On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

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On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

Freddy Delbaen, Ying Hu, Adrien Richou

To cite this version:

Freddy Delbaen, Ying Hu, Adrien Richou. On the uniqueness of solutions to quadratic BSDEs with

convex generators and unbounded terminal conditions. Annales de l’Institut Henri Poincaré (B)

Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2011, 47 (2), pp.559-574. �10.1214/10-

AIHP372�. �hal-00391112�

On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions

Freddy Delbaen Department of Mathematics

ETH-Zentrum, HG G 54.3, CH-8092 Zürich, Switzerland e-mail: [email protected]

Ying Hu

IRMAR, Université Rennes 1

Campus de Beaulieu, 35042 RENNES Cedex, France e-mail: [email protected]

Adrien Richou

IRMAR, Université Rennes 1

Campus de Beaulieu, 35042 RENNES Cedex, France e-mail: [email protected]

June 3, 2009

Abstract

In [4], the authors proved the uniqueness among the solutions which admit every exponential mo- ments. In this paper, we prove that uniqueness holds among solutions which admit some given expo- nential moments. These exponential moments are natural as they are given by the existence theorem.

Thanks to this uniqueness result we can strengthen the nonlinear Feynman-Kac formula proved in [4].

1 Introduction

In this paper, we consider the following quadratic backward stochastic differential equation (BSDE in short for the remaining of the paper)

Y

t

= ξ − Z

T

t

g(s, Y

s

, Z

s

)ds + Z

T

t

Z

s

dW

s

, 0 6 t 6 T, (1.1)

where the generator −g is a continuous real function that is concave and has a quadratic growth with respect to the variable z. Moreover ξ is an unbounded random variable (see e.g. [8] for the case of quadratic BSDEs with bounded terminal conditions). Let us recall that, in the previous equation, we are looking for a pair of process (Y, Z) which is required to be adapted with respect to the filtration generated by the R

^d

-valued Brownian motion W . In [4], the authors prove the uniqueness among the solutions which satisfy for any p > 0,

E h

e

^{psup06t6T|Yt|}

i

< ∞.

1

1 INTRODUCTION 2

The main contribution of this paper is to strengthen their uniqueness result. More precisely, we prove the uniqueness among the solutions satisfying:

∃p > ¯ γ, ∃ε > 0, E h

e

^psup06t6T

(

^Yt⁻+Rt

0α¯sds

) + e

^{εsup06t6TYt+}

i

< +∞,

where ¯ γ > 0 and (α

t

)

t∈[0,T]

is a progressively measurable nonnegative stochastic process such that, P-a.s.,

∀(t, y, z) ∈ [0, T ] × R × R

^1×d

, g(t, y, z) 6 α ¯

t

+ ¯ β |y| + γ ¯ 2 |z|

²

.

Our method is different of that in [4] where the authors apply the so-called θ-difference method, i.e.

estimating Y

¹

−θY

²

, for θ ∈ (0, 1), and then letting θ → 0. Whereas in this paper, we apply a verification method: first we define a stochastic control problem and then we prove that the first component of any solution of the BSDE is the optimal value of this associated control problem. Thus the uniqueness follows immediately. Moreover, using this representation, we are able to give a probabilistic representation of the following PDE:

∂

t

u(t, x) + Lu(t, x) − g(t, x, u(t, x), −σ

^∗

∇

x

u(t, x)) = 0, u(T, .) = h,

where h and g have a “not too high” quadratic growth with respect to the variable x. We remark that the probabilistic representation is also given by [4] under the condition that h and g are subquadratic, i.e.:

∀(t, x, y, z) ∈ [0, T ] × R

^d

× R × R

^1×d

, |h(x)| + |g(t, x, y, z)| 6 f (t, y, z) + C |x|

^p

with f > 0, C > 0 and p < 2.

The paper is organized as follows. In section 2, we prove an existence result in the spirit of [3] and [4]: here we work with generators −g such that g

⁻

has a linear growth with respect to variables y and z. As in part 5 of [3], this assumption allows us to reduce hypothesis of [4]. Section 3 is devoted to the optimal control problem from which we get as a byproduct a uniqueness result for quadratic BSDEs with unbounded terminal conditions. Finally, in the last section we derive the nonlinear Feynman-Kac formula in this framework.

Let us close this introduction by giving the notations that we will use in all the paper. For the re- maining of the paper, let us fix a nonnegative real number T > 0. First of all, (W

t

)

t∈[0,T]

is a standard Brownian motion with values in R

^d

defined on some complete probability space (Ω, F , P ). (F

t

)

t>0

is the natural filtration of the Brownian motion W augmented by the P-null sets of F. The sigma-field of predictable subsets of [0, T ] × Ω is denoted P .

As mentioned in the introduction, we will deal only with real valued BSDEs which are equations of type (1.1). The function −g is called the generator and ξ the terminal condition. Let us recall that a generator is a random function [0, T ] × Ω × R × R

^1×d

→ R which is measurable with respect to P ⊗ B(R) ⊗ B(R

^1×d

) and a terminal condition is simply a real F

T

-measurable random variable. By a solution to the BSDE (1.1) we mean a pair (Y

t

, Z

t

)

t∈[0,T]

of predictable processes with values in R×R

^1×d

such that P-a.s., t 7→ Y

t

is continuous, t 7→ Z

t

belongs to L

²

(0, T ), t 7→ g(t, Y

t

, Z

t

) belongs to L

¹

(0, T ) and P-a.s. (Y, Z) verifies (1.1). We will sometimes use the notation BSDE(ξ,g) to say that we consider the BSDE whose generator is g and whose terminal condition is ξ.

For any real p > 1, S

^p

denotes the set of real-valued, adapted and càdlàg processes (Y

t

)

t∈[0,T]

such that

kY k

_Sp

:= E

sup

06t6T

|Y

t

|

^p

1/p

< +∞.

