 0 are constant √ and h(x) = |x|, for
any x ∈ R. Then it can be easily checked that V (s, 0) = T − s and so V fails
to be Lipschitz with respect to s at s = T .
The theorem below completes the analysis of the Lipschitz regularity
of V .
Theorem 3.2. Under assumptions (S1) and (S2) the value function V is
Lipschitz continuous in [0, T − δ) × Rn , ∀δ > 0.
We give a complete proof of the above theorem using a technique that is
similar—yet simpler—to the one that will be needed later for semiconcavity.

Proof. We remark that K, during the proof, denotes a generic constant, that
may differ at different places.
Let us fix δ > 0, and let (s1 , y1 ) and (s0 , y0 ) ∈ [0, T ) × Rn be such that
min{T − s1 , T − s0 } > δ. (3.6)
We have to show that
V (s1 , y1 ) − V (s0 , y0 ) ≤ C(|s1 − s0 | + |y1 − y0 |) (3.7)
for some constant C ≥ 0. For any  > 0, there exists u0 (·) ∈ U w [s0 , T ] such
that J(s0 , y0 ; u0 ) < V (s0 , y0 ) + . Let x0 (·) be the solution of

dx0 (t) = b(t, x0 (t), u0 (t))dt + σ(t, x0 (t), u0 (t))dW (t), t ∈ [s0 , T ],
(3.8)
x0 (s0 ) = y0 .
Now, let us consider the change of time
T (s0 − s1 ) + (T − s0 )t
τ : [s1 , T ] → [s0 , T ], τ (t) = (3.9)
T − s1
Observe that τ̇ (t) is constant, more precisely
T − s0
τ̇ (t) = . (3.10)
T − s1
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 719

Define u1 (t) = u0 (τ (t)), and denote by x1 (t) the solution of


dx1 (t) = b(t, x1 (t), u1 (t))dt + σ(t, x1 (t), u1 (t))dW1 (t), t ∈ [s1 , T ],
(3.11)
x1 (s1 ) = y1 ,

where W1 (t) = 1/τ̇ W (τ (t)). Obviously, (Ω, F, P, W1 (·), u1 (·)) ∈ U w [s1 , T ].

Thus,
 t  t
x1 (t) = y1 + b(s, x1 (s), u1 (s))ds + σ(s, x1 (s), u1 (s))dW1 (s)
s1 s1
 t  t
= y1 + b(s, x1 (s), u0 (τ (s)))ds+ 1/τ̇ σ(s, x1 (s), u0 (τ (s)))dW (τ (s))
s1 s1

Next, the very definition of V yields

V (s1 , y1 ) − V (s0 , y0 ) −  ≤ J(s1 , y1 , u1 ) − J(s0 , y0 , u0 )
 T  T 
=E f (t, x1 (t), u1 (t))dt − f (t, x0 (t), u0 (t))dt + h(x1 (T ))−h(x0 (T ))
s1 s0

By the change of variable r = τ (t) in the first integral above we obtain, setting
x̃1 (r) = x1 (τ −1 (r)),
V (s1 , y1 ) − V (s0 , y0 ) − 

T
1
≤E f (τ −1 (r), x̃1 (r), u0 (r)) − f (r, x0 (r), u0 (r)) dr
s0 τ̇
+ E{h(x1 (T )) − h(x0 (T ))}
Recalling that f (t, x, u) and h(x) are bounded and Lipschitz, we have
V (s1 , y1 ) − V (s0 , y0 ) − 

T
1
≤E f (τ −1 (r), x̃1 (r), u0 (r)) − f (τ −1 (r), x̃1 (r), u0 (r)) dr
s0 τ̇

T 
+E f (τ −1 (r), x̃1 (r), u0 (r)) − f (r, x0 (r), u0 (r)) dr
s0

+ E{h(x1 (T )) − h(x0 (T ))}

   
T  1 
≤KE  −1
1 − τ̇  + |τ (r) − r| + |x̃1 (r) − x0 (r)| dr
s0

+ K E{|x1 (T ) − x0 (T )|}
So, the conclusion (3.7) will follow from the estimates
 
