First and Second Order Necessary Conditions for Stochastic Optimal Control Problems

J. Frédéric Bonnans; Francisco J. Silva

doi:10.1007/s00245-012-9162-4

Applied Mathematics & Optimization

June 2012, Volume 65, Issue 3, pp 403–439 | Cite as

First and Second Order Necessary Conditions for Stochastic Optimal Control Problems

Authors
Authors and affiliations

J. Frédéric BonnansEmail author
Francisco J. Silva

Article

First Online: 09 February 2012

Abstract

In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state.

Keywords

Stochastic optimal control Variational approach First and second order optimality conditions Polyhedric constraints Final state constraints

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Appendix

6.1 Technical Estimates in the Expansion of Solution of SDEs

Here we prove some technical estimates stated in Sect. 3.

Proof of Lemma 3.5

For notational convenience we will suppose that m=n=d=1. We have

$$ \begin{array} {@{}rcl} \mathrm{d}\delta y(t)&=& \bigl[\tilde {f}_{y}(t)\delta y(t) + \tilde{f}_{u}(t)v(t)\bigr]\mathrm{d}t + \bigl[\tilde {\sigma}_{y}(t)\delta y(t) + \tilde{\sigma}_{u}(t)v(t) \bigr]\mathrm{d}W(t),\\[2mm]\delta y (0)&=&0. \end{array} $$

(5.38)

where, for ψ=f,σ,

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Using the second assumption in (2.1), estimates (3.8), (3.8) follow from Corollary 3.1 applied to (6.1) and (3.7) respectively.

We next prove (3.9). We have that

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For ψ=f,σ, we have that $[\tilde{\psi }_{y}(t)-\psi_{y}(t) ]y_{1}(t)=O ( [|\delta y(t)|+|v(t)| ]|y_{1}(t)| )$. Also,

$$\bigl[\tilde{\sigma}_{u}(t)-\sigma_{u}(t) \bigr]v(t)=\left \{ \begin{array}{@{}l@{\quad}l}O (|\delta y(t)| |v(t)| ) & \mbox{if}\ \sigma_{uu} \equiv0,\\[1.5mm]O ( [ |\delta y(t)|+|v(t)| ] |v(t)| ) &\mathrm{otherwise}.\end{array} \right .$$

Therefore, the following equation holds for d ₁:

Open image in new window

where

$$D\bigl(\delta y(t), y_{1}(t), v(t)\bigr)=\left \{ \begin{array}{@{}l@{\quad}l} [|\delta y(t)|+|v(t)| ] [| y_{1}(t)|+|v(t)| ]-|v(t)|^{2} &\mbox{if}\ \sigma_{uu}\equiv0, \\[1.5mm][|\delta y(t)|+|v(t)| ] [| y_{1}(t)|+|v(t)| ] &\mathrm{otherwise}.\end{array} \right .$$

By (3.8) and the Cauchy Schwarz inequality

Open image in new window

(5.39)

By similar arguments,

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and (3.9) follows by Corollary 3.1, since $\|v^{2}\|_{\beta,1}^{\beta}=\|v\|_{2\beta,2}^{2\beta}$ and $\|v^{2}\|_{\beta,2}^{\beta}=\|v\|_{2\beta,4}^{2\beta}$. □

Proof of Lemma 3.11

As in the proof of Lemma 3.5 we suppose that m=n=d=1. We will use repeatedly that for every β,p,q∈[1,∞), we have

$$\bigl\| |v|^{q}\bigr\|_{\beta,p}^{\beta}=\|v\|_{q\beta,qp}^{q\beta}\quad\mbox{for all}\ v\in L^{q\beta,qp}_{\mathcal{F}} .$$

Proof of (3.18)

Recall that, by (H1), for ψ=f,σ we assume that ψ _yy, ψ _yu and ψ _uu are bounded. Using (3.8),

$$ \bigl\| y_{1}^{2} \bigr\|_{\beta ,2}^{\beta}=\mathbb {E} \biggl[ \biggl(\int_{0}^{T}\bigl|y_{1}(t)\bigr|^{4} \mathrm{d}t \biggr)^{\frac{\beta }{2}} \biggr]=O \bigl[\mathbb{E} \bigl(\sup\bigl|y_{1}(t)\bigr|^{2\beta } \bigr) \bigr]=O\bigl(\|v\|_{2\beta, 2}^{2\beta} \bigr).$$

