Weak Dynamic Programming for Generalized State Constraints

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Weak Dynamic Programming for Generalized State Constraints

Bruno Bouchard, Marcel Nutz

To cite this version:

Bruno Bouchard, Marcel Nutz. Weak Dynamic Programming for Generalized State Constraints.

SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2012,

50 (6), pp.3344-3373. �10.1137/110852942�. �hal-00590874�

Weak Dynamic Programming for Generalized State Constraints

Bruno Bouchard

Marcel Nutz

May 4, 2011

Abstract

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.

Keywords weak dynamic programming, state constraint, expectation constraint, Hamilton-Jacobi-Bellman equation, viscosity solution, comparison theorem AMS 2000 Subject Classications 93E20, 49L20, 49L25, 35K55

1 Introduction

We study the problem of stochastic optimal control under state constraints.

In the most classical case, this is the problem of maximizing an expected reward, subject to the constraint that the controlled state process has to remain in a given subset 𝒪 of the state space. There is a rich literature on the associated partial dierential equations (PDEs), going back to [10, 11, 6]

in the rst order case and [9, 8] in the second order case. The connection between the control problem and the equation is given by the dynamic pro- gramming principle. However, in the stochastic case, it is frequent practice to make this connection only formally. Of course, there are specic situations where it is indeed possible to avoid proving the state-constrained dynamic programming principle; in particular, penalization arguments can be useful to reduce to the unconstrained case. See, e.g., [5, 13] for further background.

∗CEREMADE, Université Paris Dauphine and CREST-ENSAE, bouchard@ensae.fr

†ETH Zurich, Department of Mathematics, marcel.nutz@math.ethz.ch. Research supported by Swiss National Science Foundation Grant PDFM2-120424/1.

Generally speaking, it is dicult to prove the dynamic programming principle when the regularity of the value function is not known a priori, due to certain measurable selection problems. It was observed in [3] that, in the unconstrained case, these diculties can be avoided by a weak formulation of the dynamic programming principle where the value function is replaced by a test function. This formulation, which is tailored to the derivation of the PDE in the sense of viscosity solutions, avoids the measurable selection and uses only a simple covering argument. It turns out that the latter does not extend directly to the case with state constraints. Essentially, the reason is that if 𝜈 is some admissible control for the initial condition 𝑥 ∈ 𝒪 i.e., the controlled state process 𝑋

started at 𝑥 remains in 𝒪 then 𝜈 may fail to have this property for a nearby initial condition 𝑥

∈ 𝒪 .

However, if 𝒪 is open and mild continuity assumptions are satised, then 𝑋

will violate the state constraint with at most small probability when 𝑥

is close to 𝑥 . This observation leads us to consider optimization problems with constraints in probability, and more generally expectation constraints of the form 𝐸[𝑔(𝑋

(𝑇 ))] ≤ 𝑚 for given 𝑚 ∈ ℝ. We shall see that, follow- ing the idea of [2], such problems are amenable to dynamic programming if the constraint level 𝑚 is formulated dynamically via an auxiliary family of martingales. A key idea in this paper is that relaxing the level 𝑚 by a small constant allows to prove a weak dynamic programming principle for general expectation constraint problems (Theorem 2.4), while the PDE can be derived despite the relaxation. We shall then obtain the dynamic pro- gramming principle for the classical state constraint problem (Theorem 3.1) by passing to a limit 𝑚 ↓ 0 , with a suitable choice of 𝑔 and certain regularity assumptions.

We exemplify the use of these results in the setting of controlled diusions and show how the PDEs for expectation constraints and state constraints can be derived (Theorems 4.2 and 4.6). For the latter case, we introduce an appropriate continuity condition at the boundary, under which we prove a comparison theorem. While the above concerned an open set 𝒪 and does not apply directly to the closed domain 𝒪 , we show via the comparison result that the value function for 𝒪 coincides with the one for 𝒪 , under certain conditions.

The remainder of the paper is organized as follows. In Section 2 we introduce an abstract setup for dynamic programming under expectation constraints and prove the corresponding relaxed weak dynamic programming principle. In Section 3 we deduce the dynamic programming principle for the state constraint 𝒪 . We specialize to the case of controlled diusions in Sec- tion 4, where we study the Hamilton-Jacobi-Bellman PDEs for expectation and state constraints. Appendix A provides the comparison theorem.

Throughout this paper, (in)equalities between random variables are in

the almost sure sense and relations between processes are up to evanescence,

unless otherwise stated.

2 Dynamic Programming Principle for Expectation Constraints

We x a time horizon 𝑇 ∈ (0, ∞) and a probability space (Ω, ℱ , 𝑃 ) equipped with a ltration 𝔽 = (ℱ

)

. For each 𝑡 ∈ [0, 𝑇 ] , we are given a set 𝒰

whose elements are seen as controls at time 𝑡 . Given a separable metric space 𝑆, the state space, we denote by S := [0, 𝑇 ] × 𝑆 the time-augmented state space. For each (𝑡, 𝑥) ∈ S and 𝜈 ∈ 𝒰

, we are given a càdlàg adapted process 𝑋

= {𝑋

(𝑠), 𝑠 ∈ [𝑡, 𝑇 ]} with values in 𝑆 , the controlled state process.

Finally, we are given two measurable functions 𝑓, 𝑔 : 𝑆 → ℝ. We assume that

𝐸[∣𝑓 (𝑋

(𝑇 ))∣] < ∞ and 𝐸[∣𝑔(𝑋

(𝑇 ))∣] < ∞ for all 𝜈 ∈ 𝒰

, (2.1) so that the reward and constraint functions

𝐹 (𝑡, 𝑥; 𝜈) := 𝐸[𝑓 (𝑋

(𝑇))], 𝐺(𝑡, 𝑥; 𝜈) := 𝐸[𝑔(𝑋

(𝑇 ))]

are well dened. We also introduce the value function 𝑉 (𝑡, 𝑥, 𝑚) := sup

𝜈∈𝒰(𝑡,𝑥,𝑚)

𝐹(𝑡, 𝑥; 𝜈), (𝑡, 𝑥, 𝑚) ∈ S, ˆ (2.2) where S ˆ := S × ℝ ≡ [0, 𝑇 ] × 𝑆 × ℝ and

𝒰 (𝑡, 𝑥, 𝑚) := {

𝜈 ∈ 𝒰

: 𝐺(𝑡, 𝑥; 𝜈) ≤ 𝑚 }

(2.3) is the set of controls admissible at constraint level 𝑚 . Here sup ∅ := −∞ .

The following observation is the heart of our approach: Given a control 𝜈 admissible at level 𝑚 at the point 𝑥 , relaxing the level 𝑚 will make 𝜈 ad- missible in an entire neighborhood of 𝑥 . This will be crucial for the covering arguments used below. We use the acronym u.s.c. (l.s.c.) to indicate upper (lower) semicontinuity.

Lemma 2.1. Let (𝑡, 𝑥, 𝑚) ∈ S, ˆ 𝜈 ∈ 𝒰(𝑡, 𝑥, 𝑚) and assume that the function 𝑥

7→ 𝐺(𝑡, 𝑥

; 𝜈 ) is u.s.c. at 𝑥 . For each 𝛿 > 0 there exists a neighborhood 𝐵 of 𝑥 ∈ 𝑆 such that 𝜈 ∈ 𝒰(𝑡, 𝑥

, 𝑚

+ 𝛿) for all 𝑥

∈ 𝐵 and all 𝑚

≥ 𝑚 . Proof. We have 𝐺(𝑡, 𝑥; 𝜈) ≤ 𝑚 by the denition (2.3) of 𝒰 (𝑡, 𝑥, 𝑚) . In view of the upper semicontinuity, there exists a neighborhood 𝐵 of 𝑥 such that 𝐺(𝑡, 𝑥

; 𝜈) ≤ 𝑚 + 𝛿 ≤ 𝑚

+ 𝛿 for 𝑥

∈ 𝐵 ; that is, 𝜈 ∈ 𝒰(𝑡, 𝑥

, 𝑚

+ 𝛿) .

