arXiv subjects

Mourad Lazgham

Publications and source records attributed to Mourad Lazgham.

3 recordsLinked to original sources

arXiv2016

Numerical solution of a semilinear parabolic degenerate Hamilton-Jacobi-Bellman equation with singularity

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of variables that gives rise to an equivalent HJB equation with nonsingular initial condition but irregular coefficients. We then construct explicit and implicit numerical schemes for solving the transformed HJB equation and prove their convergences by establishing an extension to the result of Barles and Souganidis (1991).

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arXiv2015

Viscosity properties with singularities in a state-constrained expected utility maximization problem

We consider the value function originating from an expected utility maximization problem with finite fuel constraint and show its close relation to a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. On one hand, we give a so-called verification argument based on the dynamic programming principle, which allows us to derive conditions under which a classical solution of the HJB equation coincides with our value function (provided that it is smooth enough). On the other hand, we establish a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.

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arXiv2015

Regularity properties in a state-constrained expected utility maximization problem

We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of optimal strategies under rather mild model assumptions. On the one hand, this result is of independent interest. On the other hand, it will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove the dynamic programming principle without appealing to the classical measurable selection arguments.

q-fin.MF↗