M

^p

denotes the set of (equivalent class of) predictable processes (Z

t

)

_t∈[0,T]

with values in R

^1×d

such that

kZk

_Mp

:= E



 Z

T

0

|Z

s

|

²

ds

!

p/2





1/p

< +∞.

2 AN EXISTENCE RESULT 3

Finally, we will use the notation Y

^∗

:= sup

_06t6T

|Y

t

| and we recall that Y belongs to the class (D) as soon as the family {Y

τ

: τ 6 T stopping time} is uniformly integrable.

2 An existence result

In this section, we prove a mere modification of the existence result for quadratic BSDEs obtained in [4]

by using a method applied in section 5 of [4]. We consider here the case where g

⁻

has a linear growth with respect to variables y and z. Let us assume the following on the generator.

Assumption (A.1). There exist three constants β > 0, γ > 0 and r > 0 together with two progressively measurable nonnegative stochastic processes (¯ α

t

)

06t6T

, (α

_t

)

06t6T

and a deterministic continuous non- decreasing function φ : R

⁺

→ R

⁺

with φ(0) = 0 such that, P-a.s.,

1. for all t ∈ [0, T ], (y, z) 7→ g(t, y, z) is continuous;

2. monotonicity in y: for each (t, z) ∈ [0, T ] × R

^1×d

,

∀y ∈ R , y(g(t, 0, z) − g(t, y, z)) 6 β |y|

²

; 3. growth condition: ∀(t, y, z) ∈ [0, T ] × R × R

^1×d

,

−α

_t

− r(|y| + |z|) 6 g(t, y, z) 6 α ¯

t

+ φ(|y|) + γ 2 |z|

²

. Theorem 2.1 Let (A.1) hold. If there exists p > 1 such that

E

"

exp γe

^βT

ξ

⁻

+ γ Z

T

0

¯ α

t

e

^βt

dt

!

+ (ξ

⁺

)

^p

+ Z

T

0

α

_t

dt

!

p

#

< +∞

then the BSDE (1.1) has a solution (Y, Z ) such that

− 1 γ log E

"

exp γe

^β(T−t)

ξ

⁻

+ γ Z

T

t

¯

α

r

e

^β(r−t)

dr

! F

t

#

6 Y

t

6 Ce

^CT

E

"

(ξ

⁺

)

^p

+ Z

T

t

α

_r

dr

!

p

F

t

#!

1/p

, with C a constant that does not depend on T .

Proof. We will fit the proof of Proposition 4 in [3] to our situation. Without loss of generality, let us assume that r is an integer. For each integer n > r, let us consider the function

g

n

(t, y, z) := inf

g(t, p, q) + n |p − y| + n |q − z| , (p, q) ∈ Q

^1+d

.

g

n

is well defined and it is globally Lipschitz continuous with constant n. Moreover (g

n

)

n>r

is increasing and converges pointwise to g. Dini’s theorem implies that the convergence is also uniform on compact sets. We have also, for all n > r,

h(t, y, z) := −α

_t

− r(|y| + |z|) 6 g

n

(t, y, z) 6 g(t, y, z).

Let (Y

ⁿ

, Z

ⁿ

) be the unique solution in S

^p

× M

^p

to BSDE(ξ,−g

n

). It follows from the classical compar- ison theorem that

Y

_tⁿ⁺¹

6 Y

_tⁿ

6 Y

_t^r

.

2 AN EXISTENCE RESULT 4

Let us prove that for each n > r Y

_tⁿ

> − 1

γ log E

"

exp γe

^β(T−t)

ξ

⁻

+ γ Z

T

t

¯

α

r

e

^β(r−t)

dr

! F

t

# := X

t

.

Let ( ˜ Y

ⁿ

, Z ˜

ⁿ

) be the unique solution S

^p

× M

^p

to BSDE(−ξ

⁻

,−g

_n⁺

). It follows from the classical com- parison theorem that Y ˜

ⁿ

6 Y

ⁿ

and Y ˜

ⁿ

6 0. Then, according to Proposition 3 in [4], we have Y ˜

ⁿ

> X and so Y

ⁿ

> X for all n > r. We set Y = inf

n>r

Y

ⁿ

and, arguing as in the proof of Proposition 3 in [4]

or Theorem 2 in [3] with a localization argument, we construct a process Z such that (Y, Z) is a solution to BSDE(ξ,−g). For the upper bound, let ( ¯ Y , Z) ¯ be the unique solution S

^p

× M

^p

to BSDE(ξ

⁺

,−f ).

Then the classical comparison theorem gives us that Y 6 Y ¯ and we apply a classical a priori estimate for

L

^p

solutions of BSDEs in [2] to Y ¯ . ⊓ ⊔

Corollary 2.2 Let (A.1) hold. We suppose that ξ

⁻

+ R

T

0

α ¯

t

dt has an exponential moment of order γe

^βT

and there exists p > 1 such that ξ

⁺

∈ L

^p

and R

T

0

α

_t

dt ∈ L

^p

.

• If ξ

⁻

+ R

T

0

α ¯

t

dt has an exponential moment of order qe

^βT

with q > γ then the BSDE (1.1) has a solution (Y, Z) such that E

e

^qA∗

< +∞ with A

t

:= Y

_t⁻

+ R

t 0

α ¯

s

ds.

• If ξ

⁺

+ R

T

0

α

_t

dt has an exponential moment of order ε then the BSDE (1.1) has a solution (Y, Z) such that E h

e

^ε(Y+)∗

i

< +∞.

Proof. Let us apply the existence result : BSDE (1.1) has a solution (Y, Z ) and we have A

t

= Y

_t⁻

+

Z

t 0

¯

α

s

ds 6 1 γ log E

"

exp γe

^βT

ξ

⁻

+ Z

T

0

¯ α

r

dr

!!

F

t

#

| {z }

:=Mt

.

