We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton- Jacobi-Bellman integro-partial differential equation in a bounded domain....
详细信息
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton- Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
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 ...
详细信息
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.
In this work, we consider the time discretization of stochastic optimal control problems. Under general assumptions on the data, we prove the convergence of the value functions associated with the discrete time proble...
详细信息
In this work, we consider the time discretization of stochastic optimal control problems. Under general assumptions on the data, we prove the convergence of the value functions associated with the discrete time problems to the value function of the original problem. Moreover, we prove that any sequence of optimal solutions of discrete problems is minimizing for the continuous one. As a consequence of the dynamic programming principle for the discrete problems, the minimizing sequence can be taken in discrete time feedback form.
In this paper we introduce a game whose value functions converge (as a parameter that measures the size of the steps goes to zero) uniformly to solutions to the second order Pucci maximal operators.
In this paper we introduce a game whose value functions converge (as a parameter that measures the size of the steps goes to zero) uniformly to solutions to the second order Pucci maximal operators.
In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we stud...
详细信息
In this paper, we consider a stochastic optimal control problem, in which the cost function is defined through a reflected backward stochastic differential equation in sublinear expectation framework. Besides, we study the regularity of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton-Jacobi-Bellman-Isaac equation.
We obtain an asymptotic Holder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where t...
详细信息
We obtain an asymptotic Holder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.
This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a c...
详细信息
This paper is concerned with the two-player zero-sum stochastic differential game in a regime switching model with an infinite horizon. The state of the system is characterized by a number of diffusions coupled by a continuous-time finite-state Markov chain. Based on the dynamic programming principle (DPP), the lower and upper value functions are shown to be the unique viscosity solutions of the associated lower and upper Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. Moreover, the lower and upper value functions coincide under the Isaacs' condition, which implies that the game admits a value. All the proofs in this paper are markedly different from those for the case when there is no regime switching. (C) 2020 Elsevier Ltd. All rights reserved.
We investigate the pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the...
详细信息
We investigate the pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing-hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglbock, Henry-Labordere, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing-hedging duality for American options.
This paper is concerned with a kind of optimal portfolio and consumption choice problem, where an investor can invest his wealth in a trade project and foreign exchange deposit. The trade project earns profit by buyin...
详细信息
ISBN:
(纸本)9789881563972
This paper is concerned with a kind of optimal portfolio and consumption choice problem, where an investor can invest his wealth in a trade project and foreign exchange deposit. The trade project earns profit by buying the merchandise and selling it with a higher price. The bank pays at an interest rate for any deposit, and vice takes at a large rate for any loan. The optimal strategy is obtained by Hamilton-Jacobi-Bellman (HJB) equation, which is derived from dynamic programming principle. For the specific Hyperbolic Absolute Risk Aversion (HARA) case, we get the explicit form of optimal portfolio and consumption solution, and we give some simulation results.
This paper deals with an optimal control problem of fully coupled forward-backward stochastic differential equations (FBSDEs), where the diffusion term does not contain the variable z and the control domain is not nec...
详细信息
ISBN:
(纸本)9781728139364
This paper deals with an optimal control problem of fully coupled forward-backward stochastic differential equations (FBSDEs), where the diffusion term does not contain the variable z and the control domain is not necessarily convex. The connection among the adjoin (variables and the value function is obtained in terms of the sub- and super-derivatives. It generalizes the result in.
暂无评论