Large Deviations for Multi-valued Stochastic Diﬀerential ...jiagang.pdf · Large Deviations for...

Large Deviations for Multi-valued Stochastic DifferentialEquations

Joint work with Siyan Xu, Xicheng Zhang

Large Deviations for Multi-valued Stochastic DifferentialEquations

Joint work with Siyan Xu, Xicheng Zhang

We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation

{dXε(t) ∈ b(Xε(t)) dt +

√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

I.e., we seek a result of the following type:

−I(F o) 6 lim infε↓0

ε log P (Xε ∈ F )

6 lim supε↓0

6 −I(F )

for F ⊂ Cx([0, 1], D(A)), where

F o = the interior of F

F = the closure of F

We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{

dXε(t) ∈ b(Xε(t)) dt +√

εσ(Xε(t)) dW (t)−A(Xε(t)) dt,

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

for F ⊂ Cx([0, 1], D(A)),

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

6 lim supε↓0

6 −I(F )

• I: a rate function on Cx([0, 1], D(A));

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

Let us begin by explaining the notion of

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).

I(A) := infx∈A

MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS

dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt

Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Difference: An additional A

What is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Difference: An additional AWhat is it?

A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Difference: An additional AWhat is it?A: a maximal monotone operator.

main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Difference: An additional AWhat is it?A: a maximal monotone operator.main features:

• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point

• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

〈x− y, u− v〉 > 0

• A is maximal:

(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).

Definition of Solution

Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of{

dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,

X(0) = x ∈ D(A).

Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of

{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,

X(0) = x ∈ D(A).

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;

(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

(Formally, this amounts to saying

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;

(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;

(iv) for any continuous processes (α, β) satisfying

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,

the measure

〈X(t)− α(t), dK(t)− β(t)dt〉 > 0

〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0

or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )

Existence and Uniqueness

Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,

X(0) = x ∈ D(A), ε ∈ (0, 1].

admits a unique solution, (X, K).

The unique solution of{dXε(t) ∈ b(Xε(t)) dt +

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

will be denoted by(Xε,Kε)

X(0) = x ∈ D(A), ε ∈ (0, 1].

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

X(0) = x ∈ D(A), ε ∈ (0, 1].

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

X(0) = x ∈ D(A), ε ∈ (0, 1].

The unique solution of

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

X(0) = x ∈ D(A), ε ∈ (0, 1].

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

X(0) = x ∈ D(A), ε ∈ (0, 1].

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

will be denoted by

(Xε,Kε)

X(0) = x ∈ D(A), ε ∈ (0, 1].

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

normally reflected SDE as a special case of MSDE

SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

The subdifferential of IO is given by

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,

whereInt(O) is the interior of OΛx is the exterior normal cone at x.

Suppose

O: a closed convex subset of Rm, IO : the indicator function of O,i.e,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

SupposeO: a closed convex subset of Rm,

IO : the indicator function of O,i.e,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

SupposeO: a closed convex subset of Rm, IO : the indicator function of O,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,

+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

∅, if x /∈ O,

{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

∅, if x /∈ O,{0}, if x ∈ Int(O),

Λx, if x ∈ ∂O,

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

Int(O) is the interior of OΛx is the exterior normal cone at x.

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

whereInt(O) is the interior of O

Λx is the exterior normal cone at x.

IO(x) ={

0, if x ∈ O,+∞, if x /∈ O.

∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}

Then ∂IO is a multivalued maximal monotone operator.

In this case, an MSDE is an SDE normally reflected at theboundary of O.

More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.

∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}

More generally,

the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.

Towards a Freidlin-Wentzell theory

We now look at the problem of small perturbation:{dXε(t) ∈ b(Xε(t)) dt +

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

We now look at the problem of small perturbation:

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

We now look at the problem of small perturbation:{dXε(t) ∈ b(Xε(t)) dt +

Xε(0) = x ∈ D(A), ε ∈ (0, 1].

Classical theory

Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +

√εσ(Xε(t)) dW (t),

Xε(0) = x, ε ∈ (0, 1].

The theory begins in the seminal paper

Wentzell-Freidlin:On small random perturbation of dynamical system, 1969

There are two standard and well known approaches to this classicalproblem.

Classical theory

Recall the classical case:

A ≡ 0{dXε(t) = b(Xε(t)) dt +

Xε(0) = x, ε ∈ (0, 1].

Classical theory

Recall the classical case: A ≡ 0

{dXε(t) = b(Xε(t)) dt +

Xε(0) = x, ε ∈ (0, 1].

Classical theory

Xε(0) = x, ε ∈ (0, 1].

Classical theory

Xε(0) = x, ε ∈ (0, 1].

Classical theory

Xε(0) = x, ε ∈ (0, 1].

Classical theory

Xε(0) = x, ε ∈ (0, 1].

1. Use of Euler Scheme

D.W.Stroock: An introduction to the theory of large deviation,1984

You look at the Euler Scheme

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

Then you have to estimate the quantity

P ( max06t61

|Xε(t, x)−Xεn| > δ)

If you have for any δ > 0

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Xεn(t) = x +

0b(Xε

n(n−1[sn])) ds +√

0σ(Xε

n(n−1[sn])) dW (s),

P ( max06t61

limn→∞

lim supε↓0

ε log P ( max06t61

|Xε(t, x)−Xεn| > δ) = −∞,

then you are done.

Unfortunately, this approach does NOT apply to MSDE case

simply because the Euler scheme cannot be defined for MSDEsince either A(Xn(t)) is not defined or may be a set.

Unfortunately, this approach does NOT apply to MSDE casesimply because the Euler scheme cannot be defined for MSDE

since either A(Xn(t)) is not defined or may be a set.