So e

^qAt

6 (M

t

)

^q/γ

with q/γ > 1. Since M

^p/γ

is a submartingale, we are able to apply the Doob’s maximal inequality to obtain

E h e

^qA∗

i

6 C

q

E h

e

^{qeβT(ξ−+R0Tα¯sds)}

i

< +∞.

To prove the second part of the corollary, we define

N

t

:= E

"

(ξ

⁺

)

^p

+

Z

T 0

α

_s

ds

p

F

t

# .

We set q > 1. There exists C

ε,p,q

> 0 such that x 7→ e

^x1/pε/q

is convex on [C

ε,p,q

, +∞[. We have e

^ε/qYt+

6 e

^{(Cε,p,q+Nt)1/pε/q}

. Since e

^{(Cε,p,q+N)1/pε/q}

is a submartingale, we are able to apply the Doob’s maximal inequality to obtain

E h

e

^ε(Y+)∗

i

6 C E h

e

^{ε(Cε,p,q+(ξ+)p+(R0Tαsds)p)1/p}

i

6 C E h

e

^{ε(ξ++R0Tαsds)}

i

< +∞.

⊓ ⊔

3 A UNIQUENESS RESULT 5

3 A uniqueness result

To prove our uniqueness result for the BSDE (1.1), we will introduce a stochastic control problem. For this purpose, we use the following assumption on g:

Assumption (A.2). There exist three constants K

g,y

> 0, β ¯ > 0 and γ > ¯ 0 together with a progres- sively measurable nonnegative stochastic process (¯ α

t

)

t∈[0,T]

such that, P-a.s.,

• for each (t, z) ∈ [0, T ] × R

^1×d

,

|g(t, y, z) − g(t, y

^′

, z)| 6 K

g,y

|y − y

^′

| , ∀(y, y

^′

) ∈ R

²

,

• for each (t, y, z) ∈ [0, T ] × R × R

^1×d

,

g(t, y, z) 6 α ¯

t

+ ¯ β |y| + γ ¯ 2 |z|

²

,

• z 7→ g(t, y, z) is a convex function ∀(t, y) ∈ [0, T ] × R.

Since g(t, y, .) is a convex function we can define the Legendre-Fenchel transformation of g : f (t, y, q) = sup

z

(zq − g(t, y, z)) , ∀t ∈ [0, T ], q ∈ R

^d

, y ∈ R.

f is a function with values in R ∪ {+∞} that verifies direct properties.

Proposition 3.1

• ∀(t, y, y

^′

, q) ∈ [0, T ] × R × R × R

^d

such that f (t, y, q) < +∞,

f (t, y

^′

, q) < +∞ and |f (t, y, z) − f (t, y

^′

, z)| 6 K

g,y

|y − y

^′

| .

• f is a convex function in q,

• f (t, y, q) > −¯ α

t

− β ¯ |y| +

_2¯¹_γ

|q|

²

. We set N ∈ N

^∗

such that

T N <

1

¯ γ − 1

p

1

β ¯ (1/p + 1/ε) . (3.1)

For i ∈ {0, ..., N} we define t

i

:=

^iT_N

and A

ti,ti+1

(η) :=

(q

s

)

s∈[ti,ti+1]

,

Z

ti+1

ti

|q

s

|

²

ds < +∞ P − a.s., (M

_tⁱ

)

t∈[ti,ti+1]

is a martingale, E

^Qi

|η| + Z

ti+1

ti

|f (s, 0, q

s

)| ds

< +∞, with M

_tⁱ

:= exp

Z

t ti

q

s

dW

s

− 1 2

Z

t ti

|q

s

|

²

ds

and dQ

ⁱ

dP := M

_tⁱ_i+1

.

Let q be in A

ti,ti+1

(η). We define dW

_t^q

:= dW

t

− q

t

dt. Thanks to the Girsanov theorem, (W

_t^q_i+h

−

W

_t^q_i

)

h∈[0,1/N]

is a Brownian motion under the probability Q

ⁱ

. So, we are able to apply Proposition 6.4 in

[2] to show this existence result:

3 A UNIQUENESS RESULT 6

Proposition 3.2 There exist two processes (Y

^η,q

, Z

^η,q

) such that (Y

_t^η,q

)

_{t∈[ti,ti+1]}

belongs to the class (D) R

ti+1

ti

|Z

_s^η,q

|

²

ds < +∞ P − a.s., R

ti+1

ti

|f (s, Y

_s^η,q

, q

s

)| ds < +∞ P − a.s. and Y

_t^η,q

= η +

Z

ti+1

t

f (s, Y

_s^η,q

, q

s

)ds + Z

ti+1

t

Z

_s^η,q

dW

_s^q

, t

i

6 t 6 t

i+1

. We are now able to define the admissible control set:

A := n

(q

s

)

s∈[0,T]

, q

|[tN−1,T]

∈ A

tN−1,T

(ξ), ∀i ∈ {N − 2, . . . , 0} , q

|[ti,ti+1]

∈ A

ti,ti+1

Y

_t^q_i+1

with Y

_t^q_i+1

:= Y

^Y

q

ti+2,q_|[ti+1,ti+2 ]

ti+1

and Y

_T^q

:= ξ

.

A is well defined by a decreasing recursion on i ∈ {0, . . . , N − 1}. For q ∈ A we can define our cost functional Y

^q

on [0, T ] by

∀i ∈ {N − 1, . . . , 0} , ∀t ∈ [t

i

, t

i+1

], Y

_t^q

:= Y

^Y

q

ti+1,q_{|[ti,ti+1 ]}

t

.

Y

^q

is also well defined by a decreasing recursion on i ∈ {0, . . . , N − 1}. Finally, the stochastic control problem consists in minimizing Y

^q

among all the admissible controls q ∈ A. Our strategy to prove the uniqueness is to prove that given a solution (Y, Z ), the first component is the optimal value.