(5.40)

Analogously, the estimates associated with the term y ₁ v is of order $\|v\|_{2\beta, 2}^{2\beta}$. Estimate (3.18) follows from Corollary 3.1 since $\| v^{2}\|_{\beta ,1}^{\beta}=\|v\|_{2\beta,2}^{2\beta}$ and $\| v^{2}\|_{\beta ,2}^{\beta }=\|v\|_{2\beta,4}^{2\beta}$. □

Proof of (3.19)

Recall that $d_{2}=\delta y - y_{1}-\frac{1}{2}y_{2}$. We have, omitting time from the arguments,

Open image in new window

where for ψ=f,σ the map r _t(ψ) is defined by

$$r_{t}(\psi):= \int_{0}^{1} (1-\theta)\bigl[ \psi_{yy}\bigl(\bar {y}(t)+\theta\delta y(t), \bar{u}(t)+\theta v(t)\bigr)- \psi_{yy}\bigl(\bar {y}(t),\bar{u}(t)\bigr) \bigr]\mathrm{d}\theta.$$

Recall that if Q is a quadratic form and a is the associated symmetric bilinear form, we have the identity Q(y)−Q(x)=a(y+x,y−x). Thus, since Dψ is Lipschitz, we obtain

Open image in new window

(5.41)

where, for ψ=f,σ,

$$\alpha_{t}(\psi):= \left \{ \begin{array}{@{}l@{\quad}l}|\delta y(t)|^{3}+ |v(t)|^{3} &\mbox{if}\ \psi_{uuu} \neq0, \\|\delta y(t)|^{3}+ |\delta y(t)||v(t)|^{2} & \mbox{if}\ \psi_{uuu}\equiv0.\end{array} \right .$$

Now, let us estimate the terms in the dW(t) part of (6.4),

Open image in new window

by (3.8) and (3.9). Analogously, estimates for the terms d ₁ y ₁ and d ₁ v are of the same order. Let us estimate the terms appearing in α _t(σ). Using (3.8),

$$ \bigl\| |\delta y|^{3}\bigr\|_{\beta ,2}^{\beta}=\mathbb {E} \biggl[ \biggl(\int_{0}^{T}\bigl|\delta y(t)\bigr|^{6} \mathrm{d}t \biggr)^{\frac{\beta }{2}} \biggr]=O \bigl[\mathbb{E} \bigl(\sup \bigl|\delta y(t)\bigr|^{3\beta } \bigr) \bigr]=O\bigl( \|v\|_{3\beta,2}^{3\beta}\bigr).$$

(5.42)

By (3.8), we obtain

Open image in new window

Also, we have that $\|v^{3}\|_{\beta,1}^{\beta}=\|v\|_{3\beta ,3}^{3\beta }$ and $\|v^{3}\|_{\beta,2}^{\beta}=\|v\|_{3\beta,6}^{3\beta}$. By the Cauchy Schwarz inequality,

$$\|v\|_{3\beta,3}^{3\beta}=\mathbb{E} \biggl[ \biggl(\int _{0}^{T} \bigl|v(t)\bigr|^{3} \mathrm{d}t\biggr)^{\beta} \biggr]\leq\mathbb{E} \biggl[ \biggl(\int _{0}^{T}\bigl|v(t)\bigr|^{2} \mathrm{d}t\biggr)^{\frac{\beta}{2}} \biggl(\int_{0}^{T}\bigl|v(t)\bigr|^{4}\mathrm{d}t \biggr)^{\frac{\beta}{2}} \biggr].$$

Using the Cauchy Schwarz inequality again, we get $\|v\|_{3\beta ,3}^{3\beta}= O(\|v\|_{2\beta,2}^{\beta} \|v\|_{4\beta,4}^{2\beta})$. Therefore, estimate (3.19) follows from Corollary 3.1. □

