Backward SDEs with superquadratic growth

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Backward SDEs with superquadratic growth

Freddy Delbaen, Ying Hu, Xiaobo Bao

To cite this version:

Freddy Delbaen, Ying Hu, Xiaobo Bao. Backward SDEs with superquadratic growth. Probability Theory and Related Fields, Springer Verlag, 2011, 150 (1-2), pp.145-192. �10.1007/s00440-010-0271-1�.

�hal-00362685�

Backward SDEs with superquadratic growth

Freddy Delbaen ^∗ , Ying Hu ^† and Xiaobo Bao ^‡ February 19, 2009

Abstract

In this paper, we discuss the solvability of backward stochastic differential equations (BSDEs) with superquadratic generators. We first prove that given a superquadratic generator, there exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. Finally, we prove the existence of a solution for Markovian BSDEs where the terminal value is a bounded continuous function of a forward stochastic differential equation.

1 Introduction.

Since the pioneer works on BSDEs of Bismut [2] and Pardoux-Peng [13], lots of works have been done in this area and the original Lipschitz assumption on the generator, i.e., the function g in the BSDE:

Y

_t

= ξ − Z

T

t

g(s, Y

_s

, Z

_s

) ds + Z

T

t

Z

_s

dB

_s

, 0 ≤ t ≤ T, (1.1)

∗

Department of Mathematics, ETH Z¨ urich, Switzerland. E-mail: [email protected] . Part of the research was done while this author was visiting Princeton, Vancouver and China. The hospitalities of Princeton University, UBC, Fudan University and Shandong University are greatly appreciated.

†

IRMAR, Universit´e Rennes 1, 35 042 RENNES Cedex, France. E-mail: [email protected] .

‡

Department of Mathematics, ETH Z¨ urich, Switzerland. E-mail: [email protected] .

has been weakened in many situations. Let us recall that, in the previous BSDE, we are looking for a pair of processes (Y, Z) which is required to be predictable with respect to the filtration generated by the Brownian motion B. One of the most important works in this direction is that of Kobylanski [12] concerning scalar-valued quadratic BSDEs with bounded terminal value. We should point out that quadratic BSDE means a BSDE whose generator has at most a quadratic growth with respect to the variable z. For these quadratic BSDEs, all the classical results, existence and uniqueness, comparison and stability of solutions, have been stated in [12] but with the restriction that the terminal conditions have to be bounded random variables. Recently, existence and uniqueness of solutions of quadratic BSDEs with unbounded terminal value were studied by Briand and Hu in [3, 4].

In this paper, we study the solvability of superquadratic BSDE (1.1) whose gener- ator g is superquadratic, i.e.,

|z|→+∞

lim g(z)

| z |

²

= ∞ .

We shall study this BSDE with bounded terminal value. And in addition, we suppose that g is a deterministic convex (or concave) function which is independent of y with g(0) = 0.

The first part of this paper shows the ill-posedness of these BSDEs. We first prove that given a superquadratic generator, there always exists a bounded terminal value, such that the associated BSDE does not admit any bounded solution. On the other hand, we prove that if the superquadratic BSDE admits a bounded solution, then there exist infinitely many bounded solutions for this BSDE. And finally, we show that the monotone stability, which plays a crucial role in quadratic BSDEs (see, e.g., [12, 3]), does not hold.

In the second part of this paper, we study BSDE (1.1) in the Markovian case, i.e., the terminal value

ξ = Φ(X

_T^t,x

),

where the diffusion process X is the solution to the SDE:

X

_s

= x + Z

s

t

b(r, X

_r

) dr + Z

s

t

σ dB

_r

, t ≤ s ≤ T. (1.2) It is by now well-known (see, e.g., [14, 12, 4] ) that, if g is Lipschitz or quadratic,

2

there exists a link between the solution of (1.1) and that of the following PDE:







u

_t

(t, x) +

¹₂

trace σσ

^T

u

_xx

(t, x)

+ u

_x

(t, x)b(t, x) − g( − u

_x

(t, x)σ) = 0, u(T, x) = Φ(x).

(1.3) This type of PDE (called viscous Hamilton-Jacobi equation) is already well studied when σ is the identity and g(z) = −| z |

^p

, see, e.g., Gilding et al. [10] and Ben-Artzi et al. [1]. In particular, in [10], they established the existence and uniqueness of classical solution to this PDE when σ is the identity.

We prove that in the Markovian case, the BSDE (1.1) admits a solution when Φ is bounded and continuous. Moreover, if we define

u(t, x) = Y

_t^t,x

,

then u is a continuous viscosity solution to PDE (1.3). We note that in our case, some kind of degeneracy of σ is allowed, whereas in [10] and [1], they assumed that σ is the identity.

A key idea to prove the existence in the Markovian case comes from the following a priori estimate of Z :

| Z

_t

| ≤ c || Φ ||

∞

(T − t)

⁻¹²

,

where c > 0 is a constant. We prove this inequality by using a stochastic argument based on BMO martingales and Jensen’s inequality. Note that Gilding et al. [10]

proved the same type of a priori estimate for u

_x

when σ is identity, by use of Bernstein’s method.

The paper is organized as follows: in the next section, we give some preliminaries about the connection between dynamic utility functions and BSDEs. Section 3 shows the ill-posedness in the general case. The last section is devoted to the proof of the existence of a solution in the Markovian case.

2 Dynamic Utility Functions and Backward SDEs.

Let { B

_t

, 0 ≤ t ≤ T } be a d-dimensional standard Brownian motion defined on a

probability space (Ω, F , P ). Let {F

t

, 0 ≤ t ≤ T } be the natural filtration of { B

_t

, t ∈

[0, T ] } , augmented by all P -null sets of F .

Before recalling the definition of dynamic utility functions, we need the following notations.

L

^∞

( F

T

) := { ξ : bounded and F

T

-measurable random variable } , L

²F

(0, T ; R

^m

) := { ϕ : R

^m

-valued, {F

t

}

0≤t≤T

-predictable and E h

R

T

0

| ϕ

_t

|

²

dt i

< ∞} . We identify random variables that are equal P a.s.

Definition 2.1. We call a dynamic utility function with the Fatou property any family of operators, indexed by stopping times σ

U

σ

: L

^∞

( F

^T

) → L

^∞

( F

^σ

) and satisfying:

• (A1) Positivity: U

σ

(0) = 0, U

σ

(ξ) ≥ 0 for all ξ ≥ 0.

• (A2) Concavity: U

_σ

(tξ + (1 − t)η) ≥ tU

_σ

(ξ) + (1 − t)U

_σ

(η), for all t, 0 ≤ t ≤ 1 and all ξ, η ∈ L

^∞

.

