HJB equation for XVA under P

The Hamilton-Jacobi-Bellman equation for XVA under the real-world measure P arises from applying the dynamic programming principle to the desk’s CARA utility maximisation problem. Unlike the standard Q-world XVA formulation, where adjustments are conditional expectations under the risk-neutral measure, the P-world formulation embeds the credit-risk premium into the hedging decision, producing materially different credit-hedge targets.

The exact HJB for the total derivative MTM $v$ under P is (eq. 5 of BUETGOLFOUSE2026):

$0={\partial }_{t}v+{\mathcal{L}}_t^{P}v+({\delta }_t-r_t^{\text{repo}})S_t{\theta }_t+(f_t-c_t)C_t-f_{t}v+{\phi }_{\text{mkt}}^{\ast }({\partial }_{\theta }v)+{\phi }_{cr}^{\ast }({\partial }_{q}v)+{\phi }_C^{\ast }({\partial }_{C}v+1)-\frac{\gamma }{2}\Vert {\Sigma }^{\top }{\nabla }_{y}v+{\theta }_t{\sigma }_{S,t}{\Vert }^2+{\lambda }_t^P\frac{e^{\gamma {\zeta }_C(v-C_t-h_C{q}_t{)}^{+}}-1}{\gamma }$

The key structural features are: (i) the generator ${\mathcal{L}}_t^P$ uses the real-world drift ${\mu }_{S,t}^P$, not the risk-neutral drift; (ii) the default jump involves ${\lambda }^P$ (actual default probability), not ${\lambda }^Q$; (iii) the discount is $-f_{t}v$ from book-funding, not the risk-free rate.

Decomposition into clean price and XVA adjustment

Writing $v=w_0+u$ where $w_0$ solves the risk-neutral PDE ${\partial }_t{w}_0+{\mathcal{L}}_t^Q{w}_0-r_0{w}_0=0$, the HJB for $u$ (eq. 8) contains:

• Book-funding discount: $-f_{t}u$
• Carry terms: $-(f_t-r_0)(w_0-C_t)-(c_t-r_0)C_t$
• Real-world drift correction: ${\beta }_t{\partial }_S(w_0+u)$
• Diffusive variance penalty: $-\frac{\gamma }{2}\Vert {\Sigma }^{\top }{\nabla }_{y}u+{\theta }_t{\sigma }_{S,t}{\Vert }^2$
• $w_0$-gradient cross-terms (vanish at leading order in perturbative expansion)
• Exponential default jump (linearised to ${\lambda }^P{\zeta }_{C}u$ at leading order)

Key Details

• The equilibrium credit-hedge level $q^{\text{tar}}$ depends on ${\lambda }^P$, not ${\lambda }^Q$ — this embeds the credit-risk premium into the hedging decision
• The real-world drift ${\beta }_t={\mu }_{S,t}^P-(r_t^{\text{repo}}-{\delta }_t)S_t={\sigma }_{S,t}^{\top }{\vartheta }_t$ enters the delta target, generating carry absent from Q-world formulations
• For investment-grade names with ${\lambda }^Q/{\lambda }^P\approx 2$: the P-world credit-hedge target is approximately half the Q-world level
• The self-consistency condition ($v$ appears on both sides of eq. 3) is resolved by the HJB, which gives $v$ as a PDE solution (Remark 2)

Critical Notes

Empirical sensitivity

The quantitative significance of working under P vs Q depends on the ratio ${\lambda }^Q/{\lambda }^P$, which requires decomposing credit spreads into default probability and risk premium. This decomposition is model-dependent (structural vs. reduced-form calibration, CDS-bond basis assumptions) and empirically contested. The paper’s illustrative ratio of 2-3 is plausible for IG names but not universal.

Relationship to existing work

The existing vault notes on xVA BSDE decomposition and GNOATTO2020 work entirely under Q. The P-world formulation here is genuinely different in structure (CARA utility, not replication), but the comparison with Q-world results needs care: the replication-based Q-world framework is model-free up to credit/funding assumptions, whereas the P-world framework requires specifying ${\lambda }^P$, risk aversion $\gamma$, and the utility function.

Textbook References

Continuous-Time Stochastic Control and Optimization with Financial Applications (Pham, 2009)

• Section 6.6.1 (pp. 162-165): The standard CARA framework that the XVA paper extends. Pham derives the BSDE for exponential utility maximisation with option payoff $\xi$ in a complete market: generator $f(t,z)=-z\cdot b_t/{\sigma }_t-|b_t/{\sigma }_t{|}^2/(2\eta )$, optimal control ${\hat{\alpha }}_t=(Z_t+b_t/(\eta {\sigma }_t))/{\sigma }_t$ (Theorem 6.6.10, p. 164). The XVA paper’s HJB (eq. 5) is the multi-dimensional, friction-laden, default-jump extension of this clean-market result
• Theorem 6.4.5 (p. 148): BSDE-control duality via Fenchel-Legendre transform — the BSDE solution $Y$ equals the value function of a stochastic control problem over linear BSDEs. The XVA paper’s use of Legendre conjugates ${\phi }_{\bullet }^{\ast }$ for the friction terms follows this pattern
• Theorem 6.4.7 (p. 151): Recovery of the HJB equation from the BSDE and stochastic maximum principle. Establishes the HJB $\to$ BSDE $\to$ control chain that the XVA paper traverses in Sections 3-4

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