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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Backward SDE with quadratic growthMagdalena Kobylanski (1999)

Jan Gairing, Plamen Turkedijev

Department of MathematicsHumboldt Universitat zu Berlin

January 13, 2009

[?] [?] [?]

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Table of Contents

1 Errata and Proof of Existence

2 Preliminary statements

3 Comparison Theorem

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

(H1)

Let α0, β0, b ∈ R and c a continous increasing function.We say F satisfies condition (H1) with α0, β0, b, c if for all(t, v, z) ∈ R+ ×R ×Rd,

F(t, v, z) = a0(t, v, z)v + F0(t, v, z)

with

β0 ≤ a0(t, v, z) ≤ α0, |F0(t, v, z)| ≤ b + c(|v|)|z|2 a.s.

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Monotone stability

Let (F, τ, ξ) and (Fn, τ, ξn)n be sets of parameters s.t.

(i) (Fn)n converges to F in the sense thatFn(t, un, zn)→ F(t, un, zn) for (un, zn)→ (u, z) ∈ R ×Rd

(ii) |Fn(t, u, z)| ≤ kt + C|z|2, k ≤ 0 ∈ L1[0,T ], C > 0

(iii) For each n we have solutions to (Fn, τ, ξn)n

(Yn,Zn) ∈ H∞τ (R) ×H2τ(R

d), ‖Yn‖∞ ≤ M,M > 0 and (Yn)n is monotone.

(iv) τ < ∞ a.s.

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Monotone stability

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDEwith (F, τ, ξ), and for all T ∈ R+

Yn → Y uniformly on [0,T ], (Zn)n → Z in H2τ(R

d)

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.

Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Existence Theorem

Let (F, τ, ξ) be a set of parameters of the BSDE. Suppose Fsatisfies (H1) with α0, β0, b ∈ R, and c : R+ → R+ continuousincreasing, ξ ∈ L∞(Ω), and either of:

(i) The terminal time τ is bounded, (τ < T a.s.), or

(ii) The τ is unbounded, α0 < 0 and E(eλτ) < ∞ for all λ > 0.

Then there exists a solution (Y,Z) ∈ H∞τ (R) ×H2τ(R

d) to the BSDE.Moreover, there exists a maximal solution (Y,Z) of (F, τ, ξ) in thesense that if (G, τ, ζ) are another set of parameters with G ≤ F andψ ≤ ξ, the YG ≤ Y for any solution (YG,ZG) of the BSDE (G, τ, ζ).

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Additional Lemmas and results

Approximation LemmaThere exists a decreasing sequence of uniformly Lipschitzcontinuous functions f n that converges to f in the same sense asin the Stability Theorem, and f ≤ f n ≤ l.

Existence Theorem for Standard ParametersLet ( f , τ, ξ) be standard parameters and β > 2L + L2. AssumeE[eβτ|ξ|2] < ∞. and that

∫ τ

0 eβs f (s, 0, 0)2ds < ∞. Then there isunique solution (y, z) ∈ H2

τ(R) ×H2τ(R

d).

LemmaLet ( f , τ, ξ) be standard parameters with bounded τ ≤ T a.s.,| f (·, ·, 0)| ≤ c and ξ bounded. Then ||Yt∧τ|| ≤ ||ξ||∞ + c(T − t) for allt ≤ T .

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Additional Lemmas and results

Approximation LemmaThere exists a decreasing sequence of uniformly Lipschitzcontinuous functions f n that converges to f in the same sense asin the Stability Theorem, and f ≤ f n ≤ l.

Existence Theorem for Standard ParametersLet ( f , τ, ξ) be standard parameters and β > 2L + L2. AssumeE[eβτ|ξ|2] < ∞. and that

∫ τ

0 eβs f (s, 0, 0)2ds < ∞. Then there isunique solution (y, z) ∈ H2

τ(R) ×H2τ(R

d).