Unfortunately, this approach does NOT apply to MSDE casesimply because the Euler scheme cannot be defined for MSDEsince either A(Xn(t)) is not defined or may be a set.

2. Use of Freidlin-Wentzell estimate

(R. Azencott: Grandes deviations et applications, 1980

P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)

Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.

where f satisfies ∫ 1

0f ′(t)2dt < ∞.

Consider an ODE

{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.

0f ′(t)2dt < ∞.

where f satisfies

0f ′(t)2dt < ∞.

If you can prove

∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that

ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0

then you have the large deviation principle.

If you can prove ∀R > 0,

ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that

If you can prove ∀R > 0, ρ > 0,

∃ε0 > 0, α > 0, r > 0 such that

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0,

α > 0, r > 0 such that

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0,

r > 0 such that

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0

such that

If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that

But this approach does NOT apply neither

since to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz(Priouret had to.).

But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directly

one has to suppose b and σ are Lipschitz(Priouret had to.).

But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz

(Priouret had to.).

But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz(Priouret had to.).

3. Use of weak convergence

This approach was developed a few years ago by

P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997

and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.

The hard core of this approach is the following observation

• Xn: a sequence of r.v., valued in a complete metric space X ;

• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if

lim supn→∞

log P (Xn ∈ F ) 6 −I(F ), F closed

lim infn→∞

log P (Xn ∈ G) > −I(G), F open

{Xn} is said to satisfy the Laplace Principle with rate function I if

limn→∞

log E{exp[−nh(Xn)]} = − infx∈X

{h(x) + I(x)}.

• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;

{Xn} is said to satisfy the LDP with rate function I if

lim supn→∞

lim infn→∞

limn→∞

{h(x) + I(x)}.

• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if

lim supn→∞

lim infn→∞

limn→∞

{h(x) + I(x)}.

lim supn→∞

lim infn→∞

limn→∞

{h(x) + I(x)}.

lim supn→∞

lim infn→∞

limn→∞

{h(x) + I(x)}.

lim supn→∞

lim infn→∞

limn→∞

{h(x) + I(x)}.

Theorem: LP⇐⇒ LDP

(⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE

dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x

where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type

6 lim supε↓0

6 −I(F )

for F ⊂ Cx([0, 1], D(A)), the story is

Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)

Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE

6 lim supε↓0

6 −I(F )

Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE.

Suppose D ⊂ Rd is an open set.Consider the following reflected SDE

6 lim supε↓0

6 −I(F )

Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE

6 lim supε↓0

6 −I(F )

6 lim supε↓0

6 −I(F )

where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.

For the result of the following type

6 lim supε↓0

6 −I(F )

6 lim supε↓0

6 −I(F )

6 lim supε↓0

6 −I(F )

6 lim supε↓0

6 −I(F )

for F ⊂ Cx([0, 1], D(A)),

the story is

6 lim supε↓0

6 −I(F )

(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;

(ii) by Doss and Priouret when σ may be degenerated;

(iii) by Dupuis for general convex sets D

Return to our original problem.

If we manage to prove a Laplace principle, then we are done.

But how to obtain the Laplace principle?

Budhiraja and Dupuis pointed out the way for us.

Things to do

SetB := C([0, 1], Rd)

H := {h ∈ B such that h is a. c. and ‖h‖2H :=

0|h(s)|2ds < ∞}

µ the standard Wiener measure on B

Then (B,H, µ) is a classical Wiener space.

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Then DN with weak topology is a metrizable compact Polish space.

AN := {h : h is an adapted process , h(·) ∈ DN a.s.}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, let

DN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

Things to do

SetB := C([0, 1], Rd)

0|h(s)|2ds < ∞}

∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}

The way pointed out by Budhiraja and Duuis

Let Yn : B 7→ X , n = 1, 2, · · · . If there exists Y0 : H 7→ X suchthat

A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some

subsequence nk, Ynk

(w + hnk

(w)√n−1

)converges in distribution to

Y0(h) in X .

B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk

, Y0(hnk) converges

weakly to Y0(h) in X .

Let Yn : B 7→ X , n = 1, 2, · · · .

If there exists Y0 : H 7→ X suchthat

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

, Y0(hnk) converges

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

, Y0(hnk) converges

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

, Y0(hnk) converges

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

, Y0(hnk) converges

Theorem: Under (A) and (B), the LP holds:

limn→∞

log E{exp[−nf(Yn)]} = − infx∈X

{g(x) + I(x)}

Verification of (A) and (B)

We replace n by ε−1 and we take

Y0 = X

Yε := Xε,hε .

Verification of (A).Now Yε satisfies:

Yε(t) = x +∫ t

0b(Yε(s)) ds +

0σ(Yε(s))hε(s) ds

0σ(Yε(s)) dW (s)−Kε(t)

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Verification of (A).

Now Yε satisfies:

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Y0 = X

Yε := Xε,hε .

Yε(t) = x +∫ t

0b(Yε(s)) ds +

Y0(t) = x +∫ t

0b(Y0(s)) ds +

0σ(Y0(s))h(s) ds−K0(t).

Note (A) is equivalent to the apparently weaker condition

A’ For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges almost surely to h ∈ AN , in H, then for some

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

This equivalence can be easily achieved by using Skorohodrepresentation.

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

subsequence nk, Ynk

(w + hnk

(w)√n−1

Y0(h) in X .

Hence, to prove (A), it suffices to prove that

Yε −→ Y0, weakly

if hε −→ h0 almost surely in H. This can be done by Ito calculus.Similarly, B can be verified.

if hε −→ h0 almost surely in H.

This can be done by Ito calculus.Similarly, B can be verified.

if hε −→ h0 almost surely in H. This can be done by Ito calculus.

Similarly, B can be verified.

if hε −→ h0 almost surely in H. This can be done by Ito calculus.Similarly, B can be verified.

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