 
1 − 1  ≤ K|s1 − s0 | (3.12)
 τ̇ 
|τ −1 (r) − r|dr ≤ K|s1 − s0 | (3.13)

T
E |x̃1 (r) − x0 (r)|dr ≤ K(|y1 − y0 | + |s1 − s0 |) (3.14)
s0
720 R. Buckdahn et al. NoDEA

and
E{|x1 (T ) − x0 (T )|} ≤ K(|y1 − y0 | + |s1 − s0 |). (3.15)

Proof of (3.12). In view of (3.10) and (3.6) we have

     
     
1 − 1  = 1 − T − s1  =  s1 − s0  ≤ 1 |s1 − s0 | ≤ K|s1 − s0 | ,
 τ̇   T − s0   T − s0  δ
which in turn yields (3.12).

Proof of (3.13). By the definition of τ (t) and assumption (3.6) we obtain

   
 (T − s1 )r − (s0 − s1 )T   (s1 − s0 )(T − r) 
−1
|τ (r) − r| =   
− r =   ≤ K|s1 − s0 |
(T − s0 ) (T − s0 ) 

Proof of (3.14). Recalling the definition of x̃1 (r) we conclude that

E{|x̃1 (r) − x0 (r)|2 } = E{|x1 (τ −1 (r)) − x0 (r)|2 }
  τ −1 (r)  r


≤ E y 1 − y 0 + b(s, x1 (s), u0 (τ (s)))ds − b(s, x0 (s), u0 (s))ds
 s1 s0
 τ −1 (r)
+ 1/τ̇ σ(s, x1 (s), u0 (τ (s)))dW (τ (s))
s1
 2
r 
− σ(s, x0 (s), u0 (s))dW (s)
s0
  r
 
= E y1 − y0 + τ̇ −1 b(τ −1 (ρ), x̃1 (ρ), u0 (ρ)) − b(ρ, x0 (ρ), u0 (ρ)) dρ
s0
 r  2

+ τ̇ −1/2
σ(τ (ρ), x̃1 (ρ), u0 (ρ)) − σ(ρ, x0 (ρ), u0 (ρ)) dW (ρ)
−1
s0
  r 
≤ K(|y1 − y0 |2 + K E |τ̇ −1 − 1||b(τ −1 (ρ), x̃1 (ρ), u0 (ρ))|
s0
2

+ |b(τ −1 (ρ), x̃1 (ρ), u0 (ρ)) − b(ρ, x0 (ρ), u0 (ρ))| dρ
 r
+K E |τ̇ −1/2 − 1|2 |σ(τ −1 (ρ), x̃1 (ρ), u0 (ρ))|2 dρ
s
 0r
+K E |σ(τ −1 (ρ), x̃1 (ρ), u0 (ρ)) − σ(ρ, x0 (ρ), u0 (ρ))|2 dρ
s0

Since b(t, x, u) and σ(t, x, u) are bounded Lipschitz functions,

E{|x̃1 (r) − x0 (r)|2 }
 r 
2 2 2

≤ K|y1 − y0 | + K E |s1 − s0 | + |x̃1 (ρ) − x̃0 (ρ)| dρ
s0
 r
2 2
≤ K(|y1 − y0 | + |s1 − s0 | ) + K E{|x̃1 (ρ) − x̃0 (ρ)|2 }dρ
s0
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 721

Finally, Gronwall’s Lemma yields

E{|x̃1 (r) − x0 (r)|2 } ≤ K(|y1 − y0 |2 + |s1 − s0 |2 ). (3.16)
From the latter estimate, (3.14) easily follows.
proof[Proof of (3.15)] Owing to (3.16),
E{|x1 (T ) − x0 (T )|} = E{|x̃1 (T ) − x0 (T )|} ≤ K(|y1 − y0 | + |s1 − s0 |).
This yields (3.15) and completes the proof.

