Continuous-Time Stochastic Control and Optimization with Financial Applications

Textbook

Authors: Huyên Pham | Publisher: Springer, 2009 | Series: Stochastic Modelling and Applied Probability, Vol. 61

Overview

A systematic treatment of continuous-time stochastic optimisation with financial applications. Covers dynamic programming (Ch. 3), viscosity solutions (Ch. 4), optimal stopping and switching (Ch. 5), BSDEs and optimal control (Ch. 6), and convex duality methods (Ch. 7). The BSDE chapter (Ch. 6) provides the foundational theory for the exponential utility / CARA framework used in the XVA HJB paper.

Topics Studied

Deep BSDE methods for XVA (studied 22-03-2026)

Chapters read: Ch. 6 (pp. 139-169)

Key Definitions

• Definition 6.2.1 (p. 140): BSDE solution space — A solution (Y,Z) to the BSDE $-d{Y}_t=f(t,Y_t,Z_t)dt-Z_{t}d{W}_t$, $Y_T=\xi$, is a pair $(Y,Z)\in {\mathcal{S}}^2(0,T)\times {\mathbb{H}}^2(0,T{)}^d$ satisfying the integral form

Key Theorems

• Theorem 6.2.1 (p. 140): Existence and uniqueness of BSDE under Lipschitz generator — Given $(\xi ,f)$ satisfying (A) $\xi \in L^2$ and (B) $f$ uniformly Lipschitz in $(y,z)$ with $f(t,0,0)\in {\mathbb{H}}^2$, there exists a unique solution. Proof via Banach fixed-point on ${\mathcal{S}}^2\times {\mathbb{H}}^2$
• Proposition 6.2.1 (p. 142): Linear BSDE explicit solution — ${\Gamma }_t{Y}_t=E[{\Gamma }_T\xi +{\int }_t^T{\Gamma }_s{C}_{s}ds|{\mathcal{F}}_t]$ where $\Gamma$ is the adjoint process solving $d{\Gamma }_t={\Gamma }_t(A_{t}dt+B_{t}d{W}_t)$
• Theorem 6.2.2 (p. 142): comparison theorem for BSDEs — If ${\xi }^1\le {\xi }^2$ a.s. and $f^1(t,Y_t^1,Z_t^1)\le f^2(t,Y_t^1,Z_t^1)$, then $Y_t^1\le Y_t^2$ for all $t$. Strict comparison: if additionally $P({\xi }^1<{\xi }^2)>0$ or $f^1<f^2$ on a set of positive measure, then $Y_0^1<Y_0^2$
• Proposition 6.3.2 (p. 145): Classical-to-BSDE — If $v\in C^{1,2}$ is a classical solution to the semilinear PDE $-{\partial }_{t}v-\mathcal{L}v-f(t,x,v,{\sigma }^{\prime }{D}_{x}v)=0$, then $(Y_t,Z_t)=(v(t,X_t),{\sigma }^{\prime }(X_t)D_{x}v(t,X_t))$ solves the BSDE
• Theorem 6.3.3 (p. 145): nonlinear Feynman-Kac formula — $v(t,x):=Y_t^{t,x}$ is a continuous viscosity solution to the semilinear PDE (6.13)-(6.14)
• Theorem 6.4.4 (p. 147): Optimisation of a family of BSDEs — If $f(t,Y_t,Z_t)={\text{ess inf}}_{\alpha }{f}^{\alpha }(t,Y_t,Z_t)$ and $\xi ={\text{ess inf}}_{\alpha }{\xi }^{\alpha }$, then $Y_t={\text{ess inf}}_{\alpha }{Y}_t^{\alpha }$
• Theorem 6.4.5 (p. 148): BSDE-control duality — The BSDE solution $Y$ equals the value function of a stochastic control problem over linear BSDEs indexed by control parameters $(\beta ,\gamma )$, with probability measure change $Q^{\gamma }$
• Theorem 6.4.6 (p. 149): Stochastic maximum principle via BSDE — If $\hat{\alpha }$ maximises the Hamiltonian $\mathcal{H}$ with concavity condition (6.27), then $\hat{\alpha }$ is optimal and the adjoint BSDE pair $(\hat{Y},\hat{Z})=(D_{x}v(t,{\hat{X}}_t),D_x^{2}v(t,{\hat{X}}_t)\sigma ({\hat{X}}_t,{\hat{\alpha }}_t))$
• Theorem 6.4.7 (p. 151): Connection between maximum principle and DPP — the HJB equation (6.32) is recovered from the BSDE adjoint equation and the stochastic maximum principle
• Theorem 6.6.10 (p. 164): exponential utility BSDE — For CARA utility $U(x)=-e^{-\eta x}$, the value function is $v(x)=-\exp (-\eta (x-Y_0))$ where $(Y,Z)$ solves $-d{Y}_t=f(t,Z_t)dt-Z_{t}d{W}_t$, $Y_T=\xi$, with generator $f(t,z)=-z\cdot b_t/{\sigma }_t-|b_t/{\sigma }_t{|}^2/(2\eta )$. Optimal control: ${\hat{\alpha }}_t=(Z_t+b_t/(\eta {\sigma }_t))/{\sigma }_t$

Key Examples

• Example (pp. 162-165): Exponential utility with option payoff in a complete market. The optimal strategy decomposes as ${\alpha }_t={\pi }_t+{\alpha }_t^0$ where ${\pi }_t=Z_t/{\sigma }_t$ hedges the option and ${\alpha }_t^0=b_t/(\eta {\sigma }_t^2)$ is the Merton strategy without option (Remark 6.6.4)
• Example (pp. 165-169): Mean-variance portfolio selection via BSDE and stochastic maximum principle. Explicit solution with deterministic ODEs for $\phi (t)$ and $\psi (t)$

Bibliographical Remarks (p. 169)

• Standard BSDE theory: Pardoux & Peng [PaPe90], Kobylanski [Ko00] for quadratic growth in $z$
• Control via BSDE: El Karoui, Peng & Quenez [ElkPQ97]
• Maximum principle: Hamadène & Lepeltier [HL95], Yong & Zhou [YZ00]
• Exponential utility BSDE: El Karoui & Rouge [ElkR00], Sekine [Se06], Hu, Imkeller & Müller [HIM05]

Atomic Notes

• nonlinear Feynman-Kac formula
• stochastic control formulation of PDEs
• comparison theorem for BSDEs
• HJB equation for XVA under P
• exponential utility BSDE

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