Proof of Lemma 3.13

As in the proof of Proposition 3.8 we denote $\delta J:= J(\bar {u}+v)-J(\bar{u})$. By definition,

$$\delta J=\mathbb{E} \biggl( \int_{0}^{T} \bigl[\ell(y_{\bar {u}+v},\bar {u}+v)-\ell(\bar{y},\bar{u}) \bigr] \mathrm{d}t+ \phi \bigl(y_{\bar {u}+v}(T)\bigr)-\phi \bigl(\bar{y}(T)\bigr) \biggr)=I_{1}+I_{2},$$

where, omitting the time argument in the integral,

$$\begin{array}{@{}rcl} I_{1} &:= &\displaystyle\mathbb{E} \Biggl( \int _{0}^{T} \biggl[\ell_{y} \delta y +\ell_{u} v + \frac{1}{2}\ell_{(y,u)^{2}} (\delta y ,v)^{2} +r_{\ell} (\delta y , v )^{2} \biggr]\mathrm{d}t\Biggr),\\[4mm]I_{2} &:= &\displaystyle\mathbb{E} \biggl[ \phi_{y}\bigl(\bar{y}(T)\bigr) \delta y(T) + \frac{1}{2} \phi_{yy}\bigl(\bar{y}(T)\bigr)\bigl(\delta y(T)\bigr)^{2}+r_{\phi}\bigl(\bar{y}(T)\bigr)\bigl(\delta y(T)\bigr)^{2} \biggr] . \end{array} $$

(5.43)

Recalling that $\delta y = y_{1}+d_{1}=y_{1}+\frac{1}{2}y_{2}+d_{2}$, assumption (2.5) in (H2) yields

Open image in new window

where, omitting time from function arguments,

$$z_{1}(v):= \mathbb{E} \bigl(\sup \bigl[ |d_{1}|^{2}+ \bigl|d_{1}(t)\bigr| |y_{1} | + |\delta y|^{3} \bigr] \bigr)+\|v\|_{1}\mathbb{E} \bigl(\sup|d_{1} | \bigr) + \|v\|_{3}^{3}.$$

On the other hand,

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where

$$z_{2}(v):= \mathbb{E} \bigl(\bigl|\delta y(T)\bigr|^{3} +\bigl|y_{1}(T)\bigr| \bigl|d_{1}(T)\bigr|+\bigl|d_{1}(T)\bigr|^{2}\bigr).$$

Denoting z(v):=z ₁(v)+z ₂(v) we get that

Open image in new window

Therefore, using (3.13) and (3.21), we get (3.22). Now, we proceed to estimate z(v). By (3.9) we have that

$$\mathbb{E} \Bigl(\sup_{t\in[0,T]} \bigl|d_{1}(t)\bigr|^{2}\Bigr)=O\bigl(\|v\|_{4}^{4}\bigr)=O\bigl(\|v\|_{\infty}^{2}\|v\|_{2}^{2}\bigr).$$

Estimates (3.8), (3.9) and the Cauchy Schwarz inequality yield

$$\mathbb{E} \Bigl(\sup_{t\in[0,T]}\bigl|d_{1}(t)\bigr| \bigl|y_{1}(t)\bigr|\Bigr)=O \bigl(\| v\|_{4}^{2} \|v\|_{2} \bigr)=O\bigl(\|v\|_{\infty}\|v\|_{2}^{2} \bigr).$$

Analogously, using (3.9), we have

$$\mathbb{E} \Bigl(\|v\|_{1}\sup_{t\in[0,T]}\bigl|d_{1}(t)\bigr|\Bigr)=O \bigl(\| v\|_{4}^{2}\|v\|_{2,1} \bigr)=O\bigl(\|v\|_{\infty}\|v\|_{2}^{2} \bigr).$$

Estimate (3.8) yields $\mathbb{E} (\sup_{t\in[0,T]}|\delta y(t)|^{3} )=O(\|v\|_{3,2}^{3})$. But

$$\|v\|_{3,2}^{3}=\mathbb{E} \biggl( \biggl[\int _{0}^{T} \bigl|v(t)\bigr|^{2}\mathrm{d}t\biggr]^{\frac{3}{2}} \biggr)=O \bigl(\|v\|_{\infty}\|v\|_{2}^{2} \bigr),$$

and $\|v\|_{3}^{3} =O (\|v\|_{\infty}\|v\|_{2}^{2} )$. Thus, $z(v)=O (\|v\|_{\infty}\|v\|_{2}^{2} )$. □