4. Semiconcavity
Theorem 4.1. In addition to (S1) and (S2), assume:
(S3)(i) h is semiconcave in Rn and f (·, ·, u) is semiconcave in [0, T ] × Rn
uniformly in u ∈ U ;
(S3)(ii) b and σ are continuously differentiable in (t, x) with derivatives bt,x
and σt,x satisfying, for some constant ≥ 0,
|bt,x (t, x, u) − bt,x (s, y, u)| + |σt,x (t, x, u) − σt,x (s, y, u)|
≤ (|x − y| + |t − s|) (4.17)
n
for all t, s ∈ [0, T ], all x, y ∈ R , and all u ∈ U .
Then the value function V is semiconcave in [0, T − δ) × Rn for every δ > 0
Proof. Once again K will denote a generic constant, that may differ from line
to line. Let δ > 0 be fixed and let (s1 , y1 ), (s0 , y0 ) ∈ [0, T ) × Rn be such that
min{T − s1 , T − s0 } > δ.
We only have to show that, for some constant C ≥ 0 and all λ ∈ [0, 1],
λV (s1 , y1 ) + (1 − λ)V (s0 , y0 ) − V (sλ , yλ )
 
≤ Cλ(1 − λ) |s1 − s0 |2 + |y1 − y0 |2 (4.18)
.
where (sλ , yλ ) = λ(s1 , y1 ) + (1 − λ)(s0 , y0 ). Let λ ∈ [0, 1] be fixed.
For any  > 0, there exists a control uλ, (·) = uλ (·) ∈ U w [s0 , T ] such that
J(sλ , yλ , uλ ) < V (sλ , yλ ) + .
Let xλ (·) be the solution of the Cauchy problem

dxλ (t) = b(t, xλ (t), uλ (t))dt + σ(t, xλ (t), uλ (t))dW (t), t ∈ [sλ , T ],
(4.19)
xλ (sλ ) = yλ .
Now, for i = 0, 1 consider the time changes
T (sλ − si ) + (T − sλ )t
τi : [si , T ] → [sλ , T ], τi (t) = (4.20)
T − si
and note that τ˙i (t) is constant, with τ˙i (t) = (T − sλ )/(T − si ), t ∈ [si , T ].
Finally, define ui (t) = uλ (τi (t)) and let xi (t) be the solution of the problem

dxi (t) = b(t, xi (t), ui (t))dt + σ(t, xi (t), ui (t))dWi (t), t ∈ [si , T ],
(4.21)
xi (si ) = yi ,
where Wi (t) = 1/τ˙i W (τi (t)).
722 R. Buckdahn et al. NoDEA

By the very definition of V , we then have:

λV (s1 , y1 ) + (1 − λ)V (s0 , y0 ) − V (sλ , yλ ) − 

≤ λJ(s1 , y1 , u1 ) + (1 − λ)J(s0 , y0 , u0 ) − J(sλ , yλ , uλ )
   T
T
=E λ f (t, x1 (t), u1 (t))dt + (1 − λ) f (t, x0 (t), u0 (t))dt
s1 s0

T
−E f (y, xλ (t), uλ (t))dt
sλ

+ E {λh(x1 (T )) + (1 − λ)h(x0 (T )) − h(xλ (T ))}

In the first two integrals above let us apply the change of variables r = τi (t),
.
defined by (4.20). Then, for x̃i (r) = xi (τi−1 (r)) (i = 0, 1), we have

λV (s1 , y1 ) + (1 − λ)V (s0 , y0 ) − V (sλ , yλ ) − 


T
λ 1−λ
≤E f (τ1−1 (r), x̃1 (r), uλ (r)) + f (τ0−1 (r), x̃0 (r), uλ (r))
sλ τ̇1 τ̇0

