Large deviation principle for SDEs with discontinuous coefficients D. - - PowerPoint PPT Presentation

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle for SDE’s with discontinuous coefficients

• D. Sobolieva1

1Department of Probability Theory

Kyiv National Taras Shevchenko University

24.10.2014

Berlin-Padova Meeting for Young Researchers

Stochastic Analysis and Applications in Biology, Finance and Physics

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Outline

1

Introduction

2

LDP for one-dimensional SDE’s without drift

3

LDP for one-dimensional SDE’s with discontinuous coefficients

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle

Let (X, ρ) be a complete separable metric space with X-valued random variables Xn, n ≥ 1. We say that family {Xn} satisfies large deviation principle (LDP) with rate function I : X → [0, ∞] if, for each opened set A, lim inf

n→∞

1 n log P {Xn ∈ A} ≥ − inf

x∈A I(x),

(1) and, for each closed set B, lim sup

n→∞

1 n log P {Xn ∈ B} ≤ − inf

x∈B I(x).

(2) If (1) holds and (2) holds for each compact set B only then family {Xn} satisfies weak LDP. If each level set {x : I(x) ≤ a} , a ≥ 0, is compact then we call rate function I “good”.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle

Let (X, ρ) be a complete separable metric space with X-valued random variables Xn, n ≥ 1. We say that family {Xn} satisfies large deviation principle (LDP) with rate function I : X → [0, ∞] if, for each opened set A, lim inf

n→∞

1 n log P {Xn ∈ A} ≥ − inf

x∈A I(x),

(1) and, for each closed set B, lim sup

n→∞

1 n log P {Xn ∈ B} ≤ − inf

x∈B I(x).

(2) If (1) holds and (2) holds for each compact set B only then family {Xn} satisfies weak LDP. If each level set {x : I(x) ≤ a} , a ≥ 0, is compact then we call rate function I “good”.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle

Let (X, ρ) be a complete separable metric space with X-valued random variables Xn, n ≥ 1. We say that family {Xn} satisfies large deviation principle (LDP) with rate function I : X → [0, ∞] if, for each opened set A, lim inf

n→∞

1 n log P {Xn ∈ A} ≥ − inf

x∈A I(x),

(1) and, for each closed set B, lim sup

n→∞

1 n log P {Xn ∈ B} ≤ − inf

x∈B I(x).

(2) If (1) holds and (2) holds for each compact set B only then family {Xn} satisfies weak LDP. If each level set {x : I(x) ≤ a} , a ≥ 0, is compact then we call rate function I “good”.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle

Let (X, ρ) be a complete separable metric space with X-valued random variables Xn, n ≥ 1. We say that family {Xn} satisfies large deviation principle (LDP) with rate function I : X → [0, ∞] if, for each opened set A, lim inf

n→∞

1 n log P {Xn ∈ A} ≥ − inf

x∈A I(x),

(1) and, for each closed set B, lim sup

n→∞

1 n log P {Xn ∈ B} ≤ − inf

x∈B I(x).

(2) If (1) holds and (2) holds for each compact set B only then family {Xn} satisfies weak LDP. If each level set {x : I(x) ≤ a} , a ≥ 0, is compact then we call rate function I “good”.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviation principle

Let (X, ρ) be a complete separable metric space with X-valued random variables Xn, n ≥ 1. We say that family {Xn} satisfies large deviation principle (LDP) with rate function I : X → [0, ∞] if, for each opened set A, lim inf

n→∞

1 n log P {Xn ∈ A} ≥ − inf

x∈A I(x),

(1) and, for each closed set B, lim sup

n→∞

1 n log P {Xn ∈ B} ≤ − inf

x∈B I(x).

(2) If (1) holds and (2) holds for each compact set B only then family {Xn} satisfies weak LDP. If each level set {x : I(x) ≤ a} , a ≥ 0, is compact then we call rate function I “good”.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

The model

Consider one-dimensional SDE’s dXn

t = a(Xn t )dt +

1 √nσ(Xn

t )dWt

(3) with initial conditions Xn

0 = x0.

Freidlin, Wentzell’79

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

The model

Consider one-dimensional SDE’s dXn

t = a(Xn t )dt +

1 √nσ(Xn

t )dWt

(3) with initial conditions Xn

0 = x0.

