Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

Abstract

In this work we study the stochastic recursive control problem, in which the aggregator (or called generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem of backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein-Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone drivers

“…It has been demonstrated in [23,27] that for certain empirically important parameters, for instance the coefficients in Table 4, which are taken from [23] and will be used for our numerical test, this driver (6.6) of the Epstein-Zin utility is non-Lipschitz but monotone in the utility y. Moreover, one can identify the value function (6.7) (with a change of time variable) as the solution to the following HJB equation: u…”

“…Now we are ready to demonstrate the existence of solutions to the discrete equation (4.19) with a general monotone driver. We shall adapt some arguments for monotone backward stochastic difference equations employed in [27], by approximating (4.19) with discrete equations with Lipschitz drivers, whose solutions subsequently enable us to construct the solution of (4.19). Proof.…”

“…As a second example, we shall address a consumption-portfolio maximization problem in terms of recursive utilities, which extend the classical additive utilities by allowing one's current wellbeing to depend on the expected future utilities in a non-risk-neutral way, and play an important role in modern mathematical finance (see e.g. [23,27] and references therein).…”

“…Such equations extend the classical HJBVIs with linear drivers, i.e., f (α, t, x, y, z, k) ≡ ℓ(α, t, x)− r(t, x)y, and play an important role in modern finance, including the following: models for American options in a market with constrained portfolios [13,12], recursive utility optimization problems [15], and robust pricing and risk measures under probability model uncertainty [30,28]. We remark that imposing merely monotonicity assumptions on the drivers allows us to consider several important non-smooth drivers stemming from robust pricing [30,14,28] and non-Lipschitz drivers arising in stochastic recursive control [23,27], while including an extra nonlinearity in the operator B α enables us to incorporate ambiguity in the jump processes [30]. As the solution to (1.1) is in general not known analytically, it is important to construct effective and robust numerical schemes for solving these fully nonlinear equations.…”

Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints

“…During the last years, there has been substantial progress in the mathematical analysis of consumption-investment problems with recursive utility and stochastic differential utility. These studies mainly utilized various stochastic control techniques, such as the Hamilton-Jacobi-Bellman (HJB) equation (see Kraft, Seiferling and Seifried (2017), Kraft, Seifried and Steffensen (2013), Pu and Zhang (2018) and Zhuo, Dong and Pu (2020)), the utility gradient approach (see Schroder and Skiadas (1999), or the convex duality method (see Matoussi and Xing (2018) and Ji and Shi (2018)). In an incomplete market, coming back to γ and ψ, we focus on the empirically relevant specification γ, ψ > 1 same as Xing (2017).…”