− f (r, xλ (r), uλ (r)) dr

+ E {λh(x1 (T )) + (1 − λ)h(x0 (T )) − h((xλ (T )))}

. .
Now, observe that λ̃1 = λ/τ̇1 and λ̃0 = (1 − λ)/τ̇0 satisfy λ̃1 + λ̃0 = 1, and
define
⎧ −1 −1
⎨t̃(r) = λ̃1 τ1 (r) + λ̃0 τ0 (r)
x̃ (r) = λ̃1 x̃1 (r) + λ̃0 x̃0 (r) (4.22)
⎩ λ
X̃λ (r) = λx1 (r) + (1 − λ)x0 (r)
By the semiconcavity and Lipschitzianity of f and h we obtain

λV (s1 , y1 ) + (1 − λ)V (s0 , y0 ) − V (sλ , yλ ) − 


T 
≤E λ̃1 f (τ1−1 (r), x̃1 (r), uλ (r)) + λ̃0 f (τ0−1 (r), x̃0 (r), uλ (r))
sλ


− f (t̃(r), x̃λ (r), uλ (r)) dr


T 
+E f (t̃(r), x̃λ (r), uλ (r)) − f (r, xλ (r), uλ (r)) dr
sλ
 
+ E λh(x1 (T )) + (1 − λ)h(x0 (T )) − h(X̃λ (T )) + h(X̃λ (T )) − h(xλ (T ))

T 
≤ K λ̃0 λ̃1 E |τ1−1 (r) − τ0−1 (r)|2 + |x̃1 (r) − x̃0 (r)|2 dr
sλ
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 723


T
+K E (|t̃(r) − r| + |x̃λ (r) − xλ (r)|)dr
sλ

+ K λ(1 − λ)E{|x1 (T ) − x0 (T )|2 } + K E{|X̃λ (T ) − xλ (T )|}

Now, the technical lemmas below will provide all the estimates that are needed
to complete the proof. 

Lemma 4.2. The following estimates hold true:

 T
λ̃0 λ̃1 |τ1−1 (r) − τ0−1 (r)|2 dr ≤ Kλ(1 − λ)|s1 − s0 |2 (4.23)
sλ

T
λ̃0 λ̃1 E |x̃1 (r) − x̃0 (r)|2 dr ≤ Kλ(1 − λ)(|s1 − s0 |2 + |y1 − y0 |2 ) (4.24)
sλ

|t̃(r) − r| ≤ Kλ(1 − λ)|s1 − s0 |2 (4.25)

2 2 2
E{|x1 (T ) − x2 (T )| } ≤ K(|s1 − s0 | + |y1 − y0 | ), (4.26)

T  
E |x̃λ (r) − xλ (r)|2 dr ≤ Kλ(1 − λ) |s1 − s0 |2 + |y1 − y0 |2 (4.27)
sλ

and

E{|X̃λ (T ) − xλ (T )|} ≤ Kλ(1 − λ)(|y1 − y0 |2 + |s1 − s0 |2 ). (4.28)

Proof of Lemma 4.2. We first observe that directly from the definitions of τi
and λ̃i it follows that
 
 (s1 − s0 )(T − r) 
|τ1−1 (r) − τ0−1 (r)| =   ≤ |s1 − s0 |
 (4.29)
T − sλ
and
λ(1 − λ)
λ̃0 λ̃1 = ≤ λ(1 − λ). (4.30)
τ̇1 τ̇0
That proves (4.23).
Inequality (4.25) can be checked by a straightforward computation that
we omit.

Proof of (4.24). By the definition of x̃i , i = 0, 1, and a change of variables we

have

E{|x̃1 (r) − x̃0 (r)|4 }

 
 r 
= E y1 − y0 + τ̇1−1 b(τ1−1 (ρ), x̃1 (ρ), uλ (ρ)) − τ̇0−1 b(τ0−1 (ρ), x̃0 (ρ), uλ (ρ)) dρ
sλ
   4
r 
dW (ρ)
−1/2 −1/2
+ τ̇1 σ(τ1−1 (ρ), x̃1 (ρ), uλ (ρ)) − τ̇0 σ(τ0−1 (ρ), x̃0 (ρ), uλ (ρ))
sλ
724 R. Buckdahn et al. NoDEA