6.2 Adapted Projections

This subsection discusses projections in $L^{2}_{\mathcal{F}}$ in the case of local constraints. In order to give the expression of the tangent cone to $\mathcal{U}$, when $\mathcal{U}$ is defined by (4.8), we need a characterization of measurable multifunctions with closed values. We call Castaing representation of a $(\mathcal{B}[0,T]\times \mathcal{F}_{T})$-measurable multifunction $U: [0,T]\times\Omega\to\mathcal{P}(\mathbb{R}^{m})$ a countable family of $(\mathcal{B}[0,T]\times\mathcal{F}_{T}) /\mathcal {B}(\mathbb{R}^{m})$-measurable functions w _k:[0,T]×Ω→ℝ^m such that U(t,ω)=clo{w _k(t,ω), k∈ℕ}. We say that the Castaing representation is adapted if each process (w _k(t))_t∈[0,T] is adapted. By a result due to C. Castaing (see e.g. [29, Thm 1B, p. 161]), any multifunction with closed values that is measurable in $L^{2}_{\mathcal{F}_{T}}$ has a Castaing representation in the same space. We next extend the result to the adapted case.

Proposition 5.14

Let $U: [0,T]\times\Omega\to\mathcal{P}(\mathbb {R}^{m})$ be an $(\mathcal{B}[0,T]\times\mathcal{F}_{T} )$-measurable, $\mathbb{F}$-adapted closed-convex-valued multifunction. Then:

(i)

For any a∈ℝ^m the map (t,ω)∈[0,T]×Ω→P _a(t,ω):=P _U(t,ω)(a)∈ℝ^m is $( \mathcal{B}[0,T]\times\mathcal{F}_{T})/\mathcal{B}(\mathbb{R}^{m})$-measurable and $\mathbb{F}$-adapted.
(ii)

For any countable dense subset {z _k}_k∈ℕ of ℝ^m we have that {P _U(t,ω)(z _k), k∈ℕ} is an $\mathbb{F}$-adapted Castaing representation of U.

Proof

First note that (ii) follows directly from (i). In order to prove (i) we essentially reproduce the proof in [29] to obtain that P _a(t,ω) is $( \mathcal{B}[0,T]\times\mathcal{F}_{T})/\mathcal{B}(\mathbb{R}^{m})$-measurable and, using that U is $\mathbb{F}$-adapted, we prove that P _a(t,ω) is $\mathbb{F}$-adapted. Let us fix a∈ℝ^m and consider the sequence of multifunctions

$$ U_k(t,{\omega}) := \bigl\{ v\in \mathbb{R}^m; \; \mathrm{dist}\bigl(v,U(t,{\omega})\bigr) < k^{-1};\; |v-a| < \mathrm{dist}\bigl(a,U(t,{\omega})\bigr) + k^{-1} \bigr\}.$$

(5.44)

Let C be a closed subset of ℝ^m. Then P _a(t,ω)∈C iff C∩U _k(t,ω)≠∅ for all k, thus

$$ P_ a^{-1}(C)=\bigcap_kU_k^{-1}(C).$$

(5.45)

Next, let D be a countable dense subset of C, which always exists. We claim that

$$ U_k^{-1}(C) = U_k^{-1}(D)= \bigcup_{d\in D} U_k^{-1}(d).$$

(5.46)

The second equality is obvious and since D is a dense subset of C, in order to establish the first equality it suffices to check that if c∈C and $(t_{0},{\omega}_{0}) \in U_{k}^{-1}(c)$, then for c′ close enough to c we have that $(t_{0},{\omega}_{0}) \in U_{k}^{-1}(c')$. But this follows directly from the definition of U _k(t,ω) in (6.7). Our claim follows.