LemmaLet ( f , τ, ξ) be standard parameters with bounded τ ≤ T a.s.,| f (·, ·, 0)| ≤ c and ξ bounded. Then ||Yt∧τ|| ≤ ||ξ||∞ + c(T − t) for allt ≤ T .

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Additional Lemmas and results

Approximation LemmaThere exists a decreasing sequence of uniformly Lipschitzcontinuous functions f n that converges to f in the same sense asin the Stability Theorem, and f ≤ f n ≤ l.

Existence Theorem for Standard ParametersLet ( f , τ, ξ) be standard parameters and β > 2L + L2. AssumeE[eβτ|ξ|2] < ∞. and that

∫ τ

0 eβs f (s, 0, 0)2ds < ∞. Then there isunique solution (y, z) ∈ H2

τ(R) ×H2τ(R

d).

LemmaLet ( f , τ, ξ) be standard parameters with bounded τ ≤ T a.s.,| f (·, ·, 0)| ≤ c and ξ bounded. Then ||Yt∧τ|| ≤ ||ξ||∞ + c(T − t) for allt ≤ T .

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Additional Lemmas and results

Approximation LemmaThere exists a decreasing sequence of uniformly Lipschitzcontinuous functions f n that converges to f in the same sense asin the Stability Theorem, and f ≤ f n ≤ l.

Existence Theorem for Standard ParametersLet ( f , τ, ξ) be standard parameters and β > 2L + L2. AssumeE[eβτ|ξ|2] < ∞. and that

∫ τ

0 eβs f (s, 0, 0)2ds < ∞. Then there isunique solution (y, z) ∈ H2

τ(R) ×H2τ(R

d).

LemmaLet ( f , τ, ξ) be standard parameters with bounded τ ≤ T a.s.,| f (·, ·, 0)| ≤ c and ξ bounded. Then ||Yt∧τ|| ≤ ||ξ||∞ + c(T − t) for allt ≤ T .

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Supersolutions

A supersolution (resp. subsolution) of a BSDE with parameters(F, τ, ξ) is an adapted triple (Y,Z,C) such that for all T > 0 and t < T

Yt = ξ +

∫ T∧τ

t∧τF(s,Ys,Zs)ds −

∫ T∧τ

t∧τZsdWs +

∫ T∧τ

t∧τdCs(

resp.Yt = ξ +

∫ T∧τ

t∧τF(s,Ys,Zs)ds −

∫ T∧τ

t∧τZsdWs −

∫ T∧τ

t∧τdCs

)C is right continuous and increasing on 0 < t < T . We will refer toall such functions as RCI(R). (Y,Z) ∈ H∞τ (R) ×H2

τ(Rd).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Supersolutions

A supersolution (resp. subsolution) of a BSDE with parameters(F, τ, ξ) is an adapted triple (Y,Z,C) such that for all T > 0 and t < T

Yt = ξ +

∫ T∧τ

t∧τF(s,Ys,Zs)ds −

∫ T∧τ

t∧τZsdWs +

∫ T∧τ

t∧τdCs(

resp.Yt = ξ +

∫ T∧τ

t∧τF(s,Ys,Zs)ds −

∫ T∧τ

t∧τZsdWs −

∫ T∧τ

t∧τdCs

)C is right continuous and increasing on 0 < t < T . We will refer toall such functions as RCI(R). (Y,Z) ∈ H∞τ (R) ×H2

τ(Rd).

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Growth Conditions

The coefficient satisfies condition (H2) on [−M,M] with l, k and Cif for all t ≥ 0, u ∈ [−M,M], and z ∈ Rd

|F(t, u, z)| ≤ l(t) + C|z|2 a.e.|∇zF| ≤ k(t) + C|z| a.e.

(H2)

The coefficient satisfies condition (H3) with cε and ε > 0 if for allt ≥ 0, u ∈ R, and z ∈ Rd

∂F∂u≤ cε(t) + ε|z|2 (H3)

l, k and cε satisfy some integrability conditions.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Growth Conditions

The coefficient satisfies condition (H2) on [−M,M] with l, k and Cif for all t ≥ 0, u ∈ [−M,M], and z ∈ Rd

|F(t, u, z)| ≤ l(t) + C|z|2 a.e.|∇zF| ≤ k(t) + C|z| a.e.