Hence, owing to the Lipschitz continuity and boundedness of b and σ,

E{|x̃1 (r) − x̃0 (r)|4 } ≤ K|y1 − y0 |4
  r
 −1
+ KE |τ̇1 − τ̇0−1 | + |τ̇0−1 ||x̃1 (ρ) − x̃0 (ρ)|
sλ
4

+ |τ̇0−1 ||τ1−1 (ρ) − τ0−1 (ρ)| dρ
 r 
−1/2 −1/2 −1/2
+ KE |τ̇1 − τ̇0 | + |τ̇0 ||x̃1 (ρ) − x̃0 (ρ)|
sλ
2 2
−1/2
+ |τ̇0 ||τ1−1 (ρ) − τ0−1 (ρ)| dρ

Using (4.23) and standard estimates, we obtain

E{|x̃1 (r) − x̃0 (r)|4 }
 r
 2
≤ K |y1 − y0 |2 + |s1 − s0 |2 + E{|x̃1 (ρ) − x̃0 (ρ)|4 }dρ .
sλ
Finally, thanks to Gronwall’s Lemma, we deduce that
 1/2  
E{|x̃1 (r) − x̃0 (r)|4 } ≤ K |y1 − y0 |2 + |s1 − s0 |2 (4.31)
which, in view of (4.30), gives the desired estimate (4.24).
Proof of (4.26). this result is a direct consequence of inequality (4.31) and the
fact that x̃i (T ) = xi (T ), i = 0, 1.
Proof of (4.27). such an estimate can be deduced from the much more techni-
cal Lemma 4.3 given at the end of the present section.
Proof of (4.28). from the fact that x̃i and xi coincide at T , for i = 0, 1, we
obtain
E{|X̃λ (T ) − xλ (T )|} ≤ E{|X̃λ (T ) − x̃λ (T )|} + E{|x̃λ (T ) − xλ (T )|}
≤ E{|λx̃1 (T ) + (1 − λ)x̃0 (T ) − λ̃1 x̃1 (T ) − λ̃0 x̃0 (T )|} + E{|x̃λ (T ) − xλ (T )|}
Thus, by the definition of λ̃i , i = 0, 1,
E{|X̃λ (T ) − xλ (T )|}
|s1 − s0 |
≤ λ(1 − λ) E{|x̃1 (T ) − x̃0 (T )|} + E{|x̃λ (T ) − xλ (T )|}
T − sλ
Then, applying Hölder’s inequality and taking into account (4.31) as well as
(4.32) in Lemma 4.3 we infer that the latter estimate is dominated by
 
Kλ(1 − λ) |y1 − y0 |2 + |s1 − s0 |2 + |s1 − s0 ||y1 − y0 | .
This proves (4.28).
The above proof is completed by the following.
Lemma 4.3. Under our standard hypothesis we have
 
(E{|x̃λ (r) − xλ (r)|2 })1/2 ≤ Kλ(1 − λ) |y1 − y0 |2 + |s1 − s0 |2 . (4.32)
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 725

Proof of Lemma 4.3. Recalling the definition of x̃λ , λ̃1 and λ̃0 we have
x̃λ (r) − xλ (r) = λ̃1 x̃1 (r) + λ̃0 x̃0 (r) − xλ (r)
 r
= λ̃1 y1 + λ̃0 y0 − yλ + λτ̇1−2 b(τ1−1 (ρ), x̃1 (ρ), uλ (ρ))
sλ
−2 −1

+ (1 − λ)τ̇0 b(τ0 (ρ), x̃0 (ρ), uλ (ρ)) − b(ρ, xλ (ρ), uλ (ρ)) dρ
 r
−3/2
+ λτ̇1 σ(τ1−1 (ρ), x̃1 (ρ), uλ (ρ))
sλ