A control problem with an expectation constraint of the form (2.3) is

not amenable to dynamic programming if we just consider a xed level 𝑚 .

Extending slightly an idea from [2, 1], we shall see that this changes if the

constraint is formulated dynamically by using auxiliary martingales. To this

end, we consider for each 𝑡 ∈ [0, 𝑇 ] a family ℳ

of càdlàg martingales 𝑀 = {𝑀 (𝑠), 𝑠 ∈ [𝑡, 𝑇 ]} with initial value 𝑀 (𝑡) = 0. We also introduce

ℳ

:= {𝑚 + 𝑀 : 𝑀 ∈ ℳ

}, 𝑚 ∈ ℝ . We assume that, for all (𝑡, 𝑥) ∈ S and 𝜈 ∈ 𝒰

,

there exists 𝑀

[𝑥] ∈ ℳ

such that 𝑀

[𝑥](𝑇 ) = 𝑔(𝑋

(𝑇 )) , (2.4) where, necessarily, 𝑚 = 𝐸[𝑔(𝑋

(𝑇 ))]. In particular, given 𝜈 ∈ 𝒰

, the set

ℳ

(𝜈 ) := {𝑀 ∈ ℳ

: 𝑀 (𝑇 ) ≥ 𝑔(𝑋

(𝑇 ))}

is nonempty for any 𝑚 ≥ 𝐸[𝑔(𝑋

(𝑇))] . More precisely, we have the follow- ing characterization.

Lemma 2.2. Let (𝑡, 𝑥) ∈ S and 𝑚 ∈ ℝ. Then 𝒰 (𝑡, 𝑥, 𝑚) = {

𝜈 ∈ 𝒰

: ℳ

(𝜈) ∕= ∅ } .

Proof. Let 𝜈 ∈ 𝒰

. If there exists some 𝑀 ∈ ℳ

(𝜈) , then taking expec- tations yields 𝐸[𝑔(𝑋

(𝑇 ))] ≤ 𝐸[𝑀(𝑇)] = 𝑚 and hence 𝜈 ∈ 𝒰 (𝑡, 𝑥, 𝑚) .

Conversely, let 𝜈 ∈ 𝒰(𝑡, 𝑥, 𝑚) ; i.e., we have 𝑚

:= 𝐸[𝑔(𝑋

(𝑇 ))] ≤ 𝑚 . With 𝑀

[𝑥] as in (2.4), 𝑀 := 𝑀

[𝑥]+𝑚−𝑚

is an element of ℳ

(𝜈) .

It will be useful in the following to consider for each 𝑡 ∈ [0, 𝑇 ] an auxiliary subltration 𝔽

= (ℱ

)

of 𝔽 such that 𝑋

and 𝑀 are 𝔽

-adapted for all 𝑥 ∈ 𝑆 , 𝜈 ∈ 𝒰

and 𝑀 ∈ ℳ

. Moreover, we denote by 𝒯

the set of 𝔽

-stopping times with values in [𝑡, 𝑇 ] .

The following assumption corresponds to one direction of the dynamic programming principle; cf. Theorem 2.4(i) below.

Assumption A. For all (𝑡, 𝑥, 𝑚) ∈ S ˆ , 𝜈 ∈ 𝒰 (𝑡, 𝑥, 𝑚) , 𝑀 ∈ ℳ

(𝜈) , 𝜏 ∈ 𝒯

and 𝑃 -a.e. 𝜔 ∈ Ω , there exists 𝜈

∈ 𝒰(𝜏 (𝜔), 𝑋

(𝜏 )(𝜔), 𝑀 (𝜏 )(𝜔)) such that

𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

(𝜔) ≤ 𝐹 (

𝜏 (𝜔), 𝑋

(𝜏 )(𝜔); 𝜈

) . (2.5)

Next, we state two variants of the assumptions for the converse direction

of the dynamic programming principle; we shall comment on the dierences

in Remark 2.5 below. In the rst variant, the intermediate time is deter-

ministic; this will be enough to cover stopping times with countably many

values in Theorem 2.4(ii) below. We recall the notation 𝑀

[𝑥] from (2.4).

Assumption B. Let (𝑡, 𝑥) ∈ S , 𝜈 ∈ 𝒰

, 𝑠 ∈ [𝑡, 𝑇 ] , ¯ 𝜈 ∈ 𝒰

and Γ ∈ ℱ

. (B1) There exists a control 𝜈 ˜ ∈ 𝒰

such that

𝑋

(⋅) = 𝑋

(⋅) on [𝑡, 𝑇 ] × (Ω ∖ Γ); (2.6) 𝑋

(⋅) = 𝑋

𝑡,𝑥(𝑠)

(⋅) on [𝑠, 𝑇 ] × Γ; (2.7) 𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

≥ 𝐹 (𝑠, 𝑋

(𝑠); ¯ 𝜈 ) on Γ. (2.8) The control 𝜈 ˜ is denoted by 𝜈 ⊗

𝜈 ¯ and called a concatenation of 𝜈 and 𝜈 ¯ on (𝑠, Γ) .

(B2) Let 𝑀 ∈ ℳ

. There exists a process 𝑀 ¯ = { 𝑀 ¯ (𝑟), 𝑟 ∈ [𝑠, 𝑇]} such that

𝑀(⋅)(𝜔) = ¯ (

𝑀

[𝑋

(𝑠)(𝜔)](⋅) )

(𝜔) on [𝑠, 𝑇 ] 𝑃 -a.s.

and

𝑀 1

+ 1

(

𝑀 1

+ [ 𝑀 ¯ − 𝑀 ¯ (𝑠) + 𝑀 (𝑠) ] 1

)

∈ ℳ

. (B3) Let 𝑚 ∈ ℝ and 𝑀 ∈ ℳ

(𝜈) . For 𝑃 -a.e. 𝜔 ∈ Ω , there exist a control

𝜈

∈ 𝒰 (𝑠, 𝑋

(𝑠)(𝜔), 𝑀 (𝑠)(𝜔)) .

In the second variant, the intermediate time is a stopping time and we have an additional assumption about the structure of the sets 𝒰

. This variant corresponds to Theorem 2.4(ii') below.

Assumption B'. Let (𝑡, 𝑥) ∈ S , 𝜈 ∈ 𝒰

, 𝜏 ∈ 𝒯

, Γ ∈ ℱ

and 𝜈 ¯ ∈ 𝒰

𝐿∞

. (B0') 𝒰

⊇ 𝒰

for all 0 ≤ 𝑠 ≤ 𝑠

≤ 𝑇 .

(B1') There exists a control 𝜈 ˜ ∈ 𝒰

, denoted by 𝜈 ⊗

𝜈 ¯ , such that 𝑋

(⋅) = 𝑋

(⋅) on [𝑡, 𝑇 ] × (Ω ∖ Γ);

𝑋

(⋅) = 𝑋

𝑡,𝑥(𝜏)

(⋅) on [𝜏, 𝑇 ] × Γ; (2.9) 𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

≥ 𝐹 (𝜏, 𝑋

(𝜏 ); ¯ 𝜈 ) on Γ. (2.10) (B2') Let 𝑀 ∈ ℳ

. There exists a process 𝑀 ¯ = { 𝑀(𝑟), 𝑟 ¯ ∈ [𝜏, 𝑇 ]} such

that

𝑀(⋅)(𝜔) = ¯ (

𝑀

[𝑋

(𝜏 )(𝜔)](⋅) )

(𝜔) on [𝜏, 𝑇 ] 𝑃 -a.s.

and

𝑀 1

+ 1

(

𝑀1

+ [ 𝑀 ¯ − 𝑀 ¯ (𝜏 ) + 𝑀(𝜏 ) ] 1

)

∈ ℳ

. (B3') Let 𝑚 ∈ ℝ and 𝑀 ∈ ℳ

(𝜈) . For 𝑃 -a.e. 𝜔 ∈ Ω , there exist a control

𝜈

∈ 𝒰 (𝜏 (𝜔), 𝑋

(𝜏 )(𝜔), 𝑀 (𝜏 )(𝜔)) .