• (A3) Translability: U

_σ

(ξ + a) = U

_σ

(ξ) + a, for all a ∈ L

^∞

( F

σ

).

• (A4) Fatou property: Given a sequence (ξ

_n

)

_n≥1

, such that sup || ξ

_n

||

∞

< ∞ , then ξ

_n

↓ ξ a.s. implies U

_σ

(ξ) = lim

_n→∞

U

_σ

(ξ

_n

) a.s.

For a lower semi-continuous convex function f : R

^d

→ R

₊

∪{∞} such that f (0) = 0 and for ξ ∈ L

^∞

( F

T

), we define

U

σ

(ξ) = ess.inf

E

Q

h ξ +

Z

_T

σ

f (q

u

) du F

^σ

i

Q ∼ P

, (2.1)

where σ ∈ [0, T ] is a stopping time and the density process E

_P

[

^dQ_dP

|F

t

] = E (q · B)

_t

= exp( R

t

0

q

_u

dB

_u

−

¹₂

R

t

0

| q

_u

|

²

du). It is easy to prove that U is a dynamic utility function. As shown by Delbaen-Peng-Rosazza Gianin [7], U is time consistent and all time consistent dynamic utility functions are of a similar form.

Set C

₀

(Q) = E

_Q

R

_T

0

f (q

_u

) du

and P = { Q | Q ≪ P } . The utility function U

₀

can be defined by P .

Lemma 2.1. For any ξ ∈ L

^∞

( F

T

), U

₀

(ξ) = inf

E

_Q

h

ξ + Z

_T

0

f (q

_u

) du i

Q ∈ P

. (2.2)

4

Proof. For any Q ∈ P with L

_t

= E

_P

_dQ

dP

F

t

= E (q · B )

_t

, using Itˆo’s lemma we get that the density process of Q

_λ

, λQ + (1 − λ)P is E (q

_λ

· B) with

q

_λ

(t) = λL

_t

q

_t

λL

_t

+ (1 − λ) 1

_{t≤τ}

, where τ = inf { t ∈ [0, T ] | L

_t

= 0 } ∧ T is a stopping time.

Then from the convexity of f :

C

₀

(Q

_λ

) = E

_Qλ

h Z

_T

0

f(q

_λ

(u)) du i

≤ E

_Qλ

h Z

τ 0

λL

t

λL

_t

+ (1 − λ) f (q(t)) dt i

= E

P

h Z

τ 0

λL

t

f (q(t)) dt i

= E

_Q

h Z

τ 0

λf(q(t)) dt i

= λC

₀

(Q),

where λ ∈ [0, 1], we deduce that lim

_λ→1

C

₀

(Q

_λ

) ≤ C

₀

(Q).

Notice that for any λ ∈ [0, 1), Q

_λ

is equivalent to P. Thus inf

E

Q

h ξ +

Z

T 0

f (q

u

) du i

Q ∈ P

≥ inf

E

Q

h ξ +

Z

T 0

f (q

u

) du i

Q ∼ P

. Since { Q | Q ∼ P } ⊆ { Q | Q ≪ P } , we have

U

₀

(ξ) = inf

E

_Q

h ξ +

Z

T 0

f (q

_u

) du i

Q ∈ P

.

Remark 2.1. The function C

₀

: P → R

₊

is lower semi-continuous (just use Fatou’s lemma) and convex. A duality argument then shows that for Q ∈ P

C

0

(Q) = sup n

E

Q

[ − ξ]

U

0

(ξ) ≥ 0 o .

In other words C

0

is the minimal penalty function as defined in F¨ ollmer-Schied [9].

We also remark that for Q ≪ P, the previous reasoning and the lower semi-continuity imply C

₀

(Q

_λ

) → C

₀

(Q).

However, for a stopping time σ, U

_σ

(ξ) cannot be the essential infimum over P P

a.s. Instead, by the similar technique as that in Lemma 2.1, we have:

Remark 2.2. For any measure Q

^∗

∈ P and ξ ∈ L

^∞

( F

T

), U

σ

(ξ) = ess.inf

E

Q

h ξ +

Z

T σ

f (q

u

) du F

^σ

i

Q ∈ P , Q ∼ P on F

^σ

, P a.s., (2.3) for any stopping time σ ∈ [0, T ] and,

U

_σ

(ξ) = ess.inf

E

_Q

h ξ +

Z

_T

σ

f (q

_u

) du F

σ

i

Q ∈ P , Q

^∗

≪ Q

, Q

^∗

a.s. (2.4) Proposition 2.1. For any ξ ∈ L

^∞

( F

^T

), the dynamic utility function U defined by (2.1) has the following properties:

1) For all Q ≪ P , we have that U

t

(ξ) + R

τ∧t

0

f(q

u

) du is a Q-submartingale where τ = inf { t ∈ [0, T ] | L

_t

= 0 } .

2) If there is a probability measure Q ≪ P with U

0

(ξ) = E

Q

[ξ + R

τ

0

f (q

u

) du], then U

_t

(ξ) + R

_τ∧t

0

f(q

_u

) du is a Q-martingale.

Proof. 1) For any s < t, it follows from Remark 2.2 that for any Q ≪ P , E

_Q

U

_t

(ξ) + Z

τ∧t

τ∧s

f (q

_u

) du F

s

= E

Q

ess.inf

_Q^′_∼P

n E

_Q^′

ξ + Z

T

t

f (q

_u^′

) du F

^t

o

+ Z

τ∧t

τ∧s

f (q

u

) du F

^s

≥ ess.inf

E

_Q^′′

h ξ +

Z

_T

τ∧s

f (q

_u^′′

) du F

^s

i

q

_u^′′

= q

^′_u

+ 1

_{{τ∧s≤u≤t}}

(q

u

− q

^′_u

)

≥ ess.inf

E

_Q^′′

h ξ +

Z

_T

τ∧s

f (q

_u^′′

) du F

s

i

Q

^′′

∈ P , Q ≪ Q

^′′

≥ U

_s

(ξ), Q a.s.

Hence,

U

_s

(ξ) + Z

_τ∧s

0

f (q

_u

) du ≤ E

_Q

h

U

_t

(ξ) + Z

_τ∧t

0

f (q

_u

) du F

s

i

, Q a.s.

Therefore, we have U

_t

(ξ) + R

_τ∧t

0

f (q

_u

) du is a Q-submartingale.

2) As Q is absolutely continuous with respect to P , it follows from the result we just proved, that

U

₀

(ξ) ≤ E

_Q

h

U

_t

(ξ) + Z

τ∧t

0

f(q

_u

) du i

. (2.5)

Combining U

₀

(ξ) = E

_Q

[ξ + R

τ

0

f (q

_u

) du] with the inequality (2.5), we have E

_Q

h

ξ + Z

_τ

τ∧t

f (q

_u

) du i

≤ E

_Q

[U

_t

(ξ)].