Theorem 3.3 We suppose that there exists a solution (Y, Z ) of the BSDE (1.1) verifying

∃p > γ, ¯ ∃ε > 0, E

exp (pA

^∗

) + exp ε(Y

⁺

)

^∗

< +∞, with A

t

:= Y

_t⁻

+ R

t

0

α ¯

s

ds. Then we have Y = ess inf

_q∈A

Y

^q

, and there exists q

^∗

∈ A such that Y = Y

^q∗

. Moreover, this implies that the solution (Y, Z) is unique among solutions verifying such condition.

Proof. Let us first prove that for any q admissible, we have Y 6 Y

^q

. To do this, we will show that Y

|[ti,ti+1]

6 Y

_|[t^q_i,ti+1]

by decreasing recurrence on i ∈ {0, N − 1}. Firstly, we have Y

T

= Y

_T^q

= ξ. Then we suppose that Y

t

6 Y

_t^q

, ∀t ∈ [t

i+1

, T ]. We set t ∈ [t

i

, t

i+1

],

τ

_nⁱ

:= inf

s > t, sup Z

s

t

|Z

u

|

²

du, Z

s

t

|Z

_u^q

|

²

du, Z

s

t

|q

u

|

²

du

> n

∧ t

i+1

, h(s, y, z) := −g(s, y, z) + zq

s

, and

h

s

:=







h(s, Y

_s^q

, Z

s

) − h(s, Y

s

, Z

s

) Y

s^q

− Y

s

if Y

_s^q

− Y

s

6= 0

0 otherwise.

We observe that |h

s

| 6 K

g,y

. Then, by applying Itô formula to the process (Y

_s^q

− Y

s

)e

^Rtshudu

we obtain Y

_t^q

−Y

t

= e

^{Rtτ inhsds}

h

Y

_τ^qi n

− Y

_τnⁱ

i

+ Z

τ_nⁱ

t

e

^Rtshudu

[f (s, Y

_s^q

, q

s

) − h(s, Y

_s^q

, Z

s

)] ds+

Z

τ_nⁱ t

e

^Rtshudu

[Z

_s^q

− Z

s

] dW

_s^q

. By definition, f (s, Y

_s^q

, q

s

) − h(s, Y

_s^q

, Z

s

) > 0, so

Y

_t^q

− Y

t

> E

^Qi

e

^R

τ in t hsds

h

Y

_τ^qi n

− Y

τ_nⁱ

i F

t

.

3 A UNIQUENESS RESULT 7

Since

Y

_τ^qi

n

e

^{Rtτ inhsds}

n

tends to Y

_t^q_i+1

e

^Rtti+1hsds

almost surely and is uniformly integrable, we have

n→+∞

lim E

^Qi

e

^{Rtτ inhsds}

Y

_τ^qi n

F

t

= E

^Qi

h

e

^Rtti+1hsds

Y

_t^q_i+1

F

t

i .

Moreover,

Y

τ_nⁱ

e

^{Rtτ inhsds}

6 (Y

⁺

)

^∗

e

^{T Kg,y}

+ (Y

⁻

)

^∗

e

^{T Kg,y}

, so, by the dominated convergence theorem we obtain

n→+∞

lim E

^Qi

e

^{Rtτ inhsds}

Y

τ_nⁱ

F

t

= E

^Qi

h

e

^Rtti+1hsds

Y

ti+1

F

t

i . Finally,

Y

_t^q

− Y

t

> lim

n→+∞

E

^Qi

e

^{Rtτ inhsds}

h Y

_τ^qi

n

− Y

τ_nⁱ

i

|F

t

= E

^Qi

h

e

^Rtti+1hsds

Y

_t^q_i+1

− Y

ti+1

F

t

i

> 0, because Y

_t^q_i+1

> Y

ti+1

by the recurrence’s hypothesis.

Now we set

^t

q

^∗_s

∈ ∂

z

g(s, Y

s

, Z

s

) with ∂

z

g(s, Y

s

, Z

s

) the subdifferential of z 7→ g(s, Y

s

, z) at Z

s

. We recall that for a convex function l : R

^1×d

→ R, the subdifferential of l at x

0

is the non-empty convex compact set of u ∈ R

^1×d

such that

l(x) − l(x

0

) > u

^t

(x − x

0

), ∀x ∈ R

^1×d

. We have f (s, Y

s

, q

_s^∗

) = zq

_s^∗

− g(s, Y

s

, Z

s

) for all s ∈ [0, T ], so

g(s, Y

s

, Z

s

) 6 Z

s

q

_s^∗

− 1

2¯ γ |q

_s^∗

|

²

+ ¯ β |Y

s

| + ¯ α

s

6 1

2 2¯ γ |Z

s

|

²

+ |q

^∗_s

|

²

2¯ γ

!

− 1

2¯ γ |q

^∗_s

|

²

+ ¯ β |Y

s

| + ¯ α

s

|q

^∗_s

|

²

4¯ γ 6 −g(s, Y

s

, Z

s

) + ¯ γ |Z

s

|

²

+ ¯ β |Y

s

| + ¯ α

s

, and finally, R

T

0

|q

^∗_s

|

²

ds < +∞, P-a.s.. Moreover, ∀t, t

^′

∈ [0, T ], Y

t

= Y

t^′

+

Z

t^′ t

f (s, Y

s

, q

^∗_s

)ds + Z

t^′

t

Z

s

(dW

s

+ q

_s^∗

ds).

Thus, we just have to show that q

^∗

is admissible to prove that q

^∗

is optimal, i.e. Y = Y

^q∗

. For this, we must prove that (q

^∗_s

)

s∈[ti,ti+1]

∈ A

ti,ti+1

(Y

ti+1

) for i ∈ {0, . . . , N − 1}. We define

M

_tⁱ

:= exp Z

t

ti

q

^∗_s

dW

s

− 1 2

Z

t ti

|q

_s^∗

|

²

ds

, dQ

^∗,i

dP := M

_tⁱ_i+1

, τ

_nⁱ

= inf

t ∈ [t

i

, t

i+1

], sup Z

t

ti

|q

^∗_s

|

²

ds, Z

t

ti

|Z

s

|

²

ds

> n

∧ t

i+1

, dQ

^∗,i_n

d P := M

_τⁱi n

. Let us show the following lemma:

Lemma 3.4 (M

_τⁱi

n

)

n

is uniformly integrable.