Remark 2.3. (i) Assumption B' implies Assumption B. Moreover, As- sumption A implies (B3').

(ii) Let D := {(𝑡, 𝑥, 𝑚) ∈ S ˆ : 𝒰 (𝑡, 𝑥, 𝑚) ∕= ∅} denote the natural domain of our optimization problem. Then (B3') can be stated as follows: for any 𝜈 ∈ 𝒰(𝑡, 𝑥, 𝑚) and 𝑀 ∈ ℳ

(𝜈) ,

( 𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ) )

∈ D 𝑃 -a.s. for all 𝜏 ∈ 𝒯

. (2.11) Under (B3), the same holds if 𝜏 takes countably many values. We remark that the invariance property (2.11) corresponds to one direction in the geometric dynamic programming of [12].

(iii) We note that (2.9) (and similarly (2.7)) states in particular that (𝑟, 𝜔) 7→ (

𝑋

𝑡,𝑥(𝜏)

(𝑟) )

(𝜔) := (

𝑋

𝑡,𝑥(𝜏)(𝜔)

(𝑟) ) (𝜔)

is a well-dened adapted process (up to evanescence) on [𝑡, 𝑇 ] × Γ . Of course, this is an implicit measurability condition on (𝑡, 𝑥) 7→ 𝑋

. (iv) For an illustration of (B1'), let us assume that the controls are pre-

dictable processes. In this case, one can often take 𝜈 ⊗

𝜈 ¯ := 𝜈1

+ 1

(

¯

𝜈1

+ 𝜈1

)

(2.12) and the condition that 𝜈 ⊗

𝜈 ¯ ∈ 𝒰

is called stability under concate- nation. The idea is that we use the control 𝜈 up to time 𝜏 ; after time 𝜏 , we continue using 𝜈 in the event Ω ∖ Γ while we switch to the control

¯

𝜈 in the event Γ . Of course, one can omit the set Γ ∈ ℱ

by observing that

𝜈 ⊗

𝜈 ¯ = 𝜈1

+ ¯ 𝜈1

for the 𝔽

-stopping time 𝜏

:= 𝜏 1

+ 𝑇 1

.

We can now state our weak dynamic programming principle for the

stochastic control problem (2.2) with expectation constraint. The formu-

lation is weak in two ways; namely, the value function is replaced by a

test function and the constraint level 𝑚 is relaxed by an arbitrarily small

constant 𝛿 > 0 (cf. (2.15) below). The exibility of choosing the set 𝒟 ap-

pearing below, will be used in Section 3. We recall the set D introduced in

Remark 2.3(ii).

Theorem 2.4. Let (𝑡, 𝑥, 𝑚) ∈ S ˆ , 𝜈 ∈ 𝒰 (𝑡, 𝑥, 𝑚) and 𝑀 ∈ ℳ

(𝜈) . Let 𝜏 ∈ 𝒯

and let 𝒟 ⊆ S ˆ be a set such that (𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) ∈ 𝒟 holds 𝑃 -a.s.

(i) Let Assumption A hold true and let 𝜑 : ˆ S → [−∞, ∞] be a measurable function such that 𝑉 ≤ 𝜑 on 𝒟 . Then 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 )

]

< ∞ and 𝐹 (𝑡, 𝑥; 𝜈) ≤ 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) ]

. (2.13)

(ii) Let 𝛿 > 0 , let Assumption B hold true and assume that 𝜏 takes count- ably many values (𝑡

)

. Let 𝜑 : ˆ S → [−∞, ∞) be a measurable func- tion such that 𝑉 ≥ 𝜑 on 𝒟 . Assume that for any xed 𝜈 ¯ ∈ 𝒰

,

(𝑥

, 𝑚

) 7→ 𝜑(𝑡

, 𝑥

, 𝑚

) is u.s.c.

𝑥

7→ 𝐹 (𝑡

, 𝑥

; ¯ 𝜈 ) is l.s.c.

𝑥

7→ 𝐺(𝑡

, 𝑥

; ¯ 𝜈) is u.s.c.

⎫

 ⎬

 ⎭

on 𝒟

(2.14)

for all 𝑖 ≥ 1 , where 𝒟

:= {(𝑥

, 𝑚

) : (𝑡

, 𝑥

, 𝑚

) ∈ 𝒟} ⊆ 𝑆 × ℝ. Then 𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) ≥ 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) ]

. (2.15)

(ii') Let 𝛿 > 0 and let Assumption B' hold true. Let 𝜑 : ˆ S → [−∞, ∞) be a measurable function such that 𝑉 ≥ 𝜑 on 𝒟 . Assume that 𝒟 ∩ D ⊆ S ˆ is 𝜎 -compact and that for any xed 𝜈 ¯ ∈ 𝒰

, 𝑡

∈ [𝑡, 𝑇 ] ,

(𝑡

, 𝑥

, 𝑚

) 7→ 𝜑(𝑡

, 𝑥

, 𝑚

) is u.s.c.

(𝑡

, 𝑥

) 7→ 𝐹 (𝑡

, 𝑥

; ¯ 𝜈 ) is l.s.c.

(𝑡

, 𝑥

) 7→ 𝐺(𝑡

, 𝑥

; ¯ 𝜈) is u.s.c.

⎫

 ⎬

 ⎭

on 𝒟 ∩ {𝑡

≤ 𝑡

}. (2.16)

Then (2.15) holds true.

The following convention is used on the right hand side of (2.15): if 𝑌 is any random variable,

we set 𝐸[𝑌 ] := −∞ if 𝐸[𝑌

] = 𝐸[𝑌

] = ∞. (2.17) We note that Assumption (B0') ensures that the expressions 𝐹(𝑡

, 𝑥

; ¯ 𝜈) and 𝐺(𝑡

, 𝑥

; ¯ 𝜈) in (2.16) are well dened for 𝑡

≤ 𝑡

.

Remark 2.5. The dierence between parts (ii) and (ii') of the theorem stems from the fact that in the proof of (ii') we consider [0, 𝑇 ] × 𝑆 × ℝ as the state space while for (ii) it suces to consider 𝑆 × ℝ and hence no assumptions on the time variable are necessary. Regarding applications, (ii) is obviously the better choice if stopping times with countably many values (and in particular deterministic times) are sucient.

There is a number of cases where the extension to a general stopping

time 𝜏 can be accomplished a posteriori by approximation, in particular if

one has a priori estimates for the value function so that one can restrict to test functions with specic growth properties. Assume for illustration that 𝑓 is bounded from above, then so is 𝑉 and one will be interested only in test functions 𝜑 which are bounded from above; moreover, it will typically not hurt to assume that 𝜑 is u.s.c. (or even continuous) in all variables. Now let (𝜏

) be a sequence of stopping time taking nitely many (e.g., dyadic) values such that 𝜏

↓ 𝜏 𝑃 -a.s. Applying (2.15) to each 𝜏

and using Fatou's lemma as well as the right-continuity of the paths of 𝑋

and 𝑀 , we then nd that (2.15) also holds for the general stopping time 𝜏 .

On the other hand, it is not always possible to pass to the limit as above and then (ii') is necessary to treat general stopping times. The compactness assumption should be reasonable provided that 𝑆 is 𝜎 -compact; e.g., 𝑆 = ℝ

. Then, the typical way to apply (ii') is to take (𝑡, 𝑥, 𝑚) ∈ int D and let 𝒟 be a open or closed neighborhood of (𝑡, 𝑥, 𝑚) such that 𝒟 ⊆ D .