Freidlin, Wentzell’79

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. The model

Consider X = C([0, ∞)), ρ(f, g) =

∞

• k=1

1 2k ( sup

t∈[0,k]

|f(t) − g(t)| ∧ 1). Let us consider SDE’s without drift, which means that coefficient a ≡ 0. In this case our model becomes of the following form: dY n

t =

1 √nσ(Y n

t )dWt

(4) with initial conditions Y n

0 = y0.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. The model

Consider X = C([0, ∞)), ρ(f, g) =

∞

• k=1

1 2k ( sup

t∈[0,k]

|f(t) − g(t)| ∧ 1). Let us consider SDE’s without drift, which means that coefficient a ≡ 0. In this case our model becomes of the following form: dY n

t =

1 √nσ(Y n

t )dWt

(4) with initial conditions Y n

0 = y0.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. The representation for the solution

The solutions to (4) can be represented as follows: Y n = F 1

nW

• .

Here function F : C([0, ∞)) → C([0, ∞)) is defined by Ft(f) = f(η−1

t

(f)) + y0, t ≥ 0, f ∈ C([0, ∞)) where the transformation η : C([0, ∞)) → C([0, ∞)) is defined by ηt(f) = t σ−2(y0 + f(s))ds, t ≥ 0, f ∈ C([0, ∞))

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. The representation for the solution

The solutions to (4) can be represented as follows: Y n = F 1

nW

• .

Here function F : C([0, ∞)) → C([0, ∞)) is defined by Ft(f) = f(η−1

t

(f)) + y0, t ≥ 0, f ∈ C([0, ∞)) where the transformation η : C([0, ∞)) → C([0, ∞)) is defined by ηt(f) = t σ−2(y0 + f(s))ds, t ≥ 0, f ∈ C([0, ∞))

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. The representation for the solution

The solutions to (4) can be represented as follows: Y n = F 1

nW

• .

Here function F : C([0, ∞)) → C([0, ∞)) is defined by Ft(f) = f(η−1

t

(f)) + y0, t ≥ 0, f ∈ C([0, ∞)) where the transformation η : C([0, ∞)) → C([0, ∞)) is defined by ηt(f) = t σ−2(y0 + f(s))ds, t ≥ 0, f ∈ C([0, ∞))

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Contraction principle and its generalizations

Contraction principle Let family {Xn} satisfy LDP with good rate function I and consider a continuous function F : X → Y, such that Y n = F(Xn). Then family {Y n} satisfies LDP with good rate function J(y) = inf{I(x) : y = F(x)} Dembo, Zeitouni’98

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Contraction principle and its generalizations

Contraction principle Let family {Xn} satisfy LDP with good rate function I and consider a continuous function F : X → Y, such that Y n = F(Xn). Then family {Y n} satisfies LDP with good rate function J(y) = inf{I(x) : y = F(x)} Dembo, Zeitouni’98

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Contraction principle and its generalizations

Contraction principle Let family {Xn} satisfy LDP with good rate function I and consider a continuous function F : X → Y, such that Y n = F(Xn). Then family {Y n} satisfies LDP with good rate function J(y) = inf{I(x) : y = F(x)} Dembo, Zeitouni’98

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

Example Consider function σ with one point of discontinuity: σ2(t) = 2, t ≥ 0 1, t < 0 and consider a sequence of functions fm(t) =    − 1

m,

t ≤ 1 − 1

m

t − 1, 1 − 1

m ≤ t ≤ c

c − 1, t ≥ c − → f(t) =    0, t ≤ 1 t − 1, 1 ≤ t ≤ c c − 1, t ≥ c

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

Example Consider function σ with one point of discontinuity: σ2(t) = 2, t ≥ 0 1, t < 0 and consider a sequence of functions fm(t) =    − 1

m,

t ≤ 1 − 1

m

t − 1, 1 − 1

m ≤ t ≤ c

c − 1, t ≥ c − → f(t) =    0, t ≤ 1 t − 1, 1 ≤ t ≤ c c − 1, t ≥ c

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

For the given functions fm the transformation η is equal to: ηt(fm) = t 2 + t ∧ 1 2 , η−1

t

(fm) = t, t ≤ 1 2t − 1, t ≥ 1 and thus the function F equals Ft(fm) = fm(t), t ≤ 1 fm(2t − 1), t ≥ 1 − → f(t), t ≤ 1 f(2t − 1), t ≥ 1 And for the function f transformation η and function F are the following: ηt(f) = t 2, η−1

t

(f) = 2t, Ft(f) = f(2t)

