A non-linear PDE for XVA by forward Monte Carlo

Vladimir Piterbarg considers a non-linear partial differentiation equation that appears in a number of XVA-related contexts, including a one-way credit-support annex, credit value adjustment with risky closeout, option pricing with differential borrowing and lending rates, accounting-consistent valuation and constrained cash supply. In showing its solution is given as the minimum of solutions of certain related but linear PDEs, he develops an efficient forward simulation algorithm for any number of dimensions

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In this article, a solution to a semi-linear PDE is obtained by taking the minimum of solutions to related linear PDEs over an infinite-dimensional space of discount boundaries. By restricting the minimum to a parameterised subset of boundaries, a practical algorithm for numerically solving the semi-linear PDE in a forward Monte Carlo is obtained. We also show how to modify the standard CVA algorithms to account for the risky closeout in the section on comparison to riskless closeout DVA