Proof of Theorem 2.4. (i) Fix (𝑡, 𝑥, 𝑚) ∈ S. With ˆ 𝜈

as in (2.5), the deni- tion (2.2) of 𝑉 and 𝑉 ≤ 𝜑 on 𝒟 imply that

𝐸 [

𝑓 (𝑋

(𝑇 ))∣ℱ

]

(𝜔) ≤ 𝐹 (

𝜏 (𝜔), 𝑋

(𝜏 )(𝜔); 𝜈

)

≤ 𝑉 (

𝜏 (𝜔), 𝑋

(𝜏 )(𝜔), 𝑀 (𝜏 )(𝜔) )

≤ 𝜑 (

𝜏 (𝜔), 𝑋

(𝜏 )(𝜔), 𝑀 (𝜏 )(𝜔) )

𝑃 -a.s.

After noting that the left hand side is integrable by (2.1), the result follows by taking expectations.

(ii) Fix (𝑡, 𝑥, 𝑚) ∈ S ˆ and let 𝜀 > 0 .

1. Countable Cover. Like 𝒟 , the set 𝒟 ∩ D has the property that ( 𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ) )

∈ 𝒟 ∩ D 𝑃-a.s.; (2.18) this follows from (2.11) since 𝜏 takes countably many values. Replacing 𝒟 by 𝒟 ∩ D if necessary, we may therefore assume that 𝒟 ⊆ D . Using also that 𝑉 ≥ 𝜑 on 𝒟 , the denition (2.2) of 𝑉 shows that there exists for each (𝑠, 𝑦, 𝑛) ∈ 𝒟 some 𝜈

∈ 𝒰(𝑠, 𝑦, 𝑛) satisfying

𝐹 (𝑠, 𝑦; 𝜈

) ≥ 𝜑(𝑠, 𝑦, 𝑛) − 𝜀. (2.19) Fix one of the points 𝑡

in time. For each (𝑦, 𝑛) ∈ 𝒟

, the semicontinuity assumptions (2.14) and Lemma 2.1 imply that there exists a neighborhood (𝑦, 𝑛) ∈ 𝐵

(𝑦, 𝑛) ⊆ 𝑆 ˆ := 𝑆 × ℝ (of size depending on 𝑦, 𝑛, 𝑖, 𝜀, 𝛿 ) such that

𝜈

∈ 𝒰(𝑡

, 𝑦

, 𝑛

+ 𝛿) 𝜑(𝑡

, 𝑦

, 𝑛

) ≤ 𝜑(𝑡

, 𝑦, 𝑛) + 𝜀 𝐹(𝑡

, 𝑦

; 𝜈

) ≥ 𝐹 (𝑡

, 𝑦; 𝜈

) − 𝜀

⎫

 ⎬

 ⎭

for all (𝑦

, 𝑛

) ∈ 𝐵

(𝑦, 𝑛) ∩ 𝒟

.

(2.20)

Here the rst inequality may read −∞ ≤ −∞ . We note that 𝒟

⊆ 𝑆 ˆ is metric separable for the subspace topology relative to the product topology on 𝑆. ˆ Therefore, since the family {𝐵

(𝑦, 𝑛) ∩ 𝒟

: (𝑦, 𝑛) ∈ 𝑆} ˆ forms an open cover of 𝒟

, there exists a sequence (𝑦

, 𝑛

)

in 𝑆 ˆ such that {𝐵

(𝑦

, 𝑛

) ∩ 𝒟

}

is a countable subcover of 𝒟

. We set 𝜈

:= 𝜈

and 𝐵

:= 𝐵

(𝑦

, 𝑛

), so that

𝒟

⊆ ∪

𝑗≥1

𝐵

. (2.21)

We can now dene, for 𝑖 still being xed, a measurable partition (𝐴

)

of

∪

𝐵

by

𝐴

:= 𝐵

, 𝐴

:= 𝐵

∖ (𝐵

∪ ⋅ ⋅ ⋅ ∪ 𝐵

), 𝑗 ≥ 1.

Since 𝐴

⊆ 𝐵

, the inequalities (2.19) and (2.20) yield that

𝐹 (𝑡

, 𝑦

; 𝜈

) ≥ 𝜑(𝑡

, 𝑦

, 𝑛

) − 3𝜀 for all (𝑦

, 𝑛

) ∈ 𝐴

∩ 𝒟

. (2.22) 2. Concatenation. Fix an integer 𝑘 ≥ 1 ; we now focus on (𝑡

)

. We may assume that 𝑡

< 𝑡

< ⋅ ⋅ ⋅ < 𝑡

, by eliminating and relabeling some of the 𝑡

. We dene the ℱ

-measurable sets

Γ

:= {

𝜏 = 𝑡

and (𝑋

(𝑡

), 𝑀 (𝑡

)) ∈ 𝐴

}

∈ ℱ

and Γ(𝑘) := ∪

1≤𝑖,𝑗≤𝑘

Γ

. Since the 𝑡

are distinct and 𝐴

∩ 𝐴

= ∅ for 𝑗 ∕= 𝑗

, we have Γ

∩ Γ

= ∅ for (𝑖, 𝑗) ∕= (𝑖

, 𝑗

) . We can then consider the successive concatenation

𝜈(𝑘) := 𝜈 ⊗

1,Γ¹₁)

𝜈

⊗

1,Γ¹₂)

𝜈

⋅ ⋅ ⋅ ⊗

1,Γ¹_𝑘)

𝜈

⊗

2,Γ²₁)

𝜈

⋅ ⋅ ⋅ ⊗

𝑘,Γ^𝑘_𝑘)

𝜈

, which is to be read from the left with 𝜈

⊗ 𝜈

⊗ 𝜈

:= (𝜈

⊗ 𝜈

) ⊗ 𝜈

. It follows from an iterated application of (B1) that 𝜈(𝑘) is well dened and in particular 𝜈(𝑘) ∈ 𝒰

. (To understand the meaning of 𝜈 (𝑘) , it may be useful to note that in the example considered in (2.12), we would have

𝜈(𝑘) = 𝜈1

+ 1

(

𝜈 1

+ ∑

1≤𝑖,𝑗≤𝑘

𝜈

1

)

;

i.e., at time 𝜏 , we switch to 𝜈

on Γ

.) We note that (2.6) implies that 𝑋

= 𝑋

on Ω ∖ Γ(𝑘) ∈ ℱ

and hence

𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

= 𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

on Ω ∖ Γ(𝑘). (2.23) Moreover, the fact that 𝜏 = 𝑡

on Γ

, repeated application of (2.6), and (2.8) show that

𝐸 [

𝑓(𝑋

(𝑇 )) ℱ

]

≥ 𝐹(𝜏, 𝑋

(𝜏 ); 𝜈

) on Γ

, for 1 ≤ 𝑖, 𝑗 ≤ 𝑘,

and we deduce via (2.22) that 𝐸 [

𝑓(𝑋

(𝑇)) ℱ

]

1

≥ ∑

1≤𝑖,𝑗≤𝑘

𝐹 (𝜏, 𝑋

(𝜏 ); 𝜈

)1

𝑗

≥ ∑

1≤𝑖,𝑗≤𝑘

( 𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) − 3𝜀 ) 1

𝑗

≥ 𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ))1

− 3𝜀. (2.24) 3. Admissibility. Next, we show that 𝜈(𝑘) ∈ 𝒰(𝑡, 𝑥, 𝑚 + 𝛿) . By (B2) there exists, for each pair 1 ≤ 𝑖, 𝑗 ≤ 𝑘 , a process 𝑀

= {𝑀

(𝑢), 𝑢 ∈ [𝑡

, 𝑇 ]}

such that

𝑀

(⋅)(𝜔) = ( 𝑀

𝑖 𝑗

𝑡𝑖

[𝑋

(𝑡

)(𝜔)](⋅) )

(𝜔) on Γ

(2.25) and such that

𝑀

:= (𝑀 + 𝛿)1

+ 1

(

(𝑀 + 𝛿)1

+ ∑

1≤𝑖,𝑗≤𝑘

[ 𝑀

− 𝑀

(𝑡

) + 𝑀 (𝑡

) + 𝛿 ] 1

𝑗

)

is an element of ℳ

. We note that 𝑀

(𝑇 ) ≥ 𝑀

(𝑇) on Γ

since 𝑀

(𝑡

) ≤ 𝑀 (𝑡

) + 𝛿 on Γ

as a result of the rst condition in (2.20). Hence, using (2.7) and (2.25), we have

𝑔(𝑋

(𝑇 )) = 𝑔 ( 𝑋

𝑖 𝑗

𝑡𝑖,𝑋_𝑡,𝑥^𝜈 (𝑡𝑖)

(𝑇 ) )

≤ 𝑀

(𝑇) ≤ 𝑀

(𝑇 ) on Γ

. (2.26) This holds for all 1 ≤ 𝑖, 𝑗 ≤ 𝑘 . On the other hand, using (2.6) and that 𝑀 ∈ ℳ

(𝜈), we have that

𝑔(𝑋

(𝑇 )) = 𝑔(𝑋

(𝑇 )) ≤ 𝑀(𝑇) ≤ 𝑀(𝑇) + 𝛿 = 𝑀

(𝑇 ) on Ω ∖ Γ(𝑘).