This implies that

U

_t

(ξ) = E

_Q

h ξ +

Z

_τ

τ∧t

f (q

_u

) du F

t

i

, Q a.s. (2.6)

Thus U

_t

(ξ) + R

_τ∧t

0

f (q

_u

) du = E

_Q

[ξ + R

_τ

0

f (q

_u

) du |F

t

] is a Q- martingale.

6

Remark 2.3. In the above proposition, τ can be replaced by T since Q[τ = T ] = 1.

Remark 2.4. In particular, we have that the process { U

_t

(ξ), t ∈ [0, T ] } is a P- submartingale. Thus there exists a c` adl` ag version.

For any ξ ∈ L

^∞

( F

T

), | U

_t

(ξ) | ≤k ξ k

∞

. So applying the Doob-Meyer decomposition theorem, there exists a unique nondecreasing predictable process { A

_t

}

0≤t≤T

with A

₀

= 0 and a continuous martingale { M

_t

}

0≤t≤T

with M

₀

= 0, such that

U

_t

(ξ) = U

₀

(ξ) + A

_t

− M

_t

. (2.7) Lemma 2.2. For all ξ ∈ L

^∞

( F

T

), the martingale part { M

_t

}

0≤t≤T

of U (ξ) induced by the Doob-Meyer decomposition theorem is a BMO-martingale.

Proof. For a given ξ ∈ L

^∞

( F

^T

), | U

t

(ξ) | ≤k ξ k

^∞

. Then applying Itˆo’s formula to (U

_t

(ξ)+ k ξ k

∞

)

²

, we get

(U

_t

(ξ)+ k ξ k

∞

)

²

+ Z

_T

t

d h M, M i

s

= (ξ+ k ξ k

^∞

)

²

− 2 Z

T

t

(U

s−

(ξ)+ k ξ k

^∞

) dA

s

− Z

T

t

dK

s

+2 Z

_T

t

(U

_s−

(ξ)+ k ξ k

∞

) dM

_s

, where

K

_s

:= X

r≤s

n

(U

_r

(ξ)+ k ξ k

∞

)

²

− (U

_r−

(ξ)+ k ξ k

∞

)

²

− 2(U

_r−

(ξ)+ k ξ k

∞

)(U

_r

(ξ) − U

_r−

(ξ)) o

= X

r≤s

U

r

(ξ) − U

r−

(ξ)

2

is an increasing process. Hence, (U

_t

(ξ)+ k ξ k

∞

)

²

+

Z

_T

t

d h M, M i

s

≤ (ξ+ k ξ k

∞

)

²

+ 2 Z

_T

t

(U

_s−

(ξ)+ k ξ k

∞

) dM

_s

from which we deduce, for any stopping time 0 ≤ σ ≤ T ,

E Z

_T

σ

d h M, M i

t

F

σ

≤ 4 k ξ k

²∞

.

Therefore, k M k

^{BM O}2

≤ 2 k ξ k

^∞

which completes the proof.

The predictable representation theorem implies that there exists a predictable pro- cess Z ∈ L

²_F

(0, T ; R

^d

) such that

M

_t

= Z

t

0

Z

_s

dB

_s

. (2.8)

So we get

U

_t

(ξ) = U

₀

(ξ) + A

_t

− Z

_t

0

Z

_s

dB

_s

. (2.9)

If g : R

^d

→ R

₊

∪ {∞} is the Fenchel-Legendre transform of f : g(z) = sup

x∈R^d

(zx − f (x)), then g is also convex and g(0) = 0.

We make the standard assumption such that both f and g are finite. We do not treat the case where f or g can take the value + ∞ . This case is similar and only requires cosmetic changes. To make the paper simpler, we dropped this more general case.

Theorem 2.1. Let U be the dynamic utility function defined by (2.1) and let U

₀

(ξ) + A

_t

− R

t

0

Z

_u

dB

_u

be its decomposition.

1) We have

dA

_t

≥ g(Z

_t

) dt, P a.s. (2.10) 2) Suppose that for some ξ ∈ L

^∞

( F

^T

) there is a probability measure Q

^∗

∼ P with U

₀

(ξ) = E

_Q^∗

[ξ + R

_T

0

f (q

_u^∗

) du], then dA

_t

= g(Z

_t

) dt and U

t

(ξ) = U

0

(ξ) +

Z

t 0

g(Z

u

) du − Z

t

0

Z

u

dB

u

. (2.11)

Proof. 1) For ξ ∈ L

^∞

( F

^T

) and any Q ∼ P , it follows from the decomposition that dU

_t

(ξ) + f (q

_t

) dt = dA

_t

− Z

_t

dB

_t

+ f (q

_t

) dt (2.12)

= dA

_t

− Z

_t

q

_t

dt + f (q

_t

) dt − Z

_t

dB

_t^Q

, (2.13) where B

^Q

is a Q − Brownian motion. This implies that dA

_t

− Z

_t

q

_t

dt + f (q

_t

) dt defines a non-negative measure since U

_t

(ξ) + R

_τ∧t

0

f (q

_u

) du is a Q-submartingale for any Q ∼ P . Hence

dA

_t

≥ Z

_t

q

_t

dt − f (q

_t

) dt.

8

By taking q

ⁿ

= g

^′

(Z)1

_{|Z|≤n}

in the above inequality and by letting n tend to infinity, we get dA

_t

≥ g(Z

_t

) dt.

2) If for ξ, there is a measure Q

^∗

∼ P with U

₀

(ξ) = E

_Q^∗

[ξ + R

T

0

f (q

^∗_u

) du], then it follows from Proposition 2.1 that U

_t

(ξ) + R

_t

0

f (q

_u^∗

) du is a Q

^∗

-martingale. Thus applying (2.13) with Q

^∗

, we get

dA

_t

= (Z

_t

q

_t^∗

− f (q

_t^∗

)) dt Q

^∗

a.s.

Since Q

^∗

∼ P, we have

dA

_t

= (Z

_t

q

_t^∗

− f (q

^∗_t

)) dt P a.s. (2.14) Finally combining (2.10) and (2.14) with the definition of g, it follows that

g(Z

_t

) dt ≥ (Z

_t

q

^∗_t

− f (q

^∗_t

)) dt = dA

_t

≥ g(Z

_t

) dt P a.s.

In general we can decompose A further and get:

Corollary 2.1. For any ξ ∈ L

^∞

( F

T

), there exists an increasing predictable process { C

_t

}

0≤t≤T

with C

₀

= 0 such that

U

_t

(ξ) = U

₀

(ξ) + Z

_t

0

g(Z

_u

) du − Z

_t

0

Z

_u

dB

_u

+ C

_t

. (2.15) Our main result is the following.