−3/2
+ (1 − λ)τ̇0 σ(τ0−1 (ρ), x̃0 (ρ), uλ (ρ)) − σ(ρ, xλ (ρ), uλ (ρ)) dW (ρ)
Taking the expectation of the square and using the Hölder as well as the
Burkholder inequality we obtain
E{|x̃λ (r) − xλ (r)|2 } ≤ K{|λ̃1 y1 + λ̃0 y0 − yλ |2
 r
 −2 −1
+E λτ̇ b(τ (ρ), x̃1 (ρ), uλ (ρ)) + (1 − λ)τ̇ −2 b(τ −1 (ρ), x̃0 (ρ), uλ (ρ))
1 1 0 0
sλ
 r 
2  −3/2
− b(ρ, xλ (ρ), uλ (ρ))| dρ + E λτ̇1 σ(τ1−1 (ρ), x̃1 (ρ), uλ (ρ))
sλ
2

σ(τ0−1 (ρ), x̃0 (ρ), uλ (ρ)) − σ(ρ, xλ (ρ), uλ (ρ))  dρ
−3/2
+ (1 − λ)τ̇0 (4.33)

In our next computations we will obtain suitable bounds for the terms in the
right-hand side of the above inequality. This will be achieved by the following
three estimates:
|λ̃1 y1 + λ̃0 y0 − yλ | ≤ Kλ(1 − λ)|s1 − s0 ||y1 − y0 | (4.34)
 
E λτ̇1−2 b(τ1−1 (ρ), x̃1 (ρ), uλ (ρ)) + (1 − λ)τ̇0−2 b(τ0−1 (ρ), x̃0 (ρ), uλ (ρ))
1/2
2
− b(ρ, xλ (ρ), uλ (ρ))|
 
≤ (E{|x̃λ (r) − xλ (r)|2 })1/2 + Kλ(1 − λ) |s1 − s0 |2 + |y1 − y0 |2 (4.35)
 
 −3/2
E λτ̇1 σ(τ1−1 (ρ), x̃1 (ρ), uλ (ρ))
2 1/2
−3/2 
+ (1 − λ)τ̇0 σ(τ0−1 (ρ), x̃0 (ρ), uλ (ρ)) − σ(ρ, xλ (ρ), uλ (ρ))
 
≤ (E{|x̃λ (r) − xλ (r)|2 })1/2 + Kλ(1 − λ) |s1 − s0 |2 + |y1 − y0 |2 (4.36)

Proof of (4.34). This is obtained by the following straightforward computation

λ(1 − λ)
|λ̃1 y1 + λ̃0 y0 − yλ |= |s1 − s0 ||y1 − y0 | ≤ Kλ(1 − λ)|s1 − s0 ||y1 − y0 |.
T − sλ

Proof of (4.35). Hereafter, to simplify the notation, we will omit the depen-
dence of functions on the independent variable: we will write, for instance, xλ
726 R. Buckdahn et al. NoDEA

instead of xλ (ρ). We also define Λ = λτ̇1−2 + (1 − λ)τ̇0−2 , λ̂1 = λτ̇1−2 /Λ and

λ̂0 = (1 − λ)τ̇0−2 /Λ, and we introduce the following notation

x̂λ = λ̂1 x̃1 + λ̂0 x̃0 , t̂(ρ) = λ̂1 τ1−1 (ρ) + λ̂0 τ0−1 (ρ).
Then, by assumption (S3) we have, recalling (4.29) and (4.25),
 −2 −1 
λτ̇ b(τ , x̃1 , uλ ) + (1 − λ)τ̇ −2 b(τ −1 , x̃0 , uλ ) − b(ρ, xλ , uλ )
1 1 0 0
 
 
= Λλ̂1 b(τ1−1 , x̃1 , uλ ) + Λλ̂0 b(τ0−1 , x̂0 , uλ ) − b(ρ, xλ , uλ )
 
 
≤ Λλ̂1 b(τ1−1 , x̃1 , uλ ) + Λλ̂0 b(τ0−1 , x̂0 , uλ ) − Λb(t̂, x̂λ , uλ )
 
+ |(Λ − 1)b(t̂, x̂λ , uλ )| + |b(t̂, x̂λ , uλ ) − b(t̃, x̃λ , uλ )|
+ |b(t̃, x̃λ , uλ ) − b(ρ, xλ , uλ )|
 
