On some recent aspects of stochastic control and their applications

Preprint, Other literature type English OPEN
Pham, Huyen;
(2005)
  • Publisher: HAL CCSD
  • Journal: issn: 1549-5787
  • Related identifiers: doi: 10.1214/154957805100000195
  • Subject: dynamic programming | 93E20, 49J20, 49L20, 60H30 | maximum principle | MSC: 93E20, 49J20, 49L20, 60H30. | [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR] | finance. | 93E20, 49J20, 49L20, 60H30 (Primary) | Controlled diffusions | Mathematics - Probability | [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] | backward stochastic differential equations | viscosity solutions | finance

This paper is a survey on some recent aspects and developments in stochastic control. We discuss the two main historical approaches, Bellman’s optimality principle and Pontryagin’s maximum principle, and their modern exposition with viscosity solutions and backward stoc... View more
  • References (91)
    91 references, page 1 of 10

    2 The problem and discussion on methodology 508 2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 508 2.2 Bellman's optimality principle . . . . . . . . . . . . . . . . . . . . 509 2.2.1 The Hamilton-Jacobi-Bellman (in)equation . . . . . . . . 511 2.2.2 The classical verification approach . . . . . . . . . . . . . 512 2.3 Pontryagin's maximum principle . . . . . . . . . . . . . . . . . . 514 2.4 Other control problems . . . . . . . . . . . . . . . . . . . . . . . 515 2.4.1 Random horizon . . . . . . . . . . . . . . . . . . . . . . . 515 2.4.2 Optimal stopping . . . . . . . . . . . . . . . . . . . . . . . 516 2.4.3 Impulse control . . . . . . . . . . . . . . . . . . . . . . . . 517 2.4.4 Ergodic control . . . . . . . . . . . . . . . . . . . . . . . . 517 2.4.5 Robust control . . . . . . . . . . . . . . . . . . . . . . . . 518 2.4.6 Partial observation control problem . . . . . . . . . . . . 518 2.4.7 Stochastic target problems . . . . . . . . . . . . . . . . . 519

    3 Dynamic programming and viscosity solutions 521 3.1 Definition of viscosity solutions . . . . . . . . . . . . . . . . . . . 521 3.2 Viscosity characterization . . . . . . . . . . . . . . . . . . . . . . 522 3.2.1 Viscosity supersolution property . . . . . . . . . . . . . . 525 3.2.2 Viscosity subsolution property . . . . . . . . . . . . . . . 526 3.2.3 Terminal condition . . . . . . . . . . . . . . . . . . . . . . 528 3.3 Some applications in finance . . . . . . . . . . . . . . . . . . . . . 532 3.3.1 Smooth-fit property of one-dimensional convex singular problem532 3.3.2 Superreplication in uncertain volatility model . . . . . . . 535 5 Numerical issues 542 5.1 Deterministic methods . . . . . . . . . . . . . . . . . . . . . . . . 542 5.2 Probabilistic methods . . . . . . . . . . . . . . . . . . . . . . . . 542 0, ∀(t, x) ∈ [s0, T ] × B¯η(x¯). (3.27) Step 2. By (3.38), v is C2 on (0, xb) and satisfies v′(x) = λ, x ∈ (0, xb). From Step 1, we have N T = (xb, ∞) = {x > 0 : v′(x) < λ}. Let us check that v is a viscosity solution of : 2 ∂2v ∂x2

    [1] Akian M. (1990) : “Analyse de l'algorithme multigrille FMGH de r´esolution d'´equations d'Hamilton-Jacobi-Bellman”, Analysis and Optimization of systems, Lect. Notes in Contr. and Inf. Sciences, 144, Springer-Verlag, pp. 113-122.

    [2] Antonelli F. (1993) : “Backward-forward stochastic differential equations”, Annals of Appl. Prob., 3, 777-793.

    [3] Artzner P., Delbaen F., Eber J.M. and D. Heath (1999) : “Coherent measures of risk”, Mathematical Finance, 9, 203-228.

    [4] Bally V. and G. Pag`es (2003) : “Error analysis of the optimal quantization algorithm for obstacle problems”, Stochastic Processes and their Applications, 106, 1-40.

    [5] Baras J., Elliott R. and M. Kohlmann (1989) : “The partially obsered stochastic minimum principle”, SIAM J. Cont. Optim., 27, 1279-1292.

    [6] Barles G. and E. Jakobsen (2004) : “Error bounds for monotone approximations schemes for Hamilton-Jacobi-Bellman equations”, to appear in SIAM J. Num. Anal..

    [7] Barles G. and P. Souganidis (1991) : : “Convergence of approximation schemes for fully non linear second-order equations”, Asymptotics Analysis, 4, pp.271-283.

    [8] Bellman R. (1957) : Dynamic programming, Princeton university press.

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