(2.27) Combining (2.26) and (2.27), we have 𝑔(𝑋

(𝑇 )) ≤ 𝑀

(𝑇) on Ω and so Lemma 2.2 yields that 𝜈(𝑘) ∈ 𝒰 (𝑡, 𝑥, 𝑚 + 𝛿) .

4. 𝜀 -Optimality. We may assume that either the positive or the negative part of 𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) is integrable, as otherwise our claim (2.15) is trivial by (2.17). Using the denition (2.2) of 𝑉 as well as (2.23) and (2.24), we have that

𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) ≥ 𝐸 [

𝑓(𝑋

(𝑇 )) ]

= 𝐸 [ 𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]]

≥ 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ))1

]

− 3𝜀 + 𝐸 [

𝑓(𝑋

(𝑇 ))1

] for every 𝑘 ≥ 1 . Letting 𝑘 → ∞ , we have Γ(𝑘) ↑ Ω 𝑃 -a.s. by (2.18) and (2.21); therefore,

𝐸 [

𝑓 (𝑋

(𝑇 ))1

]

→ 0

by dominated convergence and (2.1). Moreover, monotone convergence yields 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ))1

]

→ 𝐸 [

𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) ]

;

to see this, consider separately the cases where the positive or the negative part of 𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) are integrable. Hence we have shown that

𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) ≥ 𝐸[𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ))] − 3𝜀.

As 𝜀 > 0 was arbitrary, this completes the proof of (ii).

(ii') Fix (𝑡, 𝑥, 𝑚) ∈ S ˆ and let 𝜀 > 0 . In contrast to the proof of (ii), we shall cover a subset of S ˆ rather than 𝑆 × ℝ. By (2.11), we may again assume that 𝒟 ∩ D = 𝒟 . Since 𝑉 ≥ 𝜑 on 𝒟 , the denition (2.2) of 𝑉 shows that there exists for each (𝑠, 𝑦, 𝑛) ∈ 𝒟 some 𝜈

∈ 𝒰(𝑠, 𝑦, 𝑛) satisfying

𝐹 (𝑠, 𝑦; 𝜈

) ≥ 𝜑(𝑠, 𝑦, 𝑛) − 𝜀. (2.28) Given (𝑠, 𝑦, 𝑛) ∈ 𝒟 , the semicontinuity assumptions (2.16) and a variant of Lemma 2.1 (including the time variable) imply that there exists a set 𝐵(𝑠, 𝑦, 𝑛) ⊆ S ˆ of the form

𝐵 (𝑠, 𝑦, 𝑛) = (

(𝑠 − 𝑟, 𝑠] ∩ [0, 𝑇 ] )

× 𝐵

(𝑦, 𝑛), where 𝑟 > 0 and 𝐵

(𝑦, 𝑛) is an open ball in 𝑆 × ℝ

, such that

𝜈

∈ 𝒰 (𝑠

, 𝑦

, 𝑛

+ 𝛿) 𝜑(𝑠

, 𝑦

, 𝑛

) ≤ 𝜑(𝑠, 𝑦, 𝑛) + 𝜀 𝐹 (𝑠

, 𝑦

; 𝜈

) ≥ 𝐹 (𝑠, 𝑦; 𝜈

) − 𝜀

⎫

 ⎬

 ⎭

for all (𝑠

, 𝑦

, 𝑛

) ∈ 𝐵 (𝑠, 𝑦, 𝑛) ∩ 𝒟.

(2.29) Note that we have exploited Assumption (B0'), which forces us to use the half-closed interval for the time variable. As a result, 𝐵(𝑠, 𝑦, 𝑛) is open for the product topology on [0, 𝑇 ]×𝑆 × ℝ if 𝑆 and ℝ are given the usual topology and [0, 𝑇 ] is given the topology generated by the half-closed intervals (the upper limit topology). Let us denote the latter topological space by [0, 𝑇 ]

. Like the Sorgenfrey line, [0, 𝑇 ]

is a Lindelöf space. Let 𝒟

be the canoni- cal projection of 𝒟 ⊆ S ˆ ≡ [0, 𝑇 ]×𝑆 × ℝ to 𝑆× ℝ. Then 𝒟

is again 𝜎 -compact.

It is easy to see that the product of a Lindelöf space with a 𝜎 -compact space is again Lindelöf; in particular, [0, 𝑇 ]

× 𝒟

is Lindelöf. Moreover, since 𝒟 was 𝜎 -compact for the original topology on S ˆ and the topology of [0, 𝑇 ]

is ner than the original one, we still have that 𝒟 ⊆ [0, 𝑇 ]

× 𝒟

is a countable union of closed subsets and therefore 𝒟 is Lindelöf also for the new topology.

By the Lindelöf property, the cover {𝐵 (𝑠, 𝑦, 𝑛) ∩ 𝒟 : (𝑠, 𝑦, 𝑛) ∈ S} ˆ of 𝒟 admits a countable subcover {𝐵(𝑠

, 𝑦

, 𝑛

) ∩ 𝒟}

. We set 𝜈

:= 𝜈

and 𝐵

:= 𝐵(𝑠

, 𝑦

, 𝑛

) , then 𝒟 ⊆ ∪

𝐵

and

𝐴

:= 𝐵

, 𝐴

:= 𝐵

∖ (𝐵

∪ ⋅ ⋅ ⋅ ∪ 𝐵

), 𝑗 ≥ 1

denes a measurable partition of ∪

𝐵

. Since 𝐴

⊆ 𝐵

, the inequali- ties (2.28) and (2.29) yield that

𝐹 (𝑠

, 𝑦

; 𝜈

) ≥ 𝜑(𝑠

, 𝑦

, 𝑛

) − 3𝜀 for all (𝑠

, 𝑦

, 𝑛

) ∈ 𝐴

∩ 𝒟.

Similarly as above, we rst x 𝑘 ≥ 1 , dene the ℱ

-measurable sets Γ

:= {

(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 )) ∈ 𝐴

}

and Γ(𝑘) := ∪

1≤𝑗≤𝑘

Γ

and set 𝜈(𝑘) := (

⋅ ⋅ ⋅ ((

𝜈 ⊗

𝜈

) ⊗

𝜈

) ⋅ ⋅ ⋅ ⊗

𝜈

)

. To check that 𝐸 [

𝑓(𝑋

(𝑇)) ℱ

]

≥ 𝐹 (𝜏, 𝑋

(𝜏 ); 𝜈

) on Γ

for 1 ≤ 𝑗 ≤ 𝑘, we use that 𝜏 ≤ 𝑠

on Γ

, so that we can apply (2.10) with the stopping time

˜

𝜏 := 𝜏 ∧ 𝑠

satisfying ∥ 𝜏 ˜ ∥

≤ 𝑠

and thus 𝜈

∈ 𝒰

⊆ 𝒰

𝐿∞

; c.f. (B0').