On the other hand, for any v∈ℝ^m and α≥0 we have

$$\bigl\{(t,{\omega}) \in[0,T]\times\Omega; \ \mathrm{dist}\bigl(v,U(t,{\omega })\bigr)\leq\alpha\bigr\} = U^{-1}(v+\alpha\bar{B}),$$

(5.47)

where $\bar{B}$ is the unit ball in ℝ^m. Thus, since U is $\mathcal{B}[0,T] \times\mathcal{F}_{T}$-measurable, so is the process dist(v,U(t)). Similarly, for a.a. t∈[0,T], we have that

$$\bigl\{{\omega}\in\Omega;\mathrm{dist}\bigl(v,U(t,{\omega})\bigr)\leq\alpha\bigr\}= U^{-1}(v+\alpha\bar{B}).$$

(5.48)

Since U is $\mathbb{F}$-adapted, it follows that so is dist(v,U(t)). Therefore, from the definition (6.7), for any (t,d)∈[0,T]×∈ℝ^m we have that $U_{k}^{-1}(d)\in\mathcal{B}[0,T] \times\mathcal{F}_{T}$ and $U_{k}^{-1}(t,\cdot)(d) \in\mathcal{F}_{t}$. Using (6.8)–(6.9) we finally obtain that P _a has the desired properties. □

Lemma 5.15

Let $\mathcal{U}$ be defined by (4.8) with U being a $(\mathcal{B}[0,T]\times\mathcal{F}_{T} )/\mathcal{B}(\mathbb {R}^{m})$-measurable, $\mathbb{F}$-adapted closed-convex-valued multifunction. For $u\in L^{2}_{\mathcal{F}}$ we have that $w=P_{\mathcal{U}}(u)$ iff w(t,ω)=P _U(t,ω)(u(t,ω)) for a.a. (t,ω)∈[0,T]×Ω.

Proof

By the definition of a projection, w is characterized as the solution of the minimization problem

$$ \mathop{\mathrm{Min}}_{w\in L^2_{\mathcal{F}}} \mathbb{E} \biggl[ \int _0^T \bigl|w(t,{\omega })-u(t,{\omega})\bigr|^2\mathrm{d}t \biggr]; \quad w \in\mathcal{U}.$$

(5.49)

Let u _k (k≥1) be a $\mathbb{F}$-adapted Castaing representation of U and let the sequence $u'_{k}\in L^{2}_{\mathcal{F}}$ be defined as follows: $u'_{0}$ is an arbitrary element of $\mathcal{U}$, and for k≥0 set

$$u'_{k+1}(t,{\omega})= \left \{ \begin{array}{@{}l@{\quad}l}u_{k+1}(t,{\omega}) & \mbox{if}|u_{k+1}(t,{\omega})-u(t,{\omega})| < |u'_{k}(t,{\omega })-u(t,{\omega})|,\\[1.5mm]u'_{k}(t,{\omega}) & \mathrm{otherwise}.\end{array}\right .$$

(5.50)

Then $u'_{k}$ is a sequence of measurable and adapted functions, and since u _k is a Castaing representation, $|u'_{k}(t,{\omega})-u(t,{\omega})| \rightarrow \inf_{a\in U(t,{\omega})} |u(t,{\omega})-a|$ fort a.a. (t,ω). Therefore, since U(t,ω) is closed and convex, we obtain that $u'_{k}(t,{\omega}) \rightarrow P_{U(t,{\omega})}u(t,{\omega})$ for a.a. (t,ω). Using the dominated convergence theorem, we deduce that $u'_{k}$ has a limit in $L^{2}_{\mathcal{F}}$ equal to P _U(t,ω)(u(t,ω)) for a.a. (t,ω). It follows that the value of problem (6.12) is less or equal than $\mathbb{E} [\int_{0}^{T} |P_{U(t,{\omega})}u(t,{\omega })-u(t,{\omega})|^{2} \mathrm{d}t ]$. Since obviously it cannot be less, and the projection problem has a unique solution, the conclusion follows. □

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Authors and Affiliations

J. Frédéric Bonnans
- 1
- 2
Email author
Francisco J. Silva
- 3

1.INRIA-Saclay and CMAPÉcole PolytechniquePalaiseauFrance
2.Laboratoire de Finance des Marchés d’ÉnergieParisFrance
3.Dipartimento di Matematica Guido CastelnuovoRomeItaly

First and Second Order Necessary Conditions for Stochastic Optimal Control Problems

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