3 A UNIQUENESS RESULT 8

Proof. Firstly, from

xy 6 exp(x) + y(log(y) − 1), ∀(x, y ) ∈ R × R

^+∗

, we deduce

xy = px y

p 6 exp(px) + y

p (log y − log p − 1) . (3.2)

Thus

E

^Q∗,in

[A

^∗

] = E h M

_τⁱi

n

A

^∗

i

6 E [exp(pA

^∗

)] + 1 p E h

M

_τⁱi n

log M

_τⁱi

n

− log p − 1 i 6 C

p

+ 1

2p E

^Q∗,in

"Z

τ_nⁱ

ti

|q

s^∗

|

²

ds

# , and, in the same manner,

E

^Q∗,in

(Y

⁺

)

^∗

6 C

ε

+ 1 2ε E

^Q∗,in

"Z

τ_nⁱ

ti

|q

s^∗

|

²

ds

# .

Since g(s, Y

s

, Z

s

) = Z

s

q

_s^∗

− f (s, Y

s

, q

^∗_s

) and (M

_t∧τⁱ i

n

)

t∈[ti,ti+1]

is a martingale, we can apply the Gir- sanov theorem and we obtain

E

^Q∗,in

"

Y

τ_nⁱ

+ Z

τ_nⁱ

ti

f (s, Y

s

, q

_s^∗

)ds

#

= E

^Q∗,in

[Y

ti

] = E h M

_τⁱi

n

Y

ti

i = E [Y

ti

] .

Moreover f (t, y, q) >

_2¯¹_γ

|q|

²

− β ¯ |y| − α ¯

t

and Y

τ_nⁱ

> −Y

_τ⁻i n

, so E [Y

ti

] > −E

^Q∗,in

h

Y

_τ⁻i n

i − E

^Q∗,in

"Z

τ_nⁱ

ti

¯ α

s

ds

# + 1

2¯ γ E

^Q∗,in

"Z

τ_nⁱ

ti

|q

_s^∗

|

²

ds

#

− βE ¯

^Q∗,in

"Z

τ_nⁱ

ti

|Y

s

| ds

#

> C − E

^Q∗,in

[A

^∗

] + 1 2¯ γ E

^Q∗,in

"Z

τ_nⁱ

ti

|q

_s^∗

|

²

ds

#

− T N

β ¯ E

^Q∗,in

(Y

⁻

)

^∗

+ (Y

⁺

)

^∗

> C

p,ε

+ 1 2

1

¯ γ − 1

p − T N

β ¯ p + β ¯

ε

| {z }

>0

E

^Q∗,in

"Z

τ_nⁱ

ti

|q

_s^∗

|

²

ds

# .

This inequality explains why we take N verifying (3.1). Finally we get that 2E h

M

_τⁱi

n

log M

_τⁱi n

i = E

^Q∗,in

"Z

τ_nⁱ

ti

|q

^∗_s

|

²

ds

#

6 C

p,ε

. (3.3)

Then we conclude the proof of the lemma by using the de La Vallée Poussin lemma. ⊓ ⊔ Thanks to this lemma, we have that E[M

_tⁱ_i+1

] = 1 and so (M

_tⁱ

)

t∈[ti,ti+1]

is a Martingale. Moreover, applying Fatou’s lemma and inequality (3.3), we obtain

2E h

M

_tⁱ_i+1

log M

_tⁱ_i+1

i

= E

^Q∗,i

Z

ti+1

ti

|q

^∗_s

|

²

ds

6 lim inf

n

E

^Q∗,in

"Z

τ_nⁱ

ti

|q

_s^∗

|

²

ds

#

< +∞. (3.4)

3 A UNIQUENESS RESULT 9

So, by using this result and inequality (3.2) we easily show that E

^Q∗,i

[(Y

⁺

)

^∗

+ (Y

⁻

)

^∗

] < +∞. To conclude we have to prove that E

^Q∗,i

hR

ti+1

ti

|f (s, 0, q

^∗_s

)| ds i

< +∞:

E

^Q∗,i

Z

ti+1

ti

|f (s, 0, q

_s^∗

)| ds

6 E

^Q∗,i

Z

ti+1

ti

|f (s, Y

s

, q

_s^∗

)| + K

g,y

|Y

s

| ds

6 E

^Q∗,i

Z

ti+1

ti

|f (s, Y

s

, q

_s^∗

)| ds + K

g,y

T (Y

⁺

)

^∗

+ (Y

⁻

)

^∗

6 C + E

^Q∗,i

Z

ti+1

ti

f

⁺

(s, Y

s

, q

^∗_s

) + f

⁻

(s, Y

s

, q

_s^∗

)ds

. Firstly,

E

^Q∗,i

Z

ti+1

ti

f

⁻

(s, Y

s

, q

_s^∗

)ds

6 E

^Q∗,i

Z

ti+1

ti

¯

α

s

+ ¯ β |Y

s

| ds

< +∞.

Moreover, thanks to the Girsanov theorem we have E

^Q∗,i

[Y

ti

] = E

^Q∗,i

"

Y

τ_nⁱ

+ Z

τ_nⁱ

ti

f (s, Y

s

, q

^∗_s

)ds

# , so

E

^Q∗,i

"Z

τ_nⁱ

ti

f

⁺

(s, Y

s

, q

_s^∗

)ds

#

6 E

^Q∗,i

Y

ti

− Y

τ_nⁱ

+ E

^Q∗,i

"Z

τ_nⁱ

ti

f

⁻

(s, Y

s

, q

_s^∗

)ds

#

6 C + E

^Q∗,i

Z

ti+1

ti

f

⁻

(s, Y

s

, q

_s^∗

)ds

6 C Finally, E

^Q∗,i

hR

ti+1

ti

f

⁺

(s, Y

s

, q

^∗_s

)ds i

< +∞ and E

^Q∗,i

hR

ti+1

ti

|f (s, 0, q

_s^∗

)| ds i

< +∞. Thus, we prove that q

^∗

is optimal, i.e. Y

^q∗

= Y .