Theorem 2.2. Let U be the dynamic utility function defined by (2.1). Then the following are equivalent:

1. lim

_|x|→∞^f_|x|^(x)2

> 0;

2. lim

_|z|→∞^g(z)_|z|2

< ∞ ;

3. For all k > 0, the set { Q | C

₀

(Q) ≤ k } is weakly compact;

4. For all ξ ∈ L

^∞

( F

T

), there exists a measure Q ≪ P such that U

₀

(ξ) = E

_Q

h ξ + R

T

0

f (q

u

) du i

;

5. For all ξ ∈ L

^∞

( F

T

), there exists a measure Q ∼ P such that U

₀

(ξ) = E

_Q

h ξ + R

_T

0

f (q

_u

) du i

;

6. For all ξ ∈ L

^∞

( F

T

), the BSDE dY

_t

= g(Z

_t

) dt − Z

_t

dB

_t

has a unique bounded

solution with Y

_T

= ξ.

7. U

₀

is strictly monotone.

Proof. 1 ⇔ 2: Point 1 implies that there exist positive constants a, b ∈ R

₊

such that f (x) ≥ a | x |

²

− b. We then get

g(z) = sup

x∈R^d

(zx − f (x)) ≤ sup

x∈R^d

(zx − a | x |

²

+ b) ≤ 1

4a | z |

²

+ b

which shows that lim

_z→∞^g(z)_|z|2

< ∞ . The proof of the implication 2 ⇒ 1 is similar.

1 ⇒ 3: It suffices to verify that for any k > 0, n

dQ dP

C

₀

(Q) = E

_Q

h R

T

0

f (q

_u

) du i

≤ k o is uniformly integrable. The Dunford-Pettis theorem then shows that the set is weakly compact.

Since f (x) ≥ a | x |

²

− b, we get k ≥ E

_Q

Z

_T

0

f (q

_u

) du

≥ aE

_Q

Z

_T

0

| q

_u

|

²

du

− b.

Therefore,

1 2 E

Q

Z

T

0

| q

u

|

²

du

≤ α,

where α =

^k+b_2a

is a positive constant independent of Q. It follows from 1

2 E

_Q

Z

T

0

| q

_u

|

²

du

= E

_Q

Z

T

0

q

_u

dB

_u^Q

+ 1 2

Z

T

0

| q

_u

|

²

du

= E

_Q

Z

T

0

q

_u

dB

_u

− 1 2

Z

_T

0

| q

_u

|

²

du

= E

_Q

log dQ dP

that for any k > 0, dQ

dP

E

_Q

Z

T

0

f (q

_u

) du

≤ k

⊆ dQ

dP

E

_P

dQ

dP log dQ dP

≤ α

. (2.16)

From the de la Vall´ee Poussin theorem, we conclude that dQ

dP

E

Q

Z

T 0

f (q

u

) du

≤ k

is uniformly integrable.

3 ⇒ 1 We prove it by the contradiction. Suppose lim

_|x|→∞^f(x)_|x|2

= 0, then there exists a sequence { x

_n

}

^∞n=0

such that lim

_n→∞

| x

_n

| = ∞ and lim

_n→∞^f_|x^(xn)

n|²

= 0. Put q

_n

= x

_n

1

_[0,δn∧T]

where δ

_n

= 1 . q

f(xn)

|xn|²

| x

_n

|

²

. It follows from

C

0

(Q

n

) = E

Qn

Z

T 0

f (q

n

(u)) du

≤ s

f (x

n

)

| x

_n

|

²

→ 0, (2.17)

10

that for all k > 0, there exists N > 0 such that the sequence {

^dQ_dPⁿ

}

^∞_n=N

⊆ {

^dQ_dP

| C

₀

(Q) ≤ k } . Furthermore, we have

Z

T

0

| q

_n

|

²

(u) du = 1 s

f (x

n

)

| x

_n

|

²

!

∧ x

²_n

T

→ ∞ , (2.18)

which shows that

^dQ_dPⁿ

= E (q

_n

· B)

_T

→ 0, a.s. as n → ∞ . Thus {

^dQ_dPⁿ

}

^∞n=N

is not uniformly integrable.

3 ⇔ 4: It is a conclusion induced by the James’ theorem as shown in Jouini- Schachermayer-Touzi’s work [11].

4 ⇔ 5: It is obvious that point 5 implies point 4. For the proof of the inverse implication, we use the fact that condition 4 is equivalent to condition 2. In this case, by convexity, there exists a positive constant c such that | g

^′

(z) | ≤ c( | z | + 1). For any ξ ∈ L

^∞

( F

T

), there is a measure Q ≪ P such that U

₀

(ξ) = E

_Q

[ξ + R

T

0

f(q

_u

) du], then, by Proposition 2.1, U

t

(ξ) + R

τ∧t

0

f (q

u

) du is a Q-martingale where τ = inf { t ∈ [0, T ] | E (q · B)

_t

= 0 } ∧ T . It follows from (2.13) that

dA

_t

= (Z

_t

q

_t

− f (q

_t

)) dt m ⊗ Q a.s. on [0, τ ],

where m is the Lebesgue measure on [0, T ]. Since dA

_t

≥ g(Z

_t

) dt, m ⊗ Q a.s., we get g(Z

_t

) = Z

_t

q

_t

− f (q

_t

) m ⊗ Q a.s.,

which implies q

_t

= g

^′

(Z

_t

) on [0, τ ]. We then have Z

_τ

0

| q

_u

|

²

du = Z

_τ

0

(g

^′

(Z

_u

))

²

du

≤ c

²

Z

τ

0

(1 + | Z

_u

| )

²

du < ∞ , which means P n

dQ dP

= 0 o

= P n R

_τ

0

| q

_u

|

²

du = ∞ o

= 0. Hence Q ∼ P .

5 ⇒ 6: For a given ξ ∈ L

^∞

( F

T

), if there exists a measure Q ∼ P such that U

₀

(ξ) = E

_Q

h

ξ + R

_T

0

f (q

_u

) du i

, it follows from Lemma 2.1 that { U

_t

, Z

_t

}

0≤t≤T

is a solution of the following BSDE:



 

 

 

 

dY

_t

= g(z

_t

) dt − z

_t

dB

_t

, 0 ≤ t ≤ T ; Y

_T

= ξ, ξ ∈ L

^∞

( F

T

);

Y is bounded ,

(2.19)

where E h R

T

0

| z

t

|

²

dt i

< ∞ and E h R

T

0

g(z

t

) dt i

< ∞ . Since, as we have proved above, condition 5 implies lim

_z→∞^g(z)_|z|2

< ∞ , the BSDE has a unique bounded solution ac- cording to Kobylanski [12].

