Necessary and sufficient conditions for convexity preservation beyond $F(x,D^2u)$: the Lions–Musiela extension program for parabolic equations (general dependence, obstacle, elliptic, and nonlocal equations, boundary conditions, partial convexity, and joint state–strike convexity)

Lions and Musiela (2006) proved that convexity of the initial datum is preserved by the viscosity solution of the linear parabolic equation $\partial_t u - a^{ij}\partial_{ij}u=0$ on $\mathbb{R}^N$ if and only if $|\sigma(x+y)\cdot\xi|$ is convex in $y$ on the hyperplane $(\mathbb{R}\xi)^\perp$ for every $x$ and $\xi\neq 0$ (where $a=\frac{1}{2}\sigma\sigma^T$), and they gave an analogous necessary and sufficient condition for fully nonlinear equations $\partial_t u+F(x,D^2u)=0$. Their note explicitly defers to future work a family of extensions for which no analogous characterization is known: equations with full dependence $F(x,t,u,Du,D^2u)$, obstacle problems, stationary elliptic equations, integro-differential equations modeling jump-diffusion processes, boundary conditions other than state constraints, the classification of partial convexity when the initial datum is convex only in a subset of the variables, joint convexity in the state and strike variables for call-type payoffs, and a purely probabilistic proof of the linear theorem. Two decades of later literature have settled the following: partial progress exists — sufficient constant-rank conditions for partial convexity, quasiconvexity preservation for nonlocal equations, necessary conditions and rigidity theorems for jump diffusions, quasi-convexity for parabolic obstacle problems, boundary-value analogs for linear heat flows — but no complete necessary-and-sufficient structural characterization has been published for any of the deferred settings. The extension program therefore remains open.

Background

A classical question in the theory of second-order partial differential equations (PDEs) asks when the spatial convexity of an initial datum survives the evolution: if $u_0$ is convex on $\mathbb{R}^N$, is the solution $u(\cdot,t)$ still convex for every $t\ge 0$? Convexity of solutions is a strong geometric regularity property that implies, for instance, uniqueness of supporting hyperplanes, monotonicity of spatial derivatives, and comparison-type structure, and in mathematical finance it is a model-consistency requirement: option prices should be convex in the underlying asset price, a property tied to absence of model arbitrage and to monotonicity of prices in model parameters such as volatility (Romano and Touzi 1997; Janson and Tysk 2004).

The equations of interest here are parabolic Cauchy problems on $\mathbb{R}^N$: the linear equation $\partial_tu-a^{ij}\partial_{ij}u=0$ with diffusion matrix $a=\frac{1}{2}\sigma\sigma^T$ built from a Lipschitz map $\sigma$ into $N\times m$ matrices, and the fully nonlinear equation $\partial_tu+F(x,D^2u)=0$ with $F$ continuous, nonincreasing in the Hessian matrix variable $A\in S^N$ (the space of real symmetric $N\times N$ matrices). Because such equations may be degenerate and $F$ need not be smooth, the appropriate solution concept is the viscosity solution of Crandall, Ishii, and Lions (1992): a continuous $u$ that satisfies the equation in the sense of inequalities tested by smooth functions touching $u$ from above (subsolutions) and below (supersolutions), a weak notion under which comparison and uniqueness hold for the classes considered. Call convexity preserved for such an equation if $u(\cdot,t)$ is convex for all $t\ge 0$ whenever $u_0$ is (say uniformly continuous and) convex.

Before 2006 the literature contained essentially sufficient conditions obtained by maximum-principle and geometric methods: the concavity maximum principle of Korevaar (1983) and its power-concavity refinements (Kennington 1985; Kawohl 1985), and convexity-preserving properties for singular degenerate parabolic equations (Giga, Goto, Ishii, and Sato 1991; Ishii and Lions 1990). A more geometric method was introduced by Alvarez, Lasry, and Lions (1997): one shows that the convex envelope $u^{**}$ (the largest convex function not exceeding $u$) is a viscosity supersolution of the equation, so that the comparison principle forces $u^{**}\ge u$, hence $u^{**}=u$ and $u$ is convex; the same paper treats state constraints boundary conditions, in which the state of the associated controlled process is confined to the closure of the domain and the equation holds up to the boundary. In parallel, Janson and Tysk (2004) introduced LCP (locally convexity preserving) diffusion operators and characterized which diffusion operators, equivalently which volatility structures, preserve convexity of option payoffs.