The rest of the proof is analogous to the above.

Remark 2.6. The assumption on 𝜎 -compactness in Theorem 2.4(ii') was used only to ensure the Lindelöf property of 𝒟∩D for the topology introduced in the proof. Therefore, any other assumption ensuring this will do as well.

Let us record a slight generalization of Theorem 2.4(ii),(ii') to the case of controls which are not necessarily admissible at the given point (𝑡, 𝑥, 𝑚).

The intuition for this result is that the dynamic programming principle holds as before if we use such controls for a suciently short time (as formalized by condition (2.30) below) and then switch to admissible ones. More precisely, the proof also exploits the relaxation which is anyway present in (2.15). We use the notation of Theorem 2.4.

Corollary 2.7. Let the assumptions of Theorem 2.4(ii) hold true except that 𝜈 ∈ 𝒰

and 𝑀 ∈ ℳ

are not necessarily in 𝒰(𝑡, 𝑥, 𝑚) and ℳ

(𝜈) , respectively. In addition, assume that

( 𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ) )

∈ D 𝑃 -a.s. (2.30)

Then the conclusion (2.15) of Theorem 2.4(ii) still holds true. Moreover, the same generalization holds true for Theorem 2.4(ii').

Proof. Let us inspect the proof of Theorem 2.4(ii). Using directly (2.30) rather than appealing to (2.11), the construction of the covering in Step 1 remains unchanged and the same is true for the concatenation in Step 2. In Step 3, we proceed as above up to and including (2.26). Note that (2.27) no longer holds as it used the assumption that 𝑀 ∈ ℳ

(𝜈). However, in view of (2.26) and 𝑋

(𝑇 ) = 𝑋

(𝑇 ) on Ω ∖ Γ(𝑘) , we still have

𝐸 [

𝑔(𝑋

(𝑇 )) ]

≤ 𝐸 [

𝑀

(𝑇 )1

] + 𝐸 [

𝑔(𝑋

(𝑇 ))1

]

.

Since 𝑔(𝑋

(𝑇 )) is integrable by (2.1) and Γ(𝑘) ↑ Ω , the latter expectation is bounded by 𝛿 for large 𝑘. Moreover, Γ(𝑘) ∈ ℱ

, the martingale property of 𝑀

, and the fact that 𝑀

= 𝑀 + 𝛿 on [0, 𝜏 ] yield that

𝐸 [

𝑀

(𝑇 )1

]

= 𝐸 [

𝑀

(𝜏 )1

]

= 𝐸 [

(𝑀(𝜏 ) + 𝛿)1

] .

Since 𝐸[𝑀 (𝜏 )] = 𝑚 , the right hand side is dominated by 𝑚 + 2𝛿 for large 𝑘 . Together, we conclude that

𝐸[𝑔(𝑋

(𝑇 ))] ≤ 𝑚 + 3𝛿;

i.e., 𝜈(𝑘) ∈ 𝒰 (𝑡, 𝑥, 𝑚 + 3𝛿) for all large 𝑘 . Step 4 of the previous proof then applies as before (recall that 𝑓 (𝑋

(𝑇 )) is integrable by (2.1)), except that the changed admissibility of 𝜈(𝑘) now results in

𝑉 (𝑡, 𝑥, 𝑚 + 3𝛿) ≥ 𝐸[𝜑(𝜏, 𝑋

(𝜏 ), 𝑀 (𝜏 ))].

However, since 𝛿 > 0 was arbitrary, this is the same as (2.15). The argument to extend Theorem 2.4(ii') is analogous.

While we shall see that the relaxation 𝛿 > 0 in (2.15) is harmless for the derivation of the Hamilton-Jacobi-Bellman equation, it is nevertheless inter- esting to know when the 𝛿 can be omitted; i.e., when 𝑉 is right continuous in 𝑚 . The following sucient condition will be used when we consider state constraints.

Lemma 2.8. Let (𝑡, 𝑥, 𝑚) ∈ D. For each 𝛿 > 0 there is 𝜈

∈ 𝒰 (𝑡, 𝑥, 𝑚 + 𝛿) such that

𝐹 (𝑡, 𝑥; 𝜈

) ≥ 𝛿

∧ 𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) − 𝛿.

Let 𝛿

> 0 and assume that for all 0 < 𝛿 ≤ 𝛿

there exists 𝜈 ˜

∈ 𝒰 (𝑡, 𝑥, 𝑚) such that

lim

𝑃 {

𝑋

(𝑇 ) ∕= 𝑋

(𝑇 ) }

= 0 and such that the set

{[ 𝑓 (𝑋

(𝑇)) − 𝑓 (𝑋

(𝑇 )) ]

: 0 < 𝛿 ≤ 𝛿

}

⊆ 𝐿

(𝑃) (2.31) is uniformly integrable. Then 𝑚

7→ 𝑉 (𝑡, 𝑥, 𝑚

) is right continuous at 𝑚 . Proof. The monotonicity of 𝑚

7→ 𝑉 (𝑡, 𝑥, 𝑚

) implies that the right limit 𝑉 (𝑡, 𝑥, 𝑚+) exists and that 𝑉 (𝑡, 𝑥, 𝑚+) ≥ 𝑉 (𝑡, 𝑥, 𝑚) ; it remains to prove the opposite inequality. Since 𝑚

7→ 𝒰 (𝑡, 𝑥, 𝑚

) is increasing, we have that (𝑡, 𝑥, 𝑚) ∈ D implies 𝒰 (𝑡, 𝑥, 𝑚 + 𝛿) ∕= ∅ for all 𝛿 ≥ 0. Hence 𝜈

exists; of course, the truncation at 𝛿

is necessary only if 𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) = ∞ . Let 0 < 𝛿 ≤ 𝛿

and set 𝐴

:= {𝑋

(𝑇 ) ∕= 𝑋

(𝑇 )} , then

𝑉 (𝑡, 𝑥, 𝑚) ≥ 𝐹 (𝑡, 𝑥, 𝜈 ˜

)

= 𝐹 (𝑡, 𝑥, 𝜈

) − 𝐸 [ 1

(

𝑓 (𝑋

(𝑇 )) − 𝑓(𝑋

(𝑇 )) )]

≥ 𝛿

∧ 𝑉 (𝑡, 𝑥, 𝑚 + 𝛿) − 𝛿 − 𝐸 [ 1

( 𝑓(𝑋

(𝑇 )) − 𝑓 (𝑋

(𝑇 )) )

]

.

Letting 𝛿 ↓ 0 , we deduce by (2.31) that 𝑉 (𝑡, 𝑥, 𝑚) ≥ 𝑉 (𝑡, 𝑥, 𝑚+) .

Remark 2.9. The integrability assumption (2.31) is clearly satised if 𝑓 is bounded. In the general case, it may be useful to consider the value function for a truncated function 𝑓 in a rst step.