6 ⇒ 2 We will prove this in the next section. See Theorem 3.1.

5 ⇒ 7 For any ξ ∈ L

^∞

( F

T

), there exists an equivalent measure Q ∼ P such that U

₀

(ξ) = E

_Q

[ξ + R

_T

0

f (q

_u

) du] with

^dQ_dP

= E (q · B ).

Suppose that U

0

(η) = U

0

(ξ) for some η ∈ L

^∞

( F

^T

) with η ≤ ξ, P a.s. Since U

₀

(η) ≤ E

_Q

η +

Z

T 0

f (q

_u

) du

≤ E

_Q

ξ + Z

T

0

f (q

_u

) du

= U

₀

(ξ),

we have E

_Q

[ξ − η] = 0, hence ξ = η, Q a.s. Thus ξ = η, P a.s. and U

₀

is strictly monotone.

7 ⇒ 2 See Remark 3.2, Remark 3.5 or Example 3.1.

We have proved that in the case when the generator g is at most quadratic, the dynamic utility function U is the solution of BSDE (2.19). In general, however, we have the following inequality.

Lemma 2.3. For any ξ ∈ L

^∞

( F

^T

), if BSDE (2.19) has a bounded solution Y , then we have U (ξ) ≥ Y .

Proof. Y is bounded. The following calculation is therefore justified:

E

_Q

ξ + Z

_T

t

f (q

_u

) du F

t

= Y

_t

+ E

_Q

Z

T

t

g(Z

_u

) du − Z

_T

t

Z

_u

dB

_u

+ Z

_T

t

f(q

_u

) du F

t

= Y

_t

+ E

_Q

Z

T

t

[g(Z

_u

) − Z

_u

q

_u

+ f (q

_u

)] du F

t

≥ Y

_t

, for any Q ∼ P with E

_Q

Z

T

0

f (q

_u

) du

< ∞ .

3 Backward SDEs with superquadratic growth.

In this section, we discuss the following BSDE(g, ξ):







dY

_t

= g(Z

_t

) dt − Z

_t

dB

_t

; Y

_T

= ξ, ξ ∈ L

^∞

( F

T

),

(3.1)

12

where g : R

^d

→ R

₊

∪{ + ∞} is convex with g(0) = 0 and superquadratic lim

_|z|→∞^g(z)_|z|2

=

∞ . A pair of predictable processes (Y , Z ) is called a bounded solution to BSDE (3.1) if

Y : Ω × [0, T ] → R is bounded and Z : Ω × [0, T ] → R

^d

is such that E h

R

_T

0

g(Z

_t

) dt i

< ∞ .

Here for simplicity, we consider the BSDE with d = 1. However, the results remain valid for d > 1.

3.1 Non-existence of the solution

Different from the BSDEs with at most quadratic growth, the solution to the BSDE with super-quadratic growth does not always exist.

Theorem 3.1. (Non-existence) There exists η ∈ L

^∞

( F

T

) such that BSDE(3.1) with superquadratic growth has no bounded solution.

Proof. The proof is divided into 4 steps.

Step 1. We construct a pair of processes (X, Z), a measure Q as well as a bounded random variable ξ.

Since lim

_|z|→∞^g(z)_|z|2

= ∞ , there exists a sequence { z

_k

}

^∞k=1

such that lim

_k→∞

| z

_k

| = ∞ and g(z

_k

) ≥ k | z

_k

|

²

. Without loss of generality, we suppose z

_k

> 0. The other case is left to the reader. Thus we have

g

^′

(z

_k

) ≥ g(z

_k

)

z

_k

≥ k · z

_k

. (3.2)

We put Z

_u

, P

_∞

n=1

z

_n

1

_[^P

k<nδk,P

k≤nδk)

(u) where δ

_k

= 1

αz

_k

g

^′

(z

_k

)k

²

and we set

α =

∞

X

k=1

1

T z

_k

g

^′

(z

_k

)k

²

< ∞ , in order to have

X

k≥1

δ

_k

= X

k≥1

1

αz

_k

g

^′

(z

_k

)k

²

= T.

Then from (3.2), we have Z

_T

0

g(Z

_u

) du = X

k≥1

g(z

_k

)δ

_k

≤ X

k≥1

z

_k

g

^′

(z

_k

)δ

_k

= X

k≥1

1

αk

²

< ∞ ,

Z

T

0

| Z

_u

|

²

du = X

k≥1

| z

_k

|

²

δ

_k

≤ X

k≥1

z

_k

g

^′

(z

_k

)δ

_k

1 k = X

k≥1

1

αk

³

< ∞ . Let q

_t

= g

^′

(Z

_t

). It follows from

Z

_T

0

| q

_u

|

²

du = X

k≥1

(g

^′

(z

_k

))

²

δ

_k

≥ X

k≥1

kg

^′

(z

_k

)z

_k

δ

_k

= X

k≥1

1

αk = + ∞ , that lim

_t→T

E (q · B)

_t

= 0 and E (q · B)

_t

> 0, P a.s. for any t < T.

Let X

_t

= R

t

0

g(Z

_u

) du − R

t

0

Z

_u

dB

_u

. We stop X at a random time σ

σ , inf { t ∈ [0, T ] | E (q · B )

t

≥ n } ∧ inf { t ∈ [0, T ] | | X

t

| ≥ n } ∧ T (3.3) where n is a positive constant which is sufficiently large to ensure that P (σ = T ) > 0.

We then set a measure Q

^∗

with E

_P

h

dQ^∗ dP

F

t

i

= E (q

^∗

· B )

_t

and q

^∗_t

= g

^′

(Z

_t

)1

_{t≤σ}

. We define ξ = X

_σ

∈ L

^∞

( F

T

).

Step 2. The measure Q

^∗

≪ P but it is not equivalent to P . Set A

₁

= { σ = T } . Then

Q

^∗

(A

₁

) = Z

A1

E (q · B)

_σ

dP = Z

A1

E (q · B )

_T

dP = 0 while P (A

₁

) > 0. Thus we have Q

^∗

≁ P and Q

^∗

≪ P . However, Q

^∗

N

m ∼ P N m where m is the Lebesgue measure since Q

^∗

∼ P on F

t

for all t < T . Clearly (X

_t^σ

, Z

_t

1

_{t≤σ}

)

_0≤t≤T

is a bounded solution of BSDE (g, ξ) where X

_t^σ

= X

_σ∧t

.

