finiteness and solvability of stochastic LQ problems

Finiteness and solvability are the two fundamental well-posedness conditions for stochastic LQ optimal control problems. A problem is finite at $(s,y)$ if the value function $V(s,y)>-\infty$, meaning the cost cannot be driven to negative infinity. A problem is solvable at $(s,y)$ if an optimal control actually exists that attains the infimum. Solvability implies finiteness, but the converse fails in general — a deterministic LQ problem with $R=0$ can be finite yet unsolvable (Example 2.5 in Yong & Zhou: $\dot{x}=u$, $J=\frac{1}{2}\int x^{2}dt$ has value zero but no optimal control for $y\ne 0$).

The characterisation of these conditions relies on the bounded linear operator $N_s$ on the Hilbert space of admissible controls. Finiteness at $(s,y)$ requires $N_r\ge 0$ for all $r\in [s,T]$ (the operator must be nonnegative). Solvability is equivalent to $N_s\ge 0$ together with the existence of $\bar{u}$ satisfying $N_s\bar{u}+H_s(y)=0$. When $N_s\gg 0$ (uniformly positive), the unique minimiser is $\bar{u}=-N_s^{-1}{H}_s(y)$ and the value function is quadratic in $y$.

The stochastic case introduces a critical distinction. In deterministic LQ problems, finiteness requires $R(t)\ge 0$ a.e. (Proposition 2.4). In stochastic LQ problems, $R(t)$ can be indefinite — even negative definite — and the problem can still be finite and solvable. The necessary condition from Corollary 5.2 is $R(T)+D(T{)}^{T}GD(T)\ge 0$, which involves the diffusion coefficient $D$. The key insight is that when control enters the diffusion ($D\ne 0$), the “uncertainty cost” $D^{T}PD$ can compensate for a negative $R$. However, $R$ cannot be “too negative”: the stochastic Riccati equation requires $R(t)+D(t{)}^{T}P(t)D(t)\ge 0$ along its solution.

For the standard case ($R\gg 0$, $Q-S^T{R}^{-1}S\ge 0$, $G\ge 0$), the problem is always pathwise uniquely solvable at every initial condition. This parallels the deterministic theory. The non-standard case, where $R$ may be indefinite, is where the stochastic theory genuinely diverges and the analysis of the stochastic Riccati equation’s global solvability becomes essential.

Key Details

• Finiteness $\iff$ $V(s,y)>-\infty$ $\Rightarrow$ $N_r\ge 0$ for $r\in [s,T]$
• Solvability $\iff$ $N_s\ge 0$ and $\exists \bar{u}$: $N_s\bar{u}+H_s(y)=0$
• Deterministic necessary condition: $R(t)\ge 0$ a.e.
• Stochastic necessary condition: $R(T)+D(T{)}^{T}GD(T)\ge 0$
• Standard case: $R\gg 0$, $Q-S^T{R}^{-1}S\ge 0$, $G\ge 0$ implies $N_s\gg 0$ and unique solvability
• One-dimensional example (Example 7.8): For $dx=udt+udW$ with $J=E\{-\frac{1}{2}\int r{u}^{2}dt+\frac{1}{2}x(1{)}^2\}$, solvability on $[0,1]$ requires $r<3-2\sqrt{2}\approx 0.172$
• Connection to XVA: The condition $R+D^{T}PD\ge 0$ is the stochastic analogue of the positive definiteness requirement in Riccati system for XVA hedging; in the XVA context, “R too negative” corresponds to transaction costs being overwhelmed by the hedging benefit

Textbook References

Stochastic Controls - Hamiltonian Systems and HJB Equations (Yong & Zhou, 1999)

• Definition 2.1 (p. 285): Finiteness and solvability for deterministic LQ (Problem DLQ)
• Theorem 2.2 (p. 286): Complete characterisation via $N_s$: finiteness $\Rightarrow$ $N_r\ge 0$; solvability $\iff$ $N_s\ge 0$ plus $N_s\bar{u}+H_s(y)=0$
• Proposition 2.4 (p. 290): Finiteness of deterministic LQ implies $R(t)\ge 0$ a.e.
• Definition 3.1 (p. 301): Finiteness, solvability, and pathwise unique solvability for Problem (SLQ)
• Theorem 4.2 (p. 308): Stochastic analogue of Theorem 2.2 on the Hilbert space ${\mathcal{U}}_w[s,T]$
• Corollary 5.2 (p. 310): Necessary condition $R(T)+D(T{)}^{T}GD(T)\ge 0$ for finiteness of Problem (SLQ)
• Examples 3.2—3.4 (pp. 302—304): $R=0$ solvable (3.2), $R<0$ solvable under (3.18) (3.3), well-posed deterministic problem becomes ill-posed stochastically (3.4)

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