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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
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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.
Gairing, Turkedijev BSDE
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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.
Gairing, Turkedijev BSDE
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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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
logo1
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
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
logo1
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
logo1
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
logo1
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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