3 Application to State Constraints

We consider an open set 𝒪 ⊆ 𝑆 := ℝ

and study the stochastic control problem under the constraint that the state process has to stay in 𝒪 . Namely, we consider the value function

𝑉 ¯ (𝑡, 𝑥) := sup

𝜈∈𝒰(𝑡,𝑥)¯

𝐹 (𝑡, 𝑥; 𝜈), (𝑡, 𝑥) ∈ S, (3.1) where

𝒰 ¯ (𝑡, 𝑥) := {

𝜈 ∈ 𝒰

: 𝑋

(𝑠) ∈ 𝒪 for all 𝑠 ∈ [𝑡, 𝑇 ], 𝑃 -a.s. } . In the following discussion we assume that, for all (𝑡, 𝑥) ∈ S and 𝜈 ∈ 𝒰

,

𝑋

has continuous paths; (3.2) (𝑡, 𝑥) 7→ 𝑋

(𝑟) is continuous in probability, uniformly in 𝑟; (3.3) 𝒰 ¯ (𝑡, 𝑥) ∕= ∅ for (𝑡, 𝑥) ∈ [0, 𝑇 ] × 𝒪 . (3.4) Explicitly, the condition (3.3) means that (𝑡

, 𝑥

) → (𝑡, 𝑥) implies

sup

𝑟∈[0,𝑇]

𝑑 (

𝑋

(𝑟), 𝑋

(𝑟) )

→ 0 in probability,

where we set 𝑋

(𝑟) := 𝑥 for 𝑟 < 𝑡 , 𝑑(⋅, ⋅) denotes the Euclidean metric, and it is implicitly assumed that 𝜈 ∈ 𝒰

for all 𝑛 . We shall augment the state process so that the state constraint becomes a special case of an expectation constraint. To this end, we introduce the distance function 𝑑(𝑥) := inf {𝑑(𝑥, 𝑦) : 𝑦 ∈ 𝑆 ∖ 𝒪} for 𝑥 ∈ 𝑆 and the auxiliary process

𝑌

(𝑠) := 𝑦 ∧ inf

𝑟∈[𝑡,𝑠]

𝑑(𝑋

(𝑟)), 𝑠 ∈ [𝑡, 𝑇 ], 𝑦 ∈ [0, ∞). (3.5) By (3.2), each trajectory {𝑋

(𝑟)(𝜔), 𝑟 ∈ [𝑡, 𝑇 ]} ⊆ 𝑆 is compact; therefore, it has strictly positive distance to 𝑆 ∖ 𝒪 whenever it is contained in 𝒪 :

{𝑋

(𝑟)(𝜔), 𝑟 ∈ [𝑡, 𝑇 ]} ⊆ 𝒪 if and only if 𝑌

(𝑇 )(𝜔) > 0.

We consider the augmented state process 𝑋 ¯

(⋅) := (

𝑋

(⋅), 𝑌

(⋅) ) on the state space 𝑆 × [0, ∞), then

𝐸[𝑔( ¯ 𝑋

(𝑇))] = 𝑃{𝑌

(𝑇) ≤ 0} by setting 𝑔(𝑥, 𝑦) := 1

(𝑦)

for (𝑥, 𝑦) ∈ 𝑆 × [0, ∞) . Now the state constraint may be expressed as 𝐸[𝑔( ¯ 𝑋

(𝑇 ))] ≤ 0 and therefore

𝒰 ¯ (𝑡, 𝑥) = 𝒰 (𝑡, 𝑥, 1, 0) and 𝑉 ¯ (𝑡, 𝑥) = 𝑉 (𝑡, 𝑥, 1, 0);

of course, the value 1 may be replaced by any number 𝑦 > 0 . Here and in the sequel, we use the notation from the previous section applied to the controlled state process 𝑋 ¯ on 𝑆 × [0, ∞) ; i.e., we tacitly replace the variable 𝑥 by (𝑥, 𝑦) to dene the set 𝒰 (𝑡, 𝑥, 𝑦, 𝑚) of admissible controls and the associated value function 𝑉 (𝑡, 𝑥, 𝑦, 𝑚) .

One direction of the dynamic programming principle will be a conse- quence of the following counterpart of Assumption A.

Assumption A ¯ . For all (𝑡, 𝑥) ∈ S , 𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥) , 𝜏 ∈ 𝒯

and 𝑃 -a.e. 𝜔 ∈ Ω , there exists 𝜈

∈ 𝒰 ¯ (𝜏 (𝜔), 𝑋

(𝜏 )(𝜔)) such that

𝐸 [

𝑓 (𝑋

(𝑇 )) ℱ

]

(𝜔) ≤ 𝐹 (

𝜏 (𝜔), 𝑋

(𝜏 )(𝜔); 𝜈

) .

The more dicult direction of the dynamic programming principle will be inferred from Theorem 2.4 under a right-continuity condition; we shall exemplify in the subsequent section how to verify this condition.

Theorem 3.1. Consider (𝑡, 𝑥) ∈ S and a family {𝜏

, 𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥)} ⊆ 𝒯

. (i) Let Assumption 𝐴 ¯ hold true and let 𝜙 : S → [−∞, ∞] be a measurable

function such that 𝑉 ¯ ≤ 𝜙 . Then 𝐸 [

𝜙(𝜏

, 𝑋

(𝜏

))

]

< ∞ for all 𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥) and

𝑉 ¯ (𝑡, 𝑥) ≤ sup

𝜈∈𝒰(𝑡,𝑥)¯

𝐸 [

𝜙(𝜏

, 𝑋

(𝜏

)) ] .

(ii) Let Assumption B' hold true for the state process 𝑋 ¯ on 𝑆 × [0, ∞) and let (3.2)(3.4) hold true. Moreover, assume that

𝑉 (𝑡, 𝑥, 1, 0) = 𝑉 (𝑡, 𝑥, 1, 0+) (3.6) and that (𝑠

, 𝑥

) 7→ 𝐹 (𝑠

, 𝑥

; 𝜈) is l.s.c. on [0, 𝑡

] × 𝒪 for all 𝑡

∈ [𝑡, 𝑇 ] and 𝜈 ∈ 𝒰

. Then

𝑉 ¯ (𝑡, 𝑥) ≥ sup

𝜈∈𝒰¯(𝑡,𝑥)

𝐸 [

𝜙(𝜏

, 𝑋

(𝜏

)) ]

(3.7) for any u.s.c. function 𝜙 : S → [−∞, ∞) such that 𝑉 ¯ ≥ 𝜙 .

Proof. (i) We may assume that 𝒰 ¯ (𝑡, 𝑥) ∕= ∅ as otherwise 𝑉 ¯ (𝑡, 𝑥) = −∞ . As

in the proof of (2.13), we obtain that 𝐹 (𝑡, 𝑥; 𝜈) ≤ 𝐸[𝜑(𝜏, 𝑋

(𝜏 ))] for all

𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥) . The claim follows by taking supremum over 𝜈 .

(ii) Again, we may assume that 𝒰 ¯ (𝑡, 𝑥) ∕= ∅ as otherwise the right hand side of (3.7) equals −∞ . We set

𝒟 := [𝑡, 𝑇 ] × 𝒪 × (0, ∞) × {0}.

If 𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥) , then 𝑔( ¯ 𝑋

(𝑇 )) = 0 and hence the constant martingale 𝑀 := 0 is contained in ℳ

(𝜈) . Moreover, 𝑌

(𝑠) > 0 and hence (𝑠, 𝑋 ¯

(𝑠), 𝑀 (𝑠)) ∈ 𝒟 for all 𝑠 ∈ [𝑡, 𝑇 ] , 𝑃 -a.s. Furthermore, if we dene

𝜑(𝑡

, 𝑥

, 𝑦

, 𝑚

) :=

{ 𝜙(𝑡

, 𝑥

), (𝑡

, 𝑥

, 𝑦

, 𝑚

) ∈ 𝒟

−∞, otherwise,

then 𝜑 is u.s.c. on 𝒟 . To see that the third semicontinuity condition in (2.16) is also satised, note that for any (𝑡

, 𝑥

, 𝑦

), (𝑡

, 𝑥

, 𝑦

) ∈ [0, 𝑡

] × 𝑆 × [0, ∞) and 𝜈 ∈ 𝒰

,

∣𝑌

(𝑇)−𝑌

(𝑇)∣

≤ ∣𝑦

− 𝑦

∣ + inf

𝑟∈[𝑡^′,𝑇]

𝑑(𝑋

(𝑟)) − inf

𝑟∈[𝑡^′′,𝑇]

𝑑(𝑋

(𝑟))

≤ ∣𝑦

− 𝑦

∣ + sup

𝑟∈[0,𝑇]

𝑑 (

𝑋

(𝑟), 𝑋

(𝑟) ) .

Hence (3.3) implies that (𝑡

, 𝑥

, 𝑦

) 7→ 𝑌

(𝑇) is continuous in probability.

