We consider the infinite horizon optimal consumption-investment problem under the drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model. For a general class of utility functions, we provide the value function in explicit form, and we derive closed-form ...

Under the assumptions of the market of Black and Scholes, options are redundant since, through the classic Black-Scholes delta hedging argument, they can be replaced by an equivalent combination the risky asset underlying the option and a risk free asset. We show that options are not redundant when small proportional transaction costs of size ε are added to the model, which provides mathematica...

• The total value adjustment (XVA) under the Heston model is studied for European and American options . Linear nonlinear partial differential equations have been deduced modelling XVA. Several numerical methods suitable boundary conditions are proposed in order to solve obtained PDEs A comparison with Black-Scholes made results. Expected exposures models also computed. Since 2007/2008 financia...

Following an approach introduced by Lagnado and Osher (1997), we study Tikhonov regularization applied to an inverse problem important in mathematical finance, that of calibrating, in a generalized Black–Scholes model, a local volatility function from observed vanilla option prices. We first establish W 1,2 p estimates for the Black–Scholes and Dupire equations with measurable ingredients. Appl...

Using the symmetry group that was found in [1] and further studied in [2], in this work we study the partial differential equation of the Black-Scholes model [3] and relate such symmetries with the Mellin transform to find the price of an european like investment option. We also consider the Black-Schoes-Merton equation in the non-linear case that models investments in which volatility is a fun...

Under certain assumptions on the dependence structure of the residual lives of the insureds (independent, positively/negatively associated), in this paper we establish some laws of large numbers for the convex upper bounds, derived by the technique of comonotonicity, of the present value function of a homogenous portfolio composed of the whole-life insurance policies. Keyword: Convex order, com...

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as “the curse of dimensionality”. This paper presents a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated as a control theory prob...