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The X-valuation adjustment (XVA) problem, which is a recent topic in mathematical finance, is considered and analyzed. First, the basic properties of backward stochastic differential equations (BSDEs) with a random horizon in a progressively enlarged filtration are reviewed. Next, the pricing/hedging problem for defaultable over-the-counter (OTC) derivative securities is described using such BSDEs. An explicit sufficient condition is given to ensure the non-existence of an arbitrage opportunity for both the seller and buyer of the derivative securities. Furthermore, an explicit pricing formula is presented in which XVA is interpreted as approximated correction terms of the theoretical fair price.
Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state
This paper is concerned with solution in H{o}lder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as deterministic s
In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also
This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new f
The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $cR^p$ ($pin [1, infty)$) and backward stochastic differential equations (BSDEs) in $cR^ptimes cH^p$ ($pin (1, infty)$) and in $cR^inftyt