The uniqueness of Y is a mere consequence of the fact that Y = Y

^q∗

= ess inf

q∈A

Y

^q

. The unique-

ness of Z follows immediately. ⊓ ⊔

Remark 3.5 By taking into consideration the inequality (3.4) it is possible to restrict the admissible control set by considering

A ˜

ti,ti+1

(η) := A

ti,ti+1

(η) ∩

(q

s

)

s∈[ti,ti+1]

, E

^Qi

Z

ti+1

ti

|q

s

|

²

ds

< +∞

instead of A

ti,ti+1

(η).

Remark 3.6 If we have g(t, y, z) 6 g(t, 0, z), then f (t, y, q) > f (t, 0, q) >

_2¯¹_γ

|q|

²

− α ¯

t

and we do not have to introduce N in the proof of lemma 3.4. So we have a simpler representation theorem:

Y

t

= ess inf

q∈A0,T(ξ)

Y

_t^q

, ∀t ∈ [0, T ].

For example, when g is independent of y, we obtain Y

t

= ess inf

q∈A0,T(ξ)

E

^Q

"

ξ + Z

T

t

f (s, q

s

)ds F

t

#

, ∀t ∈ [0, T ].

4 APPLICATION TO QUADRATIC PDES 10

4 Application to quadratic PDEs

In this section we give an application of our results concerning BSDEs to PDEs which are quadratic with respect to the gradient of the solution. Let us consider the following semilinear PDE

∂

t

u(t, x) + Lu(t, x) − g(t, x, u(t, x), −σ

^∗

∇

x

u(t, x)) = 0, u(T, .) = h, (4.1) where L is the infinitesimal generator of the diffusion X

^t,x

solution to the SDE

X

_s^t,x

= x + Z

s

t

b(u, X

_u^t,x

)ds + Z

s

t

σ(u)dW

u

, t 6 s 6 T, and X

_s^t,x

= x, s 6 t. (4.2) The nonlinear Feynman-Kac formula consists in proving that the function defined by the formula

∀(t, x) ∈ [0, T ] × R

^d

, u(t, x) := Y

_t^t,x

(4.3) where, for each (t

0

, x

0

) ∈ [0, T ] × R

^d

, (Y

^t0,x0

, Z

^t0,x0

) stands for the solution to the following BSDE

Y

t

= h(X

_T^t0,x0

) − Z

T

t

g(s, X

_s^t0,x0

, Y

s

, Z

s

)ds − Z

T

t

Z

s

dW

s

, 0 6 t 6 T, (4.4) is a solution, at least a viscosity solution, to the PDE (4.1).

Assumption (A.3). Let b : [0, T ] × R

^d

→ R

^d

and σ : [0, T ] → R

^d×d

be continuous functions and let us assume that there exists K > 0 such that:

1. for all t ∈ [0, T ], |b(t, 0)| 6 K, and

∀(x, x

^′

) ∈ R

^d

× R

^d

, |b(t, x) − b(t, x

^′

)| 6 K |x − x

^′

| ; 2. σ is bounded.

Lemma 4.1

∀λ ∈

"

0, 1

2e

^2KT

kσk

²_∞

T

"

, ∃C

T

> 0, ∃C > 0, E

sup

06t6T

e

^λ

|

^Xt^t0,x0

|

²

6 C

T

e

^C|x0|2

, with T 7→ C

T

nondecreasing.

Proof. As in [4] we easily show that, for all ε > 0, we have sup

06t6T

X

_t^t0,x0

6

|x

0

| + KT + sup

06t6T

Z

t 0

1s>t0

σ(s)dW

s

e

^KT

sup

06t6T

X

_t^t0,x0

²

6 C

ε

(T

²

+ |x

0

|

²

) + (1 + ε)e

^2KT

sup

06t6T

Z

t 0

1s>t0

σ(s)dW

s

2

.

We define ˜ λ := λ(1 + ε)e

^2KT

. It follows from the Dambis-Dubins-Schwarz representation theorem and the Doob’s maximal inequality that

E

"

sup

06t6T

exp λ ˜

Z

t 0

1s>t0

σ(s)dW

s

2

!#

6 E

"

sup

06t6kσk²_∞T

e

^{λ|W˜ t|2}

# 6 4E h

e

^{˜λkσk2∞T|W1|2}

i ,

which is a finite constant if λ ˜ kσk

²_∞

T < 1/2. ⊓ ⊔

With this observation in hands, we can give our assumptions on the nonlinear term of the PDE and

the terminal condition.

4 APPLICATION TO QUADRATIC PDES 11

Assumption (A.4). Let g : [0, T ] × R

^d

× R × R

^d

→ R and h : R

^d

→ R be continuous and let us assume moreover that there exist five constants r > 0, β > 0, γ > 0, α > 0 and α

^′

> 0 such that:

1. for each (t, x, z) ∈ [0, T ] × R

^d

× R

^1×d

,

∀(y, y

^′

) ∈ R

²

, |g(t, x, y, z) − g(t, x, y

^′

, z)| 6 β |y − y

^′

| ; 2. for each (t, x, y) ∈ [0, T ] × R

^d

× R, z 7→ g(t, x, y, z) is convex on R

^1×d

; 3. for each (t, x, y, z) ∈ [0, T ] × R

^d

× R × R

^1×d

,

−r(1 + |x|

²

+ |y| + |z|) 6 g(t, x, y, z) 6 r + α |x|

²

+ β |y| + γ 2 |z|

²

,

−r − α

^′

|x|

²

6 h(x) 6 r(1 + |x|

²

);

4. for each (t, x, x

^′

, y, z) ∈ [0, T ] × R

^d

× R

^d

× R × R

^1×d

,

|g(t, x, y, z) − g(t, x

^′

, y, z)| 6 r(1 + |x| + |x

^′

|) |x − x

^′

| ,

|h(x) − h(x

^′

)| 6 r(1 + |x| + |x

^′

|) |x − x

^′

| ; 5.