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

For the given functions fm the transformation η is equal to: ηt(fm) = t 2 + t ∧ 1 2 , η−1

t

(fm) = t, t ≤ 1 2t − 1, t ≥ 1 and thus the function F equals Ft(fm) = fm(t), t ≤ 1 fm(2t − 1), t ≥ 1 − → f(t), t ≤ 1 f(2t − 1), t ≥ 1 And for the function f transformation η and function F are the following: ηt(f) = t 2, η−1

t

(f) = 2t, Ft(f) = f(2t)

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

For the given functions fm the transformation η is equal to: ηt(fm) = t 2 + t ∧ 1 2 , η−1

t

(fm) = t, t ≤ 1 2t − 1, t ≥ 1 and thus the function F equals Ft(fm) = fm(t), t ≤ 1 fm(2t − 1), t ≥ 1 − → f(t), t ≤ 1 f(2t − 1), t ≥ 1 And for the function f transformation η and function F are the following: ηt(f) = t 2, η−1

t

(f) = 2t, Ft(f) = f(2t)

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

The “badness” of the example: The function f spends a long time in the point of discontinuity of σ. Lemma If function f spends zero time in points of discontinuity of σ: λ{s : f(s) ∈ ∆σ} = 0, then function f is a point of continuity for F.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Case a ≡ 0. “Bad” example

The “badness” of the example: The function f spends a long time in the point of discontinuity of σ. Lemma If function f spends zero time in points of discontinuity of σ: λ{s : f(s) ∈ ∆σ} = 0, then function f is a point of continuity for F.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Theorem 1 Consider such measurable, bounded and separated from zero σ that the set ∆σ of discontinuity points of σ has zero Lebesgue measure. Then the family of solutions to (4) satisfies LDP with a good rate function I, which equals I(x) = 1 2 ∞ ( ˙ x(t))2 σ2(x(t))dt if x ∈ C([0, ∞)) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, ∞)), and I(x) = ∞ otherwise. Kulik, Soboleva’12 This result was obtained using semicontraction principles (Kulik’05).

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Theorem 1 Consider such measurable, bounded and separated from zero σ that the set ∆σ of discontinuity points of σ has zero Lebesgue measure. Then the family of solutions to (4) satisfies LDP with a good rate function I, which equals I(x) = 1 2 ∞ ( ˙ x(t))2 σ2(x(t))dt if x ∈ C([0, ∞)) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, ∞)), and I(x) = ∞ otherwise. Kulik, Soboleva’12 This result was obtained using semicontraction principles (Kulik’05).

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Theorem 1 Consider such measurable, bounded and separated from zero σ that the set ∆σ of discontinuity points of σ has zero Lebesgue measure. Then the family of solutions to (4) satisfies LDP with a good rate function I, which equals I(x) = 1 2 ∞ ( ˙ x(t))2 σ2(x(t))dt if x ∈ C([0, ∞)) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, ∞)), and I(x) = ∞ otherwise. Kulik, Soboleva’12 This result was obtained using semicontraction principles (Kulik’05).

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Theorem 1 Consider such measurable, bounded and separated from zero σ that the set ∆σ of discontinuity points of σ has zero Lebesgue measure. Then the family of solutions to (4) satisfies LDP with a good rate function I, which equals I(x) = 1 2 ∞ ( ˙ x(t))2 σ2(x(t))dt if x ∈ C([0, ∞)) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, ∞)), and I(x) = ∞ otherwise. Kulik, Soboleva’12 This result was obtained using semicontraction principles (Kulik’05).

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Let us consider now case with nontrivial drift a ≡ 0 on the space X = C[0, T]. Theorem 2 Consider X = C([0, T]). Suppose that assumptions of Theorem 1 hold true and, additionally,

a σ2 has bounded derivative.

Then the family of solutions to (3) satisfies LDP with a good rate function I, which equals I(x) = lim inf

y→x

1 2 T (a(y(t)) − ˙ y(t))2 σ2(y(t)) dt if x ∈ C([0, T]) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, T]), and I(x) = ∞ otherwise. Sobolieva’12

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Let us consider now case with nontrivial drift a ≡ 0 on the space X = C[0, T]. Theorem 2 Consider X = C([0, T]). Suppose that assumptions of Theorem 1 hold true and, additionally,

a σ2 has bounded derivative.

Then the family of solutions to (3) satisfies LDP with a good rate function I, which equals I(x) = lim inf

y→x

1 2 T (a(y(t)) − ˙ y(t))2 σ2(y(t)) dt if x ∈ C([0, T]) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, T]), and I(x) = ∞ otherwise. Sobolieva’12

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Let us consider now case with nontrivial drift a ≡ 0 on the space X = C[0, T]. Theorem 2 Consider X = C([0, T]). Suppose that assumptions of Theorem 1 hold true and, additionally,

a σ2 has bounded derivative.