As (−∞, 0] ⊂ ℝ is closed, we conclude by the Portmanteau theorem that (𝑡

, 𝑥

, 𝑦

) 7→ 𝑃 {

𝑌

(𝑇 ) ∈ (∞, 0] }

≡ 𝐺(𝑡

, 𝑥

, 𝑦

; 𝜈) is u.s.c.

as required. Since any open subset of a Euclidean space is 𝜎 -compact and since (3.4) implies 𝒟 ∩ D = 𝒟 , we can use (3.6) and Theorem 2.4(ii') with 𝑀 = 0 to obtain that

𝑉 ¯ (𝑡, 𝑥) = 𝑉 (𝑡, 𝑥, 1, 0)

= 𝑉 (𝑡, 𝑥, 1, 0+)

≥ 𝐸 [ 𝜑 (

𝜏

, 𝑋

(𝜏

), 𝑌

(𝜏

), 0 )]

= 𝐸[𝜙(𝜏

, 𝑋

(𝜏

))].

As 𝜈 ∈ 𝒰 ¯ (𝑡, 𝑥) was arbitrary, the result follows.

Remark 3.2. Similarly as in Theorem 2.4(ii), there is also a version of The-

orem 3.1(ii) for stopping times taking countably many values. In this case,

Assumption B replaces Assumption B', all conditions on the time variable

are superuous, and one can consider a general separable metric space 𝑆 .

4 Application to Controlled Diusions

In this section, we show how the weak formulation of the dynamic program- ming principle applies in the context of controlled Brownian stochastic dif- ferential equations and how it allows to derive the Hamilton-Jacobi-Bellman PDEs for the value functions associated to optimal control problems with expectation or state constraints. As the main purpose of this section is to illustrate the use of Theorems 2.4 and 3.1, we shall choose a fairly simple setup allowing to explain the main points, rather than seeking for maximal generality.

4.1 Setup for Controlled Diusions

From now on, we take 𝑆 = ℝ

and let Ω = 𝐶([0, 𝑇 ]; ℝ

) be the space of continuous paths, 𝑃 the Wiener measure on Ω , and 𝑊 the canonical process 𝑊

(𝜔) = 𝜔

. Let 𝔽 = (ℱ

)

be the augmentation of the ltration generated by 𝑊 ; without loss of generality, ℱ = ℱ

. For 𝑡 ∈ [0, 𝑇 ] , the auxiliary ltration 𝔽

= (ℱ

)

is chosen to be the augmentation of 𝜎(𝑊

− 𝑊

, 𝑡 ≤ 𝑟 ≤ 𝑠) ; in particular, 𝔽

is independent of ℱ

.

Consider a closed set 𝑈 ⊆ ℝ

and let 𝒰 be the set of all 𝑈 -valued pre- dictable processes 𝜈 satisfying 𝐸[ ∫

0

∣𝜈

∣

𝑑𝑡] < ∞ . Then we set 𝒰

= {

𝜈 ∈ 𝒰 : 𝜈 is 𝔽

-predictable } .

This choice will be convenient to verify Assumption B'. We remark that the restriction to 𝔽

-predictable controls entails no loss of generality, in the sense that the alternative choice 𝒰

= 𝒰 would result in the same value function.

Indeed, this follows from a well known randomization argument (see, e.g., [3, Remark 5.2]).

Let 𝕄

denote the set of 𝑑 × 𝑑 matrices. Given two Lipschitz continuous functions

𝜇 : ℝ

× 𝑈 → ℝ

, 𝜎 : ℝ

× 𝑈 → 𝕄

and (𝑡, 𝑥, 𝜈) ∈ [0, 𝑇 ] × ℝ

× 𝒰 , we dene 𝑋

(⋅) as the unique strong solution of the stochastic dierential equation (SDE)

𝑋(𝑠) = 𝑥 +

∫

𝜇(𝑋(𝑟), 𝜈

) 𝑑𝑟 +

∫

𝜎(𝑋(𝑟), 𝜈

) 𝑑𝑊

, 𝑡 ≤ 𝑠 ≤ 𝑇, (4.1) where we set 𝑋

(𝑟) = 𝑥 for 𝑟 ≤ 𝑡. As 𝑋

(𝑇 ) is square integrable for any 𝜈 ∈ 𝒰 , (2.1) is satised whenever

𝑓 and 𝑔 have quadratic growth, (4.2) which we assume from now on. In addition, we also impose that

{ 𝑓 is l.s.c. and 𝑓

has subquadratic growth,

𝑔 is u.s.c. and 𝑔

has subquadratic growth , (4.3)

where ℎ : ℝ

→ ℝ is said to have subquadratic growth if ℎ(𝑥)/∣𝑥∣

→ 0 as

∣𝑥∣ → ∞ . This will be used to obtain the semicontinuity properties (2.16).

Furthermore, we take ℳ

to be the family of all càdlàg martingales which start at 0 and are adapted to 𝔽

. By the independence of the incre- ments of 𝑊 , we see that 𝑀 ∈ ℳ

is then also a martingale in the ltration 𝔽

and that 𝑀

= 0 for 𝑟 ≤ 𝑡 . For 𝜈 ∈ 𝒰

, we have 𝑋

(𝑇 ) ∈ 𝐿

(ℱ

, 𝑃 ) and hence (2.4) is satised. It will be useful to express the martingales as stochastic integrals. Let 𝒜

denote the set of ℝ

-valued 𝔽

-predictable processes 𝛼 such that ∫

0

∣𝛼

∣

𝑑𝑡 < ∞ 𝑃 -a.s. and such that 𝑀

(⋅) :=

∫

𝛼

𝑑𝑊

is a martingale (

denotes transposition). Then the Brownian representation theorem yields that

ℳ

= {

𝑀

: 𝛼 ∈ 𝒜

} .

In the following, we also use the notation 𝑀

:= 𝑚 + 𝑀

.

Lemma 4.1. In the above setup, Assumptions A, 𝐴 ¯ , B, B' are satised and 𝐹 and 𝐺 satisfy (2.16).

Proof. Assumption (B0') is immediate from the denition of 𝒰

. We dene the concatenation of controls by (2.12).

The validity of Assumptions A, A ¯ and (B1') follows from the uniqueness and the ow property of (4.1); in particular, the control 𝜈

in Assumption A can be dened by 𝜈

(𝜔

) := 𝜈(𝜔 ⊗

𝜔

) , 𝜔

∈ Ω , where the concatenated path 𝜔 ⊗

𝜔

is given by

(𝜔 ⊗

𝜔

)

:= 𝜔

1

(𝑟) + (

𝜔

− 𝜔

(𝜔) + 𝜔

(𝜔) )

1

(𝑟).

While we refer to [3, Proposition 5.4] for a detailed proof, we emphasize that the choice of 𝒰

is crucial for the validity of (2.10): in the notation of Assumption B', (2.10) essentially requires that 𝜈 ¯ be independent of ℱ

.

Let 𝑡, 𝜏, 𝜈, 𝜈 ¯ be as in Assumption B', we show that (B2') holds. Let 𝑀 ¯ be a càdlàg version of

𝑀 ¯ (𝑟) := 𝐸[𝑔(𝑋

(𝑇 ))∣ℱ

], 𝑟 ∈ [0, 𝑇 ], where ˆ 𝜈 := 𝜈1

+ 1

𝜈. ¯ By the same argument as in [3, Proposition 5.4], we deduce from the unique- ness and the ow property of (4.1) and the fact that 𝜈 ¯ is independent of ℱ

that

𝐸[𝑔(𝑋

(𝑇 ))∣ℱ

] = 𝐸[𝑔(𝑋

𝑡,𝑥(𝜏)

(𝑇 ))∣ℱ

] = 𝑀

[𝑋

(𝜏 )](𝑟) on [𝜏, 𝑇 ].

Hence 𝑀 ¯ = 𝑀

[𝑋

(𝜏 )] on [𝜏, 𝑇 ]. The last assertion of (B2') is clear by the

denition of ℳ

. As already mentioned in Remark 2.3, Assumption (B3')

follows from Assumption A and Assumption B follows from Assumption B'.