(H2)

The coefficient satisfies condition (H3) with cε and ε > 0 if for allt ≥ 0, u ∈ R, and z ∈ Rd

∂F∂u≤ cε(t) + ε|z|2 (H3)

l, k and cε satisfy some integrability conditions.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Growth Conditions

The coefficient satisfies condition (H2) on [−M,M] with l, k and Cif for all t ≥ 0, u ∈ [−M,M], and z ∈ Rd

|F(t, u, z)| ≤ l(t) + C|z|2 a.e.|∇zF| ≤ k(t) + C|z| a.e.

(H2)

The coefficient satisfies condition (H3) with cε and ε > 0 if for allt ≥ 0, u ∈ R, and z ∈ Rd

∂F∂u≤ cε(t) + ε|z|2 (H3)

l, k and cε satisfy some integrability conditions.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Comparison Theorem

Let (Y1t ,Z

1t ,C

1t )0≤t≤τ ∈ H

∞τ (R) ×H2

τ(Rd) × RCI(R) a subsolution of

parameters (F1, τ, ξ1), and (Y2t ,Z

2t ,C

2t )0≤t≤τ is a supersolution of

parameters (F2, τ, ξ2). Let M = max(||Y1||∞, ||Y2||∞).

Assumefurther

(i) ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y1t ,Z

1t ) ≤ F2(t,Y1

t ,Z1t )

a.s.

(ii) For all ε > 0 there exist l, lε ∈ L1loc(0, ||τ||∞), k ∈ L2

loc(0, ||τ||∞),C > 0 such that F2 satisfies both conditions (H2) a.s. on[−M,M] with l, k,C and (H3) on [−M,M] a.s. with lε and ε.

(iii) For all T > 0, E∫ T∧τ

0 | f 1(s,Y1s ,Z

1s )|ds < ∞.

Then

Y1t ≤ Y2

t a.s. ∀t > 0

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Comparison Theorem

Let (Y1t ,Z

1t ,C

1t )0≤t≤τ ∈ H

∞τ (R) ×H2

τ(Rd) × RCI(R) a subsolution of

parameters (F1, τ, ξ1), and (Y2t ,Z

2t ,C

2t )0≤t≤τ is a supersolution of

parameters (F2, τ, ξ2). Let M = max(||Y1||∞, ||Y2||∞). Assumefurther

(i) ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y1t ,Z

1t ) ≤ F2(t,Y1

t ,Z1t )

a.s.

(ii) For all ε > 0 there exist l, lε ∈ L1loc(0, ||τ||∞), k ∈ L2

loc(0, ||τ||∞),C > 0 such that F2 satisfies both conditions (H2) a.s. on[−M,M] with l, k,C and (H3) on [−M,M] a.s. with lε and ε.

(iii) For all T > 0, E∫ T∧τ

0 | f 1(s,Y1s ,Z

1s )|ds < ∞.

Then

Y1t ≤ Y2

t a.s. ∀t > 0

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Comparison Theorem

Let (Y1t ,Z

1t ,C

1t )0≤t≤τ ∈ H

∞τ (R) ×H2

τ(Rd) × RCI(R) a subsolution of

parameters (F1, τ, ξ1), and (Y2t ,Z

2t ,C

2t )0≤t≤τ is a supersolution of

parameters (F2, τ, ξ2). Let M = max(||Y1||∞, ||Y2||∞). Assumefurther

(i) ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y1t ,Z

1t ) ≤ F2(t,Y1

t ,Z1t )

a.s.

(ii) For all ε > 0 there exist l, lε ∈ L1loc(0, ||τ||∞), k ∈ L2

loc(0, ||τ||∞),C > 0 such that F2 satisfies both conditions (H2) a.s. on[−M,M] with l, k,C and (H3) on [−M,M] a.s. with lε and ε.