≤ KΛλ̂1 λ̂0 |τ1−1 − τ0−1 |2 + |x̃1 − x̃0 |2 + K|(Λ − 1)b(t̂, x̂λ , uλ )|
 
+ |b(t̂, x̂λ , uλ ) − b(t̃, x̃λ , uλ )| + K |t̃ − ρ| + |x̃λ − xλ |)
Thus, from

λ̂1 λ̂0 |τ1−1 − τ0−1 | ≤ Kλ(1 − λ)|s1 − s0 |2 ,

0 ≤ Λ − 1 ≤ Kλ(1 − λ)|s1 − s0 |2 , |t̃ − ρ| ≤ Kλ(1 − λ)|s1 − s0 |2 ,
and
|t̂(ρ) − t̃(ρ)| = |(λ̂1 − λ̃1 )τ1−1 + (λ̂0 − λ̃0 )τ0−1 |
≤ λ(1 − λ)|s1 − s0 ||τ1−1 − τ0−1 |
≤ Kλ(1 − λ)|s1 − s0 |2 ,
we get
  2 1/2
E λτ̇1−2 b(τ1−1 , x̃1 , uλ ) + (1 − λ)τ̇0−2 b(τ0−1 , x̃0 , uλ ) − b(ρ, xλ , uλ )
  1/2
≤ Kλ(1 − λ)(|s1 − s0 |2 + K λ̂1 λ̂0 E |x̃1 − x̃0 |4
  1/2   1/2
+ K E |x̃λ − xλ |2 + K E |x̂λ − x̃λ |2 .
In view of (4.31) we obtain
  1/2   1/2
E |x̂λ − x̃λ |2 = E |(λ̂1 − λ̃1 )x̃1 + (λ̂0 − λ̃0 )x̃0 |2
  1/2
≤ λ(1 − λ)|s1 − s0 | E |x̃1 − x̃0 |2 ≤ Kλ(1 − λ)|s1 − s0 ||y1 − y0 |.
Consequently, using again (4.31)
  2 1/2
E λτ̇1−2 b(τ1−1 , x̃1 , uλ ) + (1 − λ)τ̇0−2 b(τ0−1 , x̃0 , uλ ) − b(ρ, xλ , uλ )
  1/2
≤ Kλ(1 − λ)(|s1 − s0 |2 + |y1 − y0 |2 ) + K E |x̃λ − xλ |2
which in turn proves (4.35).
Vol. 17 (2010) Lipschitz continuity and semiconcavity properties 727

−3/2 −3/2 −3/2

Proof of (4.36). Let us define Λ̄ = λτ̇1 + (1 − λ)τ̇0 , λ̄1 = (λ/Λ̄)τ̇1
−3/2
and λ̄0 = ((1 − λ)/Λ̄)τ̇0 . Observe that λ̄0 + λ̄1 = 1. One can easily check
that 1 − Λ̄ ≤ Kλ(1 − λ)|s1 − s0 |2 . Let us further introduce the notation
. .
x̄λ = λ̄1 x̃1 + λ̄0 x̃0 , t̂(ρ) = λ̄1 τ1−1 (ρ) + λ̄0 τ0−1 (ρ).
Then, we can rewrite the expression inside the expectation of the left-hand
side of (4.36) as follows
 
 −3/2 −3/2 
τ̇1 σ(τ1−1 , x̃1 , uλ ) + (1 − λ)τ̇0 σ(τ0−1 , x̃0 , uλ ) − σ(ρ, xλ , uλ )
 
= Λ̄λ̄1 σ(τ1−1 , x̃1 , uλ ) + Λ̄λ̄0 σ(τ0−1 , x̃0 , uλ ) − σ(ρ, xλ , uλ )
 
