Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance

Framstad, Nils Christian; Øksendal, Bernt; Sulem, Agnès

doi:10.1007/s10957-004-0949-6

Cited by 31 publications

(58 citation statements)

References 13 publications

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“…Note that (17) is a jump-diffusion SDE without delay and the mean-variance problem under this model can be solved using the classical stochastic sufficient maximum principle for control systems without delay (see, for example, Framstad et al, 2004). In what follows, we shall formulate a wealth process with delay, which may arise in various situations in practice.…”

Section: Dx(t) = [A(t)x(t)+u(t) ⊤ B(t)]dt+u(t) ⊤ σ(T)dw (T)+mentioning

confidence: 99%

“…The dynamic mean-variance portfolio selection problem has been well explored using different methods. Please see Zhou and Li (2000) for the stochastic linear-quadratic method, Framstad et al (2004) for the maximum principle, Shen and Siu (2013) Zhou and Li (2000). 20…”

Section: As Large Investors Any Decisions On Portfolio Choices Of Thmentioning

confidence: 99%

“…The early contributions to the stochastic maximum principle approach were made by Kushner (1972), Bismut (1973) and Bensoussan (1982). In the last three decades, many extensions of the stochastic maximum principle have been made, see for example, Peng (1990), Tang and Li (1994), Framstad et al (2004), Shi (2012), Haadem et al (2013) and references therein. A systematic account on the subject can be found in Yong and Zhou (1999) and Øksendal and Sulem (2007).…”

Section: Introductionmentioning

confidence: 99%

“…Particularly, the stochastic maximum principle has been widely adopted to solve the stochastic optimal control problems in finance, such as the mean-variance portfolio selection problem, the investment-consumption problem, the optimal insurance problem and others. Many examples were provided in Framstad et al (2004).…”

Section: Introductionmentioning

confidence: 99%

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Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

2014

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This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump-diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying systems. As an application, a bicriteria mean-variance portfolio selection problem with delay is studied. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean-variance problem without delays.

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Section: Dx(t) = [A(t)x(t)+u(t) ⊤ B(t)]dt+u(t) ⊤ σ(T)dw (T)+mentioning

confidence: 99%

Section: As Large Investors Any Decisions On Portfolio Choices Of Thmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

2014

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“…We have (see [19] and [32] (38) we have M(t,V (t),π(t),p(t),q(t),ŷ(t)) = sup π M(t,V (t), π,p(t),q(t),ŷ(t)),…”

Section: Stochastic Maximum Principlementioning

confidence: 99%

Problems of Mathematical Finance by Stochastic Control Methods

Stettner

2009

IFIP Advances in Information and Communication Technology

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Abstract. The purpose of this paper is to present main ideas of mathematics of finance using the stochastic control methods. There is an interplay between stochastic control and mathematics of finance. On the one hand stochastic control is a powerful tool to study financial problems. On the other hand financial applications have stimulated development in several research subareas of stochastic control in the last two decades. We start with pricing of financial derivatives and modeling of asset prices, studying the conditions for the absence of arbitrage. Then we consider pricing of defaultable contingent claims. Investments in bonds lead us to the term structure modeling problems. Special attention is devoted to historical static portfolio analysis called Markowitz theory. We also briefly sketch dynamic portfolio problems using viscosity solutions to HamiltonJacobi-Bellman equation, martingale-convex analysis method or stochastic maximum principle together with backward stochastic differential equation. Finally, long time portfolio analysis for both risk neutral and risk sensitive functionals is introduced. Pricing of Financial DerivativesOne of the fundamental problems of mathematics of finance is pricing of the derivative securities (shortly derivatives) i.e. securities the value of which depends on the basic securities such as stocks or bonds. In this section we restrict ourselves to stocks, although similar problems (unfortunately much harder) concern also derivatives of bonds. Modeling of Asset PricesWe start with modeling of asset prices (stocks). We assume that we are given d assets on the market and denote the price of the i-th asset at time t by S i (t). We shall consider in parallel way two approaches: in discrete and in continuous time. We assume a given probability space (Ω , F , (F t ), P). In the case of discrete time the asset prices satisfy the relation S i (t + 1) S i (t) = ζ i (t, z(t + 1), ξ (t + 1)),Research supported by MNiSzW grant no. 1 P03A 013 28.

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Iterative Path Integral Approach to Nonlinear Stochastic Optimal Control Under Compound Poisson Noise

Okumura

Kashima

Ohta

2016

Asian Journal of Control

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Nonlinear stochastic optimal control theory has played an important role in many fields. In this theory, uncertainties of dynamics have usually been represented by Brownian motion, which is Gaussian white noise. However, there are many stochastic phenomena whose probability density has a long tail, which suggests the necessity to study the effect of non‐Gaussianity. This paper employs Lévy processes, which cause outliers with a significantly higher probability than Brownian motion, to describe such uncertainties. In general, the optimal control law is obtained by solving the Hamilton–Jacobi–Bellman equation. This paper shows that the path‐integral approach combined with the policy iteration method is efficiently applicable to solve the Hamilton–Jacobi–Bellman equation in the Lévy problem setting. Finally, numerical simulations illustrate the usefulness of this method.

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Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance

Abstract: We give a verification theorem by employing Arrow's generalization of the Mangasarian sufficient condition to a general jump diffusion setting, and show the adjoint processes' connections to dynamic programming. The result is applied to financial optimization problems.

Cited by 31 publications

References 13 publications

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

Problems of Mathematical Finance by Stochastic Control Methods

Iterative Path Integral Approach to Nonlinear Stochastic Optimal Control Under Compound Poisson Noise

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