In the frictionless discrete time financial market of Bouchard and Nutz [Ann. Appl. Probab., 25 (2015), pp. 823{859] we consider a trader who is required to hedge xi in a risk-conservative way relative to a family of ...
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In the frictionless discrete time financial market of Bouchard and Nutz [Ann. Appl. Probab., 25 (2015), pp. 823{859] we consider a trader who is required to hedge xi in a risk-conservative way relative to a family of probability measures P. We first describe the evolution of pi(t)(xi) the superhedging price at time t of the liability xi at maturity T-via a dynamic programming principle, show that pi(t)(xi) can be seen as a concave envelope of pi(t+1)(xi) evaluated at today's prices, and prove its dual characterization. Under suitable assumptions, we show that the robust superreplication price is equal to the classical P-superhedging price for an extreme prior P is an element of P. Then we consider an optimal investment problem for the trader who is rolling over her robust superhedge and phrase this as a robust maximization problem, where the expected utility of intertemporal consumption is optimized subject to a robust superhedging constraint. This utility maximization is carried out under a subset P-u of P representing the trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique.
We develop a method for solving stochastic control problems under one-dimensional Levy processes. The method is based on the dynamic programming principle and a Fourier cosine expansion method. Local errors in the vic...
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We develop a method for solving stochastic control problems under one-dimensional Levy processes. The method is based on the dynamic programming principle and a Fourier cosine expansion method. Local errors in the vicinity of the domain boundaries may disrupt the algorithm. For efficient computation of matrix-vector products with Hankel and Toeplitz structures, we use a fast Fourier transform algorithm. An extensive error analysis provides new insights based on which we develop an extrapolation method to deal with the propagation of local errors. Copyright (c) 2013 John Wiley & Sons, Ltd.
We consider the optimal dividend distribution problem of a financial corporation whose surplus is modeled by a general diffusion process with both the drift and diffusion coefficients depending on the external economi...
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We consider the optimal dividend distribution problem of a financial corporation whose surplus is modeled by a general diffusion process with both the drift and diffusion coefficients depending on the external economic regime as well as the surplus itself through general functions. The aim is to find a dividend payout scheme that maximizes the present value of the total dividends until ruin. We show that, depending on the configuration of the model parameters, there are two exclusive scenarios: (i) the optimal strategy uniquely exists and corresponds to paying out all surpluses in excess of a critical level (barrier) dependent on the economic regime and paying nothing when the surplus is below the critical level;(ii) there are no optimal strategies. (C) 2013 Elsevier B.V. All rights reserved.
The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including me...
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The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean-Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire's (2009) functional Ito formula. This Ito formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.
We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jac...
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We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form min(u(t) vertical bar cu(x), u - g(x)) = 0, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.
We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constr...
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We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an arbitrary probability setting is equivalent to the constrained problem in weak formulation (an optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space), and thus the OSEC value is independent of a specific probabilistic setup. Using a martingale-problem formulation, we make an equivalent characterization of the probability classes in weak formulation, which implies that the OSEC value function is upper semianalytic. Then we exploit a measurable selection argument to establish a dynamic programming principle in weak formulation for the OSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon.
In this paper, we aim to develop the stochastic control theory of branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem...
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In this paper, we aim to develop the stochastic control theory of branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamicprogramming approach initiated by Nisio [J. Math. Kyoto Univ. 25 (1985) 549-575] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to derive a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.
In this paper, we study the near-optimal control for systems governed by forward-backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the vis...
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In this paper, we study the near-optimal control for systems governed by forward-backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the viscosity solution approach is employed to investigate the relationships among the value function, the adjoint equations along near-optimal trajectories. Unlike the classical case, the definition of viscosity solution contains a perturbation factor, through which the illusory differentiability conditions on the value function are dispensed properly. Moreover, we establish new relationships between variational equations and adjoint equations. As an application, a kind of stochastic recursive near-optimal control problem is given to illustrate our theoretical results.
In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller-Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman...
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In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller-Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.
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