Step 3. In this step we prove that the dynamic utility function U (ξ) is the bounded solution of BSDE (g, ξ) (3.1) and U

_t

(ξ) = E

_Q^∗

h

ξ + R

T

t

f (q

_u^∗

) du F

t

i for any t < T . As X

^σ

is a bounded solution of BSDE (g, ξ), we get

U

_t

(ξ) ≤ E

_Q^∗

h ξ +

Z

_T

t

f (q

^∗_u

) du F

t

i

= E

_Q^∗

h ξ +

Z

σ t∧σ

f (q

^∗_u

) du F

t

i

= X

_t^σ

+ E

_Q^∗

h Z

_σ

t∧σ

(f (q

_u^∗

) + g(Z

_u

)) du − Z

_σ

t∧σ

Z

_u

dB

_u

F

t

i

= X

_t^σ

+ E

_Q^∗

h Z

σ t∧σ

[f (q

^∗_u

) + g(Z

_u

) − Z

_u

q

_u^∗

] du F

t

i

= X

_t^σ

, Q

^∗

a.s. (3.4)

hence P a.s. because Q

^∗

∼ P on F

t

for t < T . Combining Lemma 2.3 with inequality (3.4), we deduce that

U

_t

(ξ) = E

_Q^∗

h ξ +

Z

_T

t

f (q

^∗_u

) du F

t

i

= X

_t^σ

, P a.s. for all t ∈ [0, T ). (3.5)

14

Set η = ξ + h where h ∈ L

^∞₊

( F

T

), P [h > 0] > 0 and h · E (q

^∗

· B )

_σ

= 0.

Step 4. We show that U

t

(ξ) = U

t

(η), P a.s. for any t < T and hence BSDE (g,η) has no solution.

It follows from η = ξ, Q

^∗

-a.s. that U

_t

(η) ≤ E

_Q^∗

h

η + Z

T

t

f (q

^∗_u

) du F

t

i

= E

_Q^∗

h ξ +

Z

_T

t

f (q

_u^∗

) du F

t

i

= U

_t

(ξ) for any t < T.

Notice that U is monotone, i.e., U

_t

(ξ) ≤ U

_t

(η), and so we have U

_t

(ξ) = U

_t

(η), P a.s. for any t < T .

Suppose Y is a bounded solution of BSDE (g, η), then we have for t < T , X

_t^σ

= U

t

(ξ) = U

t

(η) ≥ Y

t

,

and hence

η = Y

_T

= lim

t→T

Y

_t

≤ lim

t→T

X

_t^σ

= X

_T^σ

= ξ, P a.s.,

a contradiction to the fact that P [η > ξ] > 0. Therefore, BSDE (g, η) has no solution.

Remark 3.1. From this theorem, together with what we have proved in Theorem 2.2 we get that BSDE (g, ξ) has a solution for all ξ ∈ L

^∞

( F

T

) if and only if g is at most quadratic.

Remark 3.2. From the proof, we get η ≥ ξ with P (η > ξ) > 0 and U

₀

(ξ) = U

₀

(η).

Thus the utility function U

₀

is NOT strictly monotone when lim

_|x|→∞^f(x)_|x|2

= 0.

Although the BSDE (g, ξ) (3.1) does not always have a solution, in the following case it has.

Definition 3.1. We say that a random variable ξ ∈ L

^∞

( F

T

) is minimal if η ≤ ξ and P [η < ξ] > 0 imply U

₀

(η) < U

₀

(ξ).

Theorem 3.2. Let ξ ∈ L

^∞

( F

T

) be minimal. Then U (ξ) is a solution of BSDE (g, ξ).

Proof. We prove it by contradiction. Let ξ ∈ L

^∞

( F

T

) be minimal and suppose

U (ξ) is not a solution of BSDE (g, ξ). Then it follows from Corollary 2.1 that there

exists an increasing process C with C

₀

= 0 such that P [C

_T

> 0] > 0 and U

_t

(ξ) = ξ −

Z

_T

t

g(Z

_u

) du + Z

_T

t

Z

_u

dB

_u

− C

_T

+ C

_t

. (3.6) Define τ := inf { t ∈ [0, T ] | C

t

≥ k } ∧ T , where k > 0 is such that P[C

τ

> 0] > 0.

Since C may have jumps, C

_τ

can be unbounded. However, τ is predictable so there exists { τ

_n

}

^∞n=1

such that τ

_n

↑ τ and τ

_n

< τ on { τ > 0 } . It follows that C

_τn

≤ k and P [C

_τn

> 0] > 0 for n big enough. Denote by σ a stopping time τ

_n

for n big enough, then we have

U

_t

(ξ) − C

_t

= U

_σ

(ξ) − C

_σ

− Z

σ

t

g(Z

_u

) du + Z

σ

t

Z

_u

dB

_u

,

which implies that (U

_t∧σ

(ξ) − C

_t∧σ

, Z

_t

1

_{t≤σ}

)

_0≤t≤T

is a solution of BSDE (g, U

_σ

(ξ) − C

σ

). Thus by Lemma 2.3, we deduce

U

₀

(ξ) = U

₀

(ξ) − C

₀

≤ U

₀

(U

_σ

(ξ) − C

_σ

).

On the other hand, it is clear that U

₀

(ξ) ≥ U

₀

(U

_σ

(ξ) − C

_σ

). Therefore, we have U

₀

(ξ) = U

₀

(U

_σ

(ξ) − C

_σ

).

It follows from the above equality, the translability and the time-consistency of the dynamic utility function that

U

₀

(ξ) = U

₀

(U

_σ

(ξ) − C

_σ

) = U

₀

(U

_σ

(ξ − C

_σ

)) = U

₀

(ξ − C

_σ

).

This is a contradiction to the fact that ξ is minimal.

Remark 3.3. For g with at most quadratic growth lim

_|z|→∞^g(z)_|z|2

< ∞ , it follows from Theorem 2.2 that ξ is minimal for all ξ ∈ L

^∞

( F

^T

).

If g is superquadratic, there exists a bounded random variable ζ such that U (ζ) is a solution of BSDE (g, ζ) and ζ is not minimal. See Example 3.1.

3.2 Non-uniqueness of the Solution

In this subsection, we shall prove that if the BSDE has a bounded solution, the bounded solution is not unique. The main reason is that the generator g is superquadratic which makes R

t

0

g(Z

_r

)dr grow much faster than R

t

0

Z

_r

dB

_r

. Following this observation, we can construct other solutions.

16

Theorem 3.3. (Non-uniqueness) If the BSDE (g, ξ) with superquadratic growth has a bounded solution Y for a ξ ∈ L

^∞

( F

^T

), then for each y < Y

0

, there are infinitely many bounded solutions { X

_t

}

0≤t≤T

with X

₀

= y.

Proof. Suppose (Y, Z) is a bounded solution of BSDE (g, ξ). Divide the time interval [0, T ] into [T (1 − 2

⁻ⁿ

), T (1 − 2

⁻ⁿ⁻¹

)), where n = 0, 1, 2, ... and denote α

_n

= T (1 − 2

⁻ⁿ

).