(iii) For all T > 0, E∫ T∧τ

0 | f 1(s,Y1s ,Z

1s )|ds < ∞.

Then

Y1t ≤ Y2

t a.s. ∀t > 0

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Comparison Theorem

Let (Y1t ,Z

1t ,C

1t )0≤t≤τ ∈ H

∞τ (R) ×H2

τ(Rd) × RCI(R) a subsolution of

parameters (F1, τ, ξ1), and (Y2t ,Z

2t ,C

2t )0≤t≤τ is a supersolution of

parameters (F2, τ, ξ2). Let M = max(||Y1||∞, ||Y2||∞). Assumefurther

(i) ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y1t ,Z

1t ) ≤ F2(t,Y1

t ,Z1t )

a.s.

(ii) For all ε > 0 there exist l, lε ∈ L1loc(0, ||τ||∞), k ∈ L2

loc(0, ||τ||∞),C > 0 such that F2 satisfies both conditions (H2) a.s. on[−M,M] with l, k,C and (H3) on [−M,M] a.s. with lε and ε.

(iii) For all T > 0, E∫ T∧τ

0 | f 1(s,Y1s ,Z

1s )|ds < ∞.

Then

Y1t ≤ Y2

t a.s. ∀t > 0

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Comparison Theorem

Let (Y1t ,Z

1t ,C

1t )0≤t≤τ ∈ H

∞τ (R) ×H2

τ(Rd) × RCI(R) a subsolution of

parameters (F1, τ, ξ1), and (Y2t ,Z

2t ,C

2t )0≤t≤τ is a supersolution of

parameters (F2, τ, ξ2). Let M = max(||Y1||∞, ||Y2||∞). Assumefurther

(i) ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y1t ,Z

1t ) ≤ F2(t,Y1

t ,Z1t )

a.s.

(ii) For all ε > 0 there exist l, lε ∈ L1loc(0, ||τ||∞), k ∈ L2

loc(0, ||τ||∞),C > 0 such that F2 satisfies both conditions (H2) a.s. on[−M,M] with l, k,C and (H3) on [−M,M] a.s. with lε and ε.

(iii) For all T > 0, E∫ T∧τ

0 | f 1(s,Y1s ,Z

1s )|ds < ∞.

Then

Y1t ≤ Y2

t a.s. ∀t > 0

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Remarks

The result is also true if, instead of (i) and (ii) above, we have

(i)’ ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y2t ,Z

2t ) ≤ F2(t,Y2

t ,Z2t )

a.s.(ii)’ F1 satisfies (H2) and (H3) under the same conditions as in (ii).

The hypotheses of the theorem make no assumptions on theintegrability or boundedness of the terminal condition.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Remarks

The result is also true if, instead of (i) and (ii) above, we have

(i)’ ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y2t ,Z

2t ) ≤ F2(t,Y2

t ,Z2t )

a.s.(ii)’ F1 satisfies (H2) and (H3) under the same conditions as in (ii).

The hypotheses of the theorem make no assumptions on theintegrability or boundedness of the terminal condition.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Remarks

The result is also true if, instead of (i) and (ii) above, we have

(i)’ ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y2t ,Z

2t ) ≤ F2(t,Y2

t ,Z2t )

a.s.(ii)’ F1 satisfies (H2) and (H3) under the same conditions as in (ii).

The hypotheses of the theorem make no assumptions on theintegrability or boundedness of the terminal condition.

Gairing, Turkedijev BSDE

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Errata and Proof of ExistencePreliminary statements

Comparison Theorem

Remarks

The result is also true if, instead of (i) and (ii) above, we have

(i)’ ξ1 ≤ ξ2 a.s. and for all t we have F1(t,Y2t ,Z

2t ) ≤ F2(t,Y2

t ,Z2t )

a.s.(ii)’ F1 satisfies (H2) and (H3) under the same conditions as in (ii).

The hypotheses of the theorem make no assumptions on theintegrability or boundedness of the terminal condition.

Gairing, Turkedijev BSDE