Then the family of solutions to (3) satisfies LDP with a good rate function I, which equals I(x) = lim inf

y→x

1 2 T (a(y(t)) − ˙ y(t))2 σ2(y(t)) dt if x ∈ C([0, T]) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, T]), and I(x) = ∞ otherwise. Sobolieva’12

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Large deviations in case a ≡ 0

Let us consider now case with nontrivial drift a ≡ 0 on the space X = C[0, T]. Theorem 2 Consider X = C([0, T]). Suppose that assumptions of Theorem 1 hold true and, additionally,

a σ2 has bounded derivative.

Then the family of solutions to (3) satisfies LDP with a good rate function I, which equals I(x) = lim inf

y→x

1 2 T (a(y(t)) − ˙ y(t))2 σ2(y(t)) dt if x ∈ C([0, T]) is an absolutely continuous function with x(0) = x0, ˙ x ∈ L2([0, T]), and I(x) = ∞ otherwise. Sobolieva’12

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Remark. Rate function in Theorem 2 differs from standart Freidlin-Wentzell result. The

reason for this is that we require rate function to be lower semicontinuous and function Q(y) = 1 2 T (a(y(t)) − ˙ y(t))2 σ2(y(t)) dt, in general, is not lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Example. Let T = 1, x0 = 0 and

σ(y) =

• c1,

y < 0 c2, y ≥ 0 , 0 < c1 < c2, a(y) = σ2(y). Then conditions of Theorem 2 are satisfied. Consider a sequence yn(t) =

• − t

n,

t ∈ [0, 1

2]

− 1

2n,

t ∈ [ 1

2, 1] −

→ y0 ≡ 0, as n → ∞ For this functions Q is equal to Q(yn) = 1 2

• c2

1 + 1

n + 1 2n2 1 c2

1

• → c2

1

2 as n → ∞, Q(y0) = c2

2

2 . Thus, we obtain that lim inf

n→∞ Q(yn) < Q( lim n→∞ yn), which means that function Q is not

lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Example. Let T = 1, x0 = 0 and

σ(y) =

• c1,

y < 0 c2, y ≥ 0 , 0 < c1 < c2, a(y) = σ2(y). Then conditions of Theorem 2 are satisfied. Consider a sequence yn(t) =

• − t

n,

t ∈ [0, 1

2]

− 1

2n,

t ∈ [ 1

2, 1] −

→ y0 ≡ 0, as n → ∞ For this functions Q is equal to Q(yn) = 1 2

• c2

1 + 1

n + 1 2n2 1 c2

1

• → c2

1

2 as n → ∞, Q(y0) = c2

2

2 . Thus, we obtain that lim inf

n→∞ Q(yn) < Q( lim n→∞ yn), which means that function Q is not

lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Example. Let T = 1, x0 = 0 and

σ(y) =

• c1,

y < 0 c2, y ≥ 0 , 0 < c1 < c2, a(y) = σ2(y). Then conditions of Theorem 2 are satisfied. Consider a sequence yn(t) =

• − t

n,

t ∈ [0, 1

2]

− 1

2n,

t ∈ [ 1

2, 1] −

→ y0 ≡ 0, as n → ∞ For this functions Q is equal to Q(yn) = 1 2

• c2

1 + 1

n + 1 2n2 1 c2

1

• → c2

1

2 as n → ∞, Q(y0) = c2

2

2 . Thus, we obtain that lim inf

n→∞ Q(yn) < Q( lim n→∞ yn), which means that function Q is not

lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Example. Let T = 1, x0 = 0 and

σ(y) =

• c1,

y < 0 c2, y ≥ 0 , 0 < c1 < c2, a(y) = σ2(y). Then conditions of Theorem 2 are satisfied. Consider a sequence yn(t) =

• − t

n,

t ∈ [0, 1

2]

− 1

2n,

t ∈ [ 1

2, 1] −

→ y0 ≡ 0, as n → ∞ For this functions Q is equal to Q(yn) = 1 2

• c2

1 + 1

n + 1 2n2 1 c2

1

• → c2

1

2 as n → ∞, Q(y0) = c2

2

2 . Thus, we obtain that lim inf

n→∞ Q(yn) < Q( lim n→∞ yn), which means that function Q is not

lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Difference from Freidlin-Wentzell result