Suppose the new solution (X, Z

^′

) has been constructed on [0, α

_n

] with X

₀

= y where y < Y

₀

such that X

_αn

≤ Y

_αn

P a.s. Let us construct (X, Z

^′

) on the time interval [α

_n

, α

_n+1

).

Our idea is the following. Since g is superquadratic, we can construct a process X

_αn

+V

_t

, t ∈ [α

_n

, α

_n+1

) such that lim

_t→αn+1

V

_t

= + ∞ , P a.s. and for any 0 < ε < 1, V

_t

exceeds downwards − 2

⁻ⁿ⁻¹

ε with a very small probability. The fact that the solution Y is bounded implies that it is touched by the process X

_αn

+ V

_t

because X

_αn

≤ Y

_αn

. We then get a new solution X

t

on this time interval [α

n

, α

n+1

] by stopping X

αn

+ V

t

when it reaches Y .

First, let us construct the process V

t

.

It follows from lim

_z→∞^g(z)_|z|2

= ∞ that there exists a sequence { x

_k

}

^∞k=0

such that for any k ≥ 0,

1. g(x

_k

) ≥ 4

ⁿ

x

²_k

;

2. x

²_k

≥

_(θ^k_−θk+1¹ )θ^kδn

where θ ∈ (0, 1) is a constant and δ

_n

= α

_n+1

− α

_n

= 2

⁻ⁿ⁻¹

T.

Set b

_t

= P

_∞

k=0

x

_k

1

_[αn+1−θkδn, αn+1−θ^k+1δn)

(t) and V

_t

= R

_t

αn

g(b

_u

) du − R

_t

αn

b

_u

dB

_u

for any t ∈ [α

n

, α

n+1

).

We then have for t ∈ [α

_n+1

− θ

^N+1

δ

_n

, α

_n+1

− θ

^N+2

δ

_n

), Z

_t

αn

b

²_u

du ≥

N

X

k=0

x

²_k

(θ

^k

− θ

^k+1

)δ

_n

≥

N

X

k=0

1

θ

^k

, (3.7)

Z

_t

αn

g(b

_u

) du ≥ 4

ⁿ

Z

_t

αn

b

²_u

du ≥ 4

ⁿ

N

X

k=0

1

θ

^k

. (3.8)

Thus lim

_t→αn+1

R

t

αn

b

²_u

du = ∞ and lim

_t→αn+1

R

t

αn

g(b

_u

) du = ∞ . Step 1. We have lim

t→αn+1

V

t

= + ∞ P a.s.

Define φ(t) = R

t

αn

b

²_u

du for t ∈ [α

_n

, α

_n+1

). Then φ is strictly increasing with φ(α

n

) = 0 and

t→α

lim

n+1

φ(t) = + ∞ .

Setting B

_t^∗

, R

φ⁻¹(t)

αn

b

_u

dB

_u

, we get a time changed Brownian motion with respect to the filtration {F

^B∗

} . It follows from the construction of V that

V

_t

≥ 4

ⁿ

φ(t) − B

_φ(t)^∗

= φ(t)

4

ⁿ

− B

_φ(t)^∗

φ(t)

, which implies that

t→α

lim

n+1

V

t

= + ∞ , P a.s. (3.9)

since

t→α

lim

n+1

B

_φ(t)^∗

φ(t) = 0, P a.s.

Now we estimate the probability that V

_t

reaches a small negative number − 2

⁻ⁿ⁻¹

ε.

Step 2. Calculate the probability

P( { ω ∈ Ω | ∃ t ∈ [α

_n

, α

_n+1

) such that V

_t

(ω) < − 2

⁻ⁿ⁻¹

ε } ).

Applying the submartingale inequality, we deduce that

P ( { ω ∈ Ω | ∃ t ∈ [α

n

, α

n+1

) such that V

t

(ω) < − 2

⁻ⁿ⁻¹

ε } )

= P ( { ω ∈ Ω | ∃ t ∈ [α

_n

, α

_n+1

) such that 4

ⁿ

φ(t) − B

_φ(t)^∗

< − 2

⁻ⁿ⁻¹

ε } )

= P ( { ω ∈ Ω | ∃ s ∈ [0, ∞ ) such that 4

ⁿ

s − B

_s^∗

< − 2

⁻ⁿ⁻¹

ε } )

≤ exp {− 2

ⁿ

ε } . (3.10)

Step 3. Construct the new solution (X

_t

, Z

_t^′

) for all t ∈ [α

_n

, α

_n+1

].

Define

τ

₁

, inf { t ≥ α

_n

| V

_t

= − 2

⁻ⁿ⁻¹

ε } ∧ α

_n+1

and

τ

₂

, inf { t ≥ α

_n

| X

_αn

+ V

_t

≥ Y

_t

} ∧ α

_n+1

which are the stopping times when the process X

_αn

+ V

_t

touches X

_αn

− 2

⁻ⁿ⁻¹

ε and Y

_t

respectively. It follows from lim

t→αn+1

V

t

= + ∞ P a.s. that P [τ

2

< α

n+1

] = 1. Define

τ

₃

, inf { t ≥ τ

₁

| X

_αn

− 2

⁻ⁿ⁻¹

ε = Y

_t

} ∧ α

_n+1

. Now we have three cases



 

 

 

 

τ

1

< τ

2

, τ

3

< α

n+1

, put Z

_t^′

(ω) = b

t

1

_{t≤τ1}

+ Z

t

1

_{t>τ3}

; τ

₁

< τ

₂

, τ

₃

= α

_n+1

, put Z

_t^′

(ω) = b

_t

1

_{t≤τ1}

;

τ

1

≥ τ

2

, put Z

_t^′

(ω) = b

t

1

_{t≤τ2}

+ Z

t

1

_{t>τ2}

,

(3.11)

18

where (Y, Z) is the original bounded solution of the BSDE (g, ξ).

Thus we get

Z

_t^′

= 1

_{{τ1<τ2,τ3<αn+1}}

(b

_t

1

_{t≤τ1}

+ Z

_t

1

_{t>τ3}

) + 1

_{{τ1<τ2,τ3=αn+1}}

b

_t

1

_{t≤τ1}

+ 1

_{τ1≥τ2}

(b

_t

1

_{t≤τ2}

+ Z

_t

1

_{t>τ2}

) (3.12)

= 1

_{{t≤τ1∧τ2}}

b

_t

+ [1

_{{τ1<τ2,τ3<αn+1,t>τ3}}

+ 1

_{{τ1≥τ2,t>τ2}}

]Z

_t

. (3.13) Obviously, Z

^′

is a predictable process.

Set

X

_t

, X

_αn

+ Z

t

αn

g(Z

_u^′

) du − Z

t

αn

Z

_u^′

dB

_u

(3.14)

for all t ∈ [α

_n

, α

_n+1

].