α

^′

+ T α < 1

2γe

^3βT

kσk

²_∞

T .

Thanks to Lemma 4.1, we see that there exist q > γe

^βT

and ε > 0 such that h

⁻

(X

_T^t0,x0

) + R

T 0

C + α

X

_t^t0,x0

²

dt has an exponential moment of order q and h

⁺

(X

_T^t0,x0

) + R

T 0

r + r

X

_t^t0,x0

²

dt has an exponential moment of order ε. So we are able to apply Corollary 2.2 and Theorem 3.3 to construct a unique solution (Y

^t0,x0

, Z

^t0,x0

) to the BSDE (4.4). Let us prove that u is a viscosity solution to the PDE (4.1).

Proposition 4.2 Let assumptions (A.3) and (A.4) hold. The function u defined by (4.3) is continuous on [0, T ] × R

^d

and satisfies

∀(t, x) ∈ [0, T ] × R

^d

, |u(t, x)| 6 C(1 + |x|

²

).

Moreover u is a viscosity solution to the PDE (4.1).

Before giving a proof of this result, we will show some auxiliary results about admissible control sets. We have already notice in Remark 3.6 that we have a simpler representation theorem when T is small enough to take N = 1 in (3.1). So we define a constant T

1

> 0 such that for all T ∈ [0, T

1

] we are allowed to set N = 1. We will reuse notations of section 3. By using Remark 3.5, for all T ∈ [0, T

1

], t ∈ [0, T ], x ∈ R

^d

, we define the admissible control set

A

0,T

(t, x) :=

(

(q

s

)

_s∈[0,T]

, Z

T

0

|q

s

|

²

ds < +∞ P − a.s., E

^Q

"Z

T

0

|q

s

|

²

ds

#

< +∞,

(M

t

)

t∈[0,T]

is a martingale, E

^Q

"

h(X

_T^t,x

) +

Z

T 0

f (s, X

_s^t,x

, 0, q

s

) ds

#

< +∞, with M

t

:= exp

Z

t 0

q

s

dW

s

− 1 2

Z

t 0

|q

s

|

²

ds

and dQ dP := M

T

.

We will prove a first lemma and then we will use it to show that this admissible control set does not

depend on t and x.

4 APPLICATION TO QUADRATIC PDES 12

Lemma 4.3 ∃C > 0 such that ∀T ∈ [0, T

1

], ∀t ∈ [0, T ], ∀x ∈ R

^d

, ∀q ∈ A

0,T

(t, x), ∀s ∈ [t, T ], E

^Q

h

X

_s^t,x

²

i 6 C

1 + |x|

²

+ T Z

s

t

E

^Q

h

|q

u

|

²

i du

. Remark 4.4 q and Q depend on x and t but we do not write it to simplify notations.

Proof. For all s ∈ [t, T ] we have an obvious inequality X

_s^t,x

²

6 C



1 + |x|

²

+ Z

s

t

X

_u^t,x

du

2

+ sup

t6t^′6T

Z

t^′ t

σ(u)dW

_u^q

2

+ Z

s

t

|q

u

| du

2



 . Then, by applying Cauchy-Schwarz’s inequality and Doob’s maximal inequality, we obtain

E

^Q

h X

_s^t,x

²

i

6 C



1 + |x|

²

+ T Z

s

t

E

^Q

h X

_u^t,x

²

i du + E

^Q





Z

T t

σ(u)dW

_u^q

2





+T E

^Q

Z

s

t

|q

u

|

²

du

.

Finally, the Gronwall’s Lemma gives us the result. ⊓ ⊔

Proposition 4.5 A

0,T

(t, x) is independent of t and x, so we will write it A

0,T

.

Proof. Let x, x

^′

∈ R

^d

, t, t

^′

∈ [0, T ] and q ∈ A

0,T

(t, x). We will show that q ∈ A

0,T

(t

^′

, x

^′

). Firstly, E

^Q

h

h(X

_T^t′,x′

) i

6 C

1 + E

^Q

X

_T^t′,x′

2

6 C 1 + Z

T

t^′

E

^Q

h

|q

u

|

²

i du

!

< +∞.

Moreover

−C(1 + X

_s^t′,x′

2

) 6 1

2γ |q

s

|

²

− C(1 + X

_s^t′,x′

2

) 6 f (s, X

_s^t′,x′

, 0, q

s

), and

f (s, X

_s^t′,x′

, 0, q

s

) 6 f (s, X

_s^t,x

, 0, q

s

) + C(

X

_s^t,x

²

+

X

_s^t′,x′

2

).

So,

f (s, X

_s^t′,x′

, 0, q

s

)

6 |f (s, X

_s^t,x

, 0, q

s

)| + C(|X

_s^t,x

|

²

+ X

_s^t′,x′

2

) and finally

E

^Q

"Z

T

0

f (s, X

_s^t′,x′

, 0, q

s

) ds

#

< +∞.

⊓ ⊔ Now we will do a new restriction of the admissible control set.

Proposition 4.6 ∃T

2

∈]0, T

1

], ∃ C > ˜ 0, such that, ∀T ∈ [0, T

2

], ∀t ∈ [0, T ], ∀s ∈ [0, T ], ∀x ∈ R

^d

, Y

_s^t,x

6 C(1 + ˜ |x|

²

) and E

^Q∗

"Z

T

t

|q

^∗_u

|

²

du

#

6 C(1 + ˜ |x|

²

).