• Example. Let T = 1, x0 = 0 and

σ(y) =

• c1,

y < 0 c2, y ≥ 0 , 0 < c1 < c2, a(y) = σ2(y). Then conditions of Theorem 2 are satisfied. Consider a sequence yn(t) =

• − t

n,

t ∈ [0, 1

2]

− 1

2n,

t ∈ [ 1

2, 1] −

→ y0 ≡ 0, as n → ∞ For this functions Q is equal to Q(yn) = 1 2

• c2

1 + 1

n + 1 2n2 1 c2

1

• → c2

1

2 as n → ∞, Q(y0) = c2

2

2 . Thus, we obtain that lim inf

n→∞ Q(yn) < Q( lim n→∞ yn), which means that function Q is not

lower semicontinuous.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Varadhan lemma and Bryc formula

Let {Xn} be a sequence of X-valued random variables and consider the rate transform Λ(f) = lim

n→∞

1 n log Eenf(Xn) Varadhan Lemma. Suppose that {Xn} satisfies large deviation principle with good rate function I. Then for each f ∈ Cb(X) Λ(f) = sup

x∈X

{f(x) − I(x)}. Bryc formula Suppose that the sequence {Xn} is exponentially tight and that the rate transform Λ(f) exists for each f ∈ Cb(X). Then {Xn} satisfies large deviation principle with good rate function I(x) = sup

f∈Cb(X)

{f(x) − Λ(f)} Proofs of these statements one can find in, e.g., Feng, Kurtz’06.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Varadhan lemma and Bryc formula

Let {Xn} be a sequence of X-valued random variables and consider the rate transform Λ(f) = lim

n→∞

1 n log Eenf(Xn) Varadhan Lemma. Suppose that {Xn} satisfies large deviation principle with good rate function I. Then for each f ∈ Cb(X) Λ(f) = sup

x∈X

{f(x) − I(x)}. Bryc formula Suppose that the sequence {Xn} is exponentially tight and that the rate transform Λ(f) exists for each f ∈ Cb(X). Then {Xn} satisfies large deviation principle with good rate function I(x) = sup

f∈Cb(X)

{f(x) − Λ(f)} Proofs of these statements one can find in, e.g., Feng, Kurtz’06.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Varadhan lemma and Bryc formula

Let {Xn} be a sequence of X-valued random variables and consider the rate transform Λ(f) = lim

n→∞

1 n log Eenf(Xn) Varadhan Lemma. Suppose that {Xn} satisfies large deviation principle with good rate function I. Then for each f ∈ Cb(X) Λ(f) = sup

x∈X

{f(x) − I(x)}. Bryc formula Suppose that the sequence {Xn} is exponentially tight and that the rate transform Λ(f) exists for each f ∈ Cb(X). Then {Xn} satisfies large deviation principle with good rate function I(x) = sup

f∈Cb(X)

{f(x) − Λ(f)} Proofs of these statements one can find in, e.g., Feng, Kurtz’06.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Varadhan lemma and Bryc formula

Let {Xn} be a sequence of X-valued random variables and consider the rate transform Λ(f) = lim

n→∞

1 n log Eenf(Xn) Varadhan Lemma. Suppose that {Xn} satisfies large deviation principle with good rate function I. Then for each f ∈ Cb(X) Λ(f) = sup

x∈X

{f(x) − I(x)}. Bryc formula Suppose that the sequence {Xn} is exponentially tight and that the rate transform Λ(f) exists for each f ∈ Cb(X). Then {Xn} satisfies large deviation principle with good rate function I(x) = sup

f∈Cb(X)

{f(x) − Λ(f)} Proofs of these statements one can find in, e.g., Feng, Kurtz’06.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

Some comments

The rate transform in Bryc formula for the family of solutions to (3) one can write with the help of Girsanov transformation through the solutions to (4) in the following form: Λ(f) = lim

n→∞

1 n log E exp{ng(Y n) − 1 2 T σ2(Y n

s )( a

σ2 )′(Y n

s )ds}.

D.Sobolieva

Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients

References

A.M. Kulik, D.D. Soboleva Large deviations for one-dimensional SDE with discontinuous diffusion coefficient, Theory of Stochastic Processes, 18(34)(2012),

• no. 1, 101–110.

D.D. Sobolieva, Large deviation principle for one-dimensional SDE with discontinuous coefficients, 18(34)(2012), no. 2, 102–108.

D.Sobolieva