≤ Λ̄ λ̄1 σ(τ −1 , x̃1 , uλ ) + λ̄0 σ(τ −1 , x̃0 , uλ ) − σ(t̂, x̄λ , uλ ) 
1 0
+ |(Λ̄ − 1)σ(t̂, x̄λ , uλ )| + |σ(t̂, x̄λ , uλ ) − σ(t̃, x̃λ , uλ )|
+ |σ(t̃, x̃λ , uλ ) − σ(ρ, xλ , uλ )|.
From now on, we proceed in the same way as in the proof of (4.35), using first
the regularity assumption (S3) and the boundedness of σ, and then equations
(4.25) and (4.31). We can thus show that the previous expression is smaller or
equal than
⎧ .   
⎨ξ = K Λ̄λ̄1 λ̄0 |τ1−1 − τ0−1 |2 + |x̃1 − x̃0 |2 + |t̃ − ρ| + (Λ̄ − 1)
+   (4.37)
⎩ .
η = K |t̂ − t̃| + |x̄λ − x̃λ | + |x̃λ − xλ |
Using the same ideas as in the proof of (4.35), we obtain
(E[ξ 2 ])1/2 ≤ Kλ(1 − λ)(|s1 − s0 |2 + |y1 − y0 |2 )
while
η = |(λ̄1 − λ̃1 )τ1−1 + (λ̄0 − λ̃0 )τ0−1 | + |(λ̄1 − λ̃1 )x̃1 + (λ̄0 − λ̃0 )x̃0 | + |x̃λ − xλ |.
Recalling the definition of λ̄i , λ̃i , x̃i for i = 0, 1 a direct computation yields
η ≤ K[ λ(1 − λ)(|s0 − s1 |2 + |s0 − s1 ||x̃1 − x̃0 |) + |x̃λ − xλ |].
Consequently, from the estimates in the proof of (4.35)
  2 1/2
 −3/2 −3/2 
E τ̇1 σ(τ1−1 , x̃1 , uλ ) + (1 − λ)τ̇0 σ(τ0−1 , x̃0 , uλ ) − σ(ρ, xλ , uλ )
 1/2
≤ K[λ(1 − λ)(|s0 − s1 |2 + |y0 − y1 |2 ) + E{|x̃λ − xλ |2 } .
This completes the proof of (4.36).

References

[1] Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton–Jacobi equations,

and optimal control. In: Progress in Nonlinear Differential Equations and their
Applications, vol. 58. Birkhauser, Boston (2004)

[2] Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of
Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282, 452–502 (1984)
728 R. Buckdahn et al. NoDEA

[3] Fleming, W.H., Soner, H.M.: Controlled Markov processes and viscosity solu-
tions. 2nd edn. Springer, New York (2006)

[4] Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity
preserving properties for singular degenerate parabolic equations on unbounded
domains. Indiana Univ. Math. J. 40(2), 443–469 (1991)

[5] Ishii, H., Lions, P.L.: Viscosity solutions of fully nonlinear second order elliptic
partial differential equations. J. Differ. Equ. 83, 26–78 (1990)

[6] Krylov, N.V.: Controlled diffusion processes. Applications of Mathemat-

ics. Vol. 14, Springer, New York-Berlin (1980)

[7] Wang, L.: On the regularity of fully nonlinear parabolic equations II. Comm. Pure
Appl. Math. 45, 141–178 (1992)

[8] Yong, J., Zhou, X.Y.: Stochastic controls, Hamiltonian systems and HJB equa-
tions. Springer, New York (1999)

R. Buckdahn, M. Quincampoix
Laboratoire de Mathématiques, UMR6205,
Université de Bretagne Occidentale,
6 Avenue Le Gorgeu, 29200 Brest, France
e-mail: Marc.Quincampoix@univ-brest.fr
R. Buckdahn
e-mail: Rainer.Buckdahn@univ-brest.fr
P. Cannarsa
Dipartimento di Matematica,
Università di Roma Tor Vergata,
Via della Ricerca Scientifica 1,
00133 Roma, Italy
e-mail: cannarsa@mat.uniroma2.it

Received: 31 August 2009.

Accepted: 10 May 2010.