4 APPLICATION TO QUADRATIC PDES 13

Proof. We are able to use estimations of the existence Theorem 2.1 and Lemma 4.1:

−C log E

sup

06s6T

exp

C + γe

^βT

(α

^′

+ T α) X

_s^t,x

²

6 Y

_s^t,x

6 C

1 + E

sup

06s6T

X

_s^t,x

⁴

1/2

− C(1 + ˜ |x|

²

) 6 Y

_s^t,x

6 C(1 + ˜ |x|

²

).

Then, according to the representation theorem, we have Y

₀^t,x

= E

^Q∗

"

h(X

_T^t,x

) + Z

T

0

f (s, X

_s^t,x

, Y

_s^t,x

, q

_s^∗

)ds

#

> −C − α

^′

E

^Q∗

h X

_T^t,x

²

i

+ 1 2γ E

^Q∗

"Z

T

0

|q

^∗_u

|

²

du

#

− αE

^Q∗

"Z

T

0

X

_s^t,x

²

ds

#

− βE

^Q∗

"Z

T

0

Y

_s^t,x

ds

# .

But, thanks to the uniqueness, we have Y

_s^t,x

= Y

s^s,Xst,x

for s > t, so E

^Q∗

[|Y

_s^t,x

|] 6 C

1 + E

^Q∗

h

|X

_s^t,x

|

²

i . Moreover, we are allowed to use Lemma 4.3,

Y

₀^t,x

> −C(1 + |x|

²

) − C(α

^′

+ T α + βC) 1 + |x|

²

+ T Z

T

t

E

^Q∗

h

|q

_u^∗

|

²

i du

!

+ 1 2γ E

^Q∗

"Z

T

0

|q

_u^∗

|

²

du

# ,

> −C(1 + |x|

²

) + 1

2γ − CT

E

^Q∗

"Z

T

0

|q

_u^∗

|

²

du

# . We set 0 < T

2

6 T

1

such that

_2γ¹

− CT > 0 for all T ∈ [0, T

2

]. Finally,

E

^Q∗

"Z

T

0

|q

^∗_u

|

²

du

#

6 C(1 + |x|

²

) + Y

₀^t,x

6 C(1 + ˜ |x|

²

).

⊓ ⊔ According to the Proposition 4.6 we know that E

^Q∗

hR

T

t

|q

^∗_u

|

²

du i

6 C(1 + ˜ |x|

²

) so we are allowed to restrict A

0,T

: for all r > 0 we define

A

^r_0,T

= A

0,T

∩ (

(q

s

)

_s∈[0,T]

, E

^Q

"Z

T

t

|q

u

|

²

du

#

6 C(1 + ˜ r

²

) )

. (4.5)

With this new admissible control set we will prove a last inequality:

Proposition 4.7 ∃C > 0, ∀T ∈ [0, T

2

], ∀t, t

^′

∈ [0, T ], ∀x, x

^′

∈ R

^d

, ∀q ∈ A

^|x|∨|x_0,T ^′|

, ∀s ∈ [0, T ], E

^Q

X

_s^t,x

− X

_s^t′,x′

2

6 C

|x − x

^′

|

²

+ (1 + |x|

²

+ |x

^′

|

²

) |t − t

^′

|

.

4 APPLICATION TO QUADRATIC PDES 14

Proof.

E

^Q

X

_s^t,x

− X

_s^t′,x′

2

6 E

^Q

X

_s^t,x

− X

_s^t,x′

2

+ E

^Q

X

_s^t,x′

− X

_s^t′,x′

2

. We have, for s > t,

X

_s^t,x

− X

_s^t,x′

= x − x

^′

+ Z

s

t

b(u, X

_u^t,x

) − b(u, X

_u^t,x′

) du.

So,

E

^Q

X

_s^t,x

− X

_s^t,x′

2

6 C

|x − x

^′

|

²

+ Z

s

t

E

^Q

X

_u^t,x

− X

_u^t,x′

2

du

. We apply Gronwall’s Lemma to obtain that

E

^Q

X

_s^t,x

− X

_s^t,x′

2

6 C |x − x

^′

|

²

.

Now we deal with the second term. Let us assume that t 6 t

^′

. For s 6 t, X

_s^t,x′

− X

_s^t′,x′

= 0. When t 6 s 6 t

^′

, we have

X

_s^t,x′

− X

_s^t′,x′

= Z

s

t

b(u, X

_u^t,x′

)du + Z

s

t

σ(u)dW

_u^q

+ Z

s

t

σ(u)q

u

du.

So,

E

^Q

X

_s^t,x′

− X

_s^t′,x′

2

6 C



E

^Q



 Z

t^′

t

b(u, X

_u^t,x′

) du

!

2



 + Z

t^′

t

|σ(u)|

²

du + E

^Q



 Z

t^′

t

|σ(u)q

u

| du

!

2









6 C |t

^′

− t| + |t

^′

− t|

Z

t^′ t

E

^Q

X

_u^t,x′

2

du + |t

^′

− t|

Z

t^′ t

E

^Q

h

|q

u

|

²

i du

!

6 C |t

^′

− t| 1 + |x

^′

|

²

+ Z

T

0

E

^Q

h

|q

u

|

²

i du

!

6 C(1 + |x|

²

+ |x

^′

|

²

) |t

^′

− t| . Lastly, when t

^′

6 s,

X

_s^t,x′

− X

_s^t′,x′

= X

_t^t,x′ ^′

− X

_t^t′^′,x′

+ Z

s

t^′

b(u, X

_u^t,x′

) − b(u, X

_u^t′,x′

)du.

So,

E

^Q

X

_s^t,x′

− X

_s^t,x′

2

6 C(1 + |x|

²

+ |x

^′

|

²

) |t

^′

− t| + Z

s

t^′

E

^Q

X

_u^t,x′

− X

_u^t′,x′

2

du

, and according to Gronwall’s Lemma,

E

^Q

X

_s^t,x′

− X

_s^t,x′

2

6 C(1 + |x|

²

+ |x

^′

|

²

) |t

^′

− t| .

⊓ ⊔