exponential utility BSDE

The exponential utility BSDE is the backward stochastic differential equation that arises when an agent with CARA (constant absolute risk aversion) utility $U(x)=-e^{-\eta x}$ maximises expected utility of terminal wealth minus a contingent claim $\xi$. The key structural insight is that the exponential form of the utility function allows a multiplicative decomposition $J_t^{\alpha }=U(X_t^{x,\alpha }-Y_t)C_t^{\alpha }$, where $Y$ solves a BSDE and $C^{\alpha }$ is a supermartingale deflator that becomes a martingale at the optimum.

In a complete market with one risky asset $d{S}_t=S_t(b_{t}dt+{\sigma }_{t}d{W}_t)$ and wealth process $X_t^{x,\alpha }=x+{\int }_0^t{\alpha }_u(b_{u}du+{\sigma }_{u}d{W}_u)$, the value function takes the form $v(x)=-\exp (-\eta (x-Y_0))$ where $(Y,Z)$ solves:

$-d{Y}_t=f(t,Z_t)dt-Z_{t}d{W}_t,Y_T=\xi$

with generator $f(t,z)=-z\cdot b_t/{\sigma }_t-|b_t/{\sigma }_t{|}^2/(2\eta )$. The optimal control is ${\hat{\alpha }}_t=(Z_t+b_t/(\eta {\sigma }_t))/{\sigma }_t$, which decomposes into a hedging component $Z_t/{\sigma }_t$ and the Merton strategy $b_t/(\eta {\sigma }_t^2)$ (Remark 6.6.4 of Pham 2009).

In incomplete markets (option $\xi$ not perfectly replicable), the generator acquires a quadratic term in $z$ (El Karoui and Rouge 2000), making the BSDE quadratic. This is the stepping stone toward the Qexp BSDEs (quadratic in $Z$ with exponential growth in $u$) that arise in the XVA setting of BUETGOLFOUSE2026.

Key Details

• The cash-additivity of CARA utility gives $V(t,x,...)=x+v(t,y,\theta ,q,C)$ — the value function separates into cash + a position-dependent component. This is eq. (4) of the XVA HJB paper
• The generator $f(t,z)$ is linear in $z$ in the complete market case — much simpler than the XVA paper’s driver which has quadratic growth in $Z$, exponential default jump, book-funding discount, and friction terms
• The XVA paper extends this framework by adding: (i) book-funding cost $-f_{t}u$, (ii) counterparty default jump ${\lambda }^P[e^{\gamma {\zeta }_C{E}^{+}}-1]/\gamma$, (iii) collateral terms, (iv) market-impact frictions ${\phi }_{\bullet }^{\ast }$, (v) credit-hedge controls $q_t$
• The entropic transformation (log of the exponential) converts the utility maximisation to a quadratic running cost in the HJB — this is the mechanism behind eq. (5) of the XVA paper

Textbook References

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

• Theorem 6.6.10 (p. 164): Value function $v(x)=U(x-Y_0)$ where $(Y,Z)$ solves the BSDE with generator (6.75). Optimal control given by (6.76)
• Remark 6.6.4 (p. 164): Decomposition of optimal strategy into hedging + Merton components. In a complete market, the hedging strategy ${\pi }_t=Z_t/{\sigma }_t$ replicates $\xi$ and ${\alpha }_t^0=b_t/(\eta {\sigma }_t^2)$ is the pure Merton strategy
• Section 6.7 (p. 169): For incomplete markets with quadratic generator, see El Karoui and Rouge [ElkR00]; for power utility via BSDEs, see Hu, Imkeller, Müller [HIM05]

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