Singular stochastic control and optimal stopping with partial information of jump diffusions Bernt Øksendal∗
Agn`es Sulem†
20 August 2010
Abstract We study partial information, possibly non-Markovian, singular stochastic control of jump diffusions and obtain general maximum principles. The results are used to find connections between singular stochastic control, reflected BSDEs and optimal stopping in the partial information case. Mathematics Subject Classification 2010: 93E20, 60H07, 60H10, 60HXX, 60J75 Key words: Singular stochastic control, maximum principle, jump diffusion, partial information, backward stochastic differential equations, optimal stopping
1
Introduction
The aim of this paper is to establish stochastic maximum principles for singular control problems of jump diffusions and to study relations with some associated optimal stopping problems. Maximum principles for singular stochastic control problems have been studied in [1, 2, 3, 4]. Here, we study general singular control problems of Itˆo-L´evy processes, in which the controller has only partial information and the system is not necessarily Markovian. The first part of the paper (section 2) is dedicated to the statement of stochastic maximum principles. Two different approaches are considered: (i) by using Malliavin calculus, leading to generalized variational inequalities for partial information singular control of possibly non-Markovian systems (subsection 2.2), (ii) by introducing a singular control version of the ∗
Center of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, email:
[email protected] and Norwegian School of Economics and Business Administration (NHH), Helleveien 30, N–5045 Bergen, Norway. This work was done partially while one of the authors was visiting the Institute for Mathematical Sciences, National University of Singapore in 2009. The visit was supported by the Institute. The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087]. † INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex, 78153, France, email:
[email protected]
1
Hamiltonian and using backward stochastic differential equations (BSDEs) to obtain both a necessary and a sufficient partial information maximum principle for such problems (subsections 2.3 and 2.5). We show that the two methods are related, and we find a connection between them. In the second part of the paper (Section 3), we study the relations between optimal singular control for jumps diffusions with partial information with general reflected backward stochastic differential equations (RBSDEs) and optimal stopping. We first give a connection between the generalized variational inequalities found in Section 2 and RBSDEs (subsection (3.1)). These are shown to be equivalent to general optimal stopping problems for such processes (subsection (3.2)). Combining this, a connection between singular control and optimal stopping is obtained in subsection 3.3. A illustrating example is provided in subsection 3.4.
2 2.1
Maximum principles for optimal singular control Formulation of the singular control problem
Consider a controlled singular jump diffusion X(t) = X ξ (t) of the form X(0− ) = x ∈ R and dX(t) =b(t, X(t), ω)dt + σ(t, X(t), ω)dB(t) Z ˜ (dt, dz) + λ(t, ω)dξ(t) ; t ∈ [0, T ], + θ(t, X(t− ), z, ω)N
(2.1)
R0
defined on a probability space (Ω, F, (Ft )t≥0 , P ), where t → b(t, x), t → σ(t, x) and t → θ(t, x, z) are given Ft -predictable processes for each x ∈ R, z ∈ R0 ≡ R\{0}. We assume that b, σ and θ are C 1 with respect to x and that there exists > 0 such that ∂θ (t, x, z, ω) ≥ −1 + ∂x
a.s. for all (t, x, z) ∈ [0, T ] × R × R0 .
(2.2)
˜ (dt, dz) is a compensated jump measure defined as N ˜ (dt, dz) = N (dt, dz) − ν(dz)dt Here N where ν is the L´evy measure of a L´evy process η with jump measure R N2 , and B is a Brownian 2 ˜ motion (independant of N ). We assume E[η (t)] < ∞ ∀t , (i.e. R0 z ν(dz) < ∞). Let Et ⊆ Ft ; t ∈ [0, T ] be a given subfiltration of Ft satisfying the usual assumptions. We assume that the process λ(t) is Et -adapted and continuous. The process ξ(t) = ξ(t, ω) is our control process, assumed to be c`adl`ag and non-decreasing for each ω, with ξ(0− ) = 0. We require that the control ξ(t) is Et -adapted. The set of such controls is denoted by AE . Let t → f (t, x) and t → h(t, x) be given Ft -predictable processes and g(x) an FT -measurable random variable for each x. Define the performance functional Z T Z T − J(ξ) = E f (t, X(t), ω)dt + g(X(T ), ω) + h(t, X(t ), ω)dξ(t) . (2.3) 0
0
2
We want to find an optimal control ξ ∗ ∈ AE such that Φ := sup J(ξ) = J(ξ ∗ ).
(2.4)
ξ∈AE
For ξ ∈ AE we let V(ξ) denote the set of Et -adapted processes ζ of finite variation such that there exists δ = δ(ξ) > 0 such that ξ + yζ ∈ AE for all y ∈ [0, δ]. For ξ ∈ AE and ζ ∈ V(ξ) we have Z T ∂f 1 (t, X(t))Y(t)dt + g 0 (X(T ))Y(T ) lim+ (J(ξ + yζ) − J(ξ)) = E y→0 y ∂x 0 Z T Z T ∂h − − − + h(t, X(t ))dζ(t) (t, X(t ))Y(t )dξ(t) + 0 ∂x 0
(2.5)
(2.6)
where Y(t) is the derivative process defined by 1 Y(t) = lim+ (X ξ+yζ (t) − X ξ (t)) ; t ∈ [0, T ]. y→0 y
(2.7)
We have
∂b ∂σ dY(t) = Y(t ) (t)dt + (t)dB(t) + ∂x ∂x −
Z R0
∂θ ˜ (dt, dz) + λ(t)dζ(t), (t, z)N ∂x
(2.8)
where we here (and in the following) are using the abbreviated notation ∂b ∂b ∂σ ∂σ (t) = (t, X(t)), (t) = (t, X(t)) etc. ∂x ∂x ∂x ∂x Note that
d 1 Y(0) = lim+ (X ξ+yζ (0) − X ξ (0)) = x |y=0 = 0. y→0 y dy
Lemma 2.1 The solution of equation (2.8) is "Z # t X Y(t) = Z(t) Z −1 (s− )λ(s)dζ(s) + Z −1 (s− )λ(s)α(s)∆ζ(s) , t ∈ [0, T ] 0
(2.9)
(2.10)
0