Step 4. Some properties of X.

It follows from the construction that { X

_t

}

αn≤t≤αn+1

has the following properties:

1. X

_t

1

_{{τ2≤τ1,t≥τ2}}

= Y

_t

1

_{{τ2≤τ1,t≥τ2}}

;

2. X

_t

1

_{{τ1<τ2,τ3≥t≥τ1}}

= (X

_αn

− 2

⁻ⁿ⁻¹

ε)1

_{{τ1<τ2,τ3≥t≥τ1}}

; 3. X

_t

1

_{{τ1<τ2,t>τ3}}

= Y

_t

1

_{{τ1<τ2,t>τ3}}

.

Therefore, we have

X

_αn+1

= Y

_αn+1

(1

_{τ2≤τ1}

+ 1

_{{τ1<τ2,τ3<αn+1}}

)

+ (X

_αn

− 2

⁻ⁿ⁻¹

ε)1

_{{τ1<τ2,τ3=αn+1}}

. (3.15) So the induction assumption X

_αn

≤ Y

_αn

implies X

_αn+1

≤ Y

_αn+1

. It is also clear that the new solution X is bounded by k Y k

^∞

+ | y | + ε.

Set A

_n

, { ω ∈ Ω | τ

₁

< τ

₂

, τ

₃

= α

_n+1

} . Then P (A

_n

) is the probability that X

_αn+1

is not equal to Y

αn+1

. From (3.10), we get

P (A

_n

) ≤ P ( { ω ∈ Ω | ∃ t ∈ [α

_n

, α

_n+1

) such that V

_t

(ω) < − 2

⁻ⁿ⁻¹

ε } ) ≤ exp {− 2

ⁿ

ε } . Since P

∞

n=0

exp {− 2

ⁿ

ε } < + ∞ , the Borel-Cantelli Lemma implies that

P ( ∩

^∞n=0

∪

k≥n

A

_k

) = 0, (3.16)

which shows X

_T

= Y

_T

= ξ, P a.s.

To sum up, (X, Z

^′

) is indeed a new bounded solution with X

₀

= y.

The construction used many different constants. It is clear that this yields infinitely many different solutions.

Notice that in the proof we only use the fact that g is superquadratic to guarantee that the new solution X is bounded below. This shows if g is at least quadratic , i.e.

lim

_|z|→∞^g(z)_|z|2

> 0, we can construct a process V

_t

such that lim

_t→αn+1

V

_t

= + ∞ as well.

Thus we have the following conclusion.

Corollary 3.1. Suppose g is at least quadratic lim

_|z|→∞^g(z)_|z|2

> 0 and, for ξ ∈ L

^∞

( F

T

), Y is a bounded solution of the BSDE (g, ξ), then for each y < Y

₀

, there exists infinitely many solutions X which are bounded above with X

₀

= y.

3.3 Non-stability of the solutions

The monotone stability plays an important role in the study of quadratic BSDEs (See, e.g., [12, 3]). Here we shall show that the same type of monotone stability does not hold.

Theorem 3.4. (Non-stability) Suppose lim

_z→∞^g(z)_|z|2

= ∞ . Then there exists a se- quence of solutions { Y

^k

}

^∞k=1

of BSDEs (g, ξ

_k

) which increasingly and boundedly con- verges to Y such that Y is not a solution of BSDE (g, ξ), where ξ is the L

^∞

limit of { ξ

_k

}

^∞k=1

.

Proof. It follows from lim

_z→∞^g(z)_|z|2

= ∞ that there exists a sequence { z

_k

}

^∞k=1

with | z

_k

| → + ∞ such that g(z

_k

) ≥ max { 16

^k

T | z

_k

|

²

, 2

^k+1

T } . W.l.o.g., we suppose that z

_k

> 0.

Denote α

_k

:= ⌈ g(z

_k

) ⌉ where ⌈·⌉ is the ceiling function and put Z

^k

(t) , z

_k

1

_∪αk i=1[^T

αki−^T

α2 k

,^T

αki]

(t) for all 0 ≤ t ≤ T . Then it follows that

E Z

t

0

Z

_u^k

dB

_u

2

≤ Z

T

0

(Z

_u^k

)

²

du

= (z

_k

)

²

T

α

_k

≤ 16

^−k

→ 0, for k → ∞ . However, we have

Z

t 0

g(Z

_u^k

) du = Z

t

0

g(z

_k

)1

_∪αk

i=1[_αk^Ti−^T

α2 k

,_αk^Ti]

(u) du ∈ h g(z

_k

) α

_k

t −

T α

_k

− T

α

²_k

, g(z

_k

) α

_k

t i

,

(3.17)

20

which implies

sup

0≤t≤T

Z

t 0

g(Z

_u^k

) du − t

≤ 2

^−k

→ 0, (3.18)

as k → ∞ .

Define stopping times

ν

_k

, inf n t ≥ 0

Z

t 0

Z

_u^k

dB

_u

> 2

^−k

o

∧ T (3.19)

and

ν = inf

_k≥1

ν

_k

. (3.20)

Applying the submartingale inequality, we get P [ν

_k

< T ] = P

"

sup

0≤t≤T

Z

t 0

Z

_u^k

dB

_u

> 2

^−k

#

≤ 4

^k

E Z

T

0

Z

_u^k

dB

_u

2

≤ 4

^−k

. Thus we get

P [ν = T ] = 1 − P [ ∪

k≥1

{ ν

_k

< T } ]

≥ 1 − X

k≥1

P [ν

_k

< T ]

≥ 2 3 > 0,

which is due to the selection of sufficient large z

_k

, k ≥ 1.

Since P

k≥1

P [ν

_k

< T ] < ∞ , it follows from the Borel-Cantelli Lemma that P

∩

n≥1

∪

k≥n

{ ν

_k

< T }

= 0,

which means P [ ∪

^n≥1

∩

k≥n

{ ν

_k

= T } ] = 1. It implies that, for almost all ω ∈ Ω, there exists N (ω) such that for any k > N (ω), ν

_k

(ω) = T. Thus we have P [ν > 0] = 1.

Define y

_t^k

= R

t∧ν

0

g(Z

_u^k

) du − R

t∧ν

0

Z

_u^k

dB

_u

. We then deduce that sup

0≤t≤T

| y

_t^k

− t ∧ ν |

≤ sup

0≤t≤T

Z

t∧ν 0

g(Z

_u^k

) du − t ∧ ν

+ sup

0≤t≤T

Z

t∧ν 0

Z

_u^k

dB

_u

≤ 2 · 2

^−k

, (3.21)

which implies that

k→∞

lim sup

0≤t≤T

| y

^k_t

− t ∧ ν | = 0. (3.22)