Lions and Musiela (2006) proved the first necessary and sufficient conditions. For the linear equation, convexity is preserved if and only if

$$\left(2\,a(x+y)\cdot(\xi,\xi)\right)^{1/2}=\lvert\sigma(x+y)\cdot\xi\rvert\quad\text{is convex in } y\in(\mathbb{R}\xi)^\perp\text{ for all }x\in\mathbb{R}^N\text{ and all }\xi\neq 0,$$

where $(\mathbb{R}\xi)^\perp$ is the hyperplane orthogonal to $\xi$. For the fully nonlinear equation, convexity is preserved if and only if, for every $\xi\neq 0$, $x\in\mathbb{R}^N$ and every matrix $A\in X_\xi$ — where $X_\xi$ consists of the symmetric $A$ with $A\xi=0$, $A$ positive definite on $(\mathbb{R}\xi)^\perp$, and $B=A^{-1}$ on $(\mathbb{R}\xi)^\perp$ with $B\xi=0$ — the map $y\mapsto F(x+y,B)$ is concave on $(\mathbb{R}\xi)^\perp$. Necessity is proved by an explicit construction: if the structural condition fails at a point, one builds a convex initial datum (a fourth-order perturbation polynomial, extended to a global convex function) whose second directional derivative immediately becomes negative along the evolution. The conditions are rigid: for $N\ge 2$, if $a$ grows strictly slower than quadratically at infinity, convexity is preserved only for constant $a$. The note also propagates semi-convexity (convexity of $u+\frac{C_0}{2}\lvert x\rvert^2$) and $C^{1,1}$ bounds, i.e. bounds forcing the first spatial derivatives to be Lipschitz continuous (Theorem 4.1).

Section 4 of the note then announces, without proofs, an extension program: equations depending on $t$, $u$ and $Du$ (that is, $F=F(x,t,u,Du,D^2u)$), obstacle problems (in which the solution is constrained to stay above a given obstacle function, leading to a variational inequality with a free boundary), stationary elliptic equations, integro-differential equations with jump terms (operators of the form $\int (u(x+z)-u(x)-\nabla u(x)\cdot z)\,\nu(dz)$ associated with jump-diffusion processes, for which the natural solution concept is again the viscosity one, adapted to nonlocal operators), boundary conditions other than state constraints, joint convexity in $(x,K)$ for call payoffs $u_0(x)=(x-K)^+$, and partial convexity questions; the detailed proofs were deferred to later publication, and no sequel carrying them has appeared.

Two further lines of later work provide context. First, the microscopic or constant-rank program (Caffarelli, Guan, and Ma 2007; Bian and Guan 2009; Bian, Guan, Ma, and Xu 2011; Chen and Hu 2013; Chen 2014; Székelyhidi and Weinkove 2021) studies when the rank of the Hessian cannot drop along solutions — a constant rank theorem asserts that the concavity/curvature structure forces nonnegative curvature to spread to all directions — and yields sufficient conditions for convexity of solutions; it does not aim at the Lions–Musiela necessary-and-sufficient classification. Second, the power- and F-concavity program of the Ishige–Salani school (Ishige, Liu, and Salani 2020; Ishige, Salani, and Takatsu 2024) characterizes which concavity transforms are preserved by heat-type flows on domains — F-concavity meaning concavity of a fixed monotone transform $\phi(u)$ of the solution rather than of $u$ itself, power concavity being the model case $\phi(s)=s^p$ — a boundary-value analog for linear equations.

Problem Statement

Let $u$ denote the viscosity solution of the Cauchy problem $\partial_tu+F=0$ on $\mathbb{R}^N\times(0,\infty)$ with convex initial datum $u_0$, and call convexity preserved if $u(\cdot,t)$ is convex for all $t\ge 0$ whenever $u_0$ is convex. Lions and Musiela (2006) solved the characterization problem for $F=F(x,D^2u)$ (and the linear subclass $\partial_tu-a^{ij}\partial_{ij}u=0$; their Theorems 2.1 and 3.1) and explicitly deferred, in Section 4 of their note, the analogous characterizations. The problem is to carry out that program: for each of the following settings, determine necessary and sufficient structural conditions on the operator under which convexity is preserved, in the same iff sense and for the same admissible data as the 2006 theorems, or prove that only an explicitly identified trivial subclass of operators preserves convexity (a rigidity/triviality statement):

1. General dependence: $F=F(x,t,u,Du,D^2u)$, including $u$- and gradient-dependence.
2. Obstacle problems: variational inequalities in which the solution is constrained to stay above a given obstacle function, in both the parabolic and the stationary case.
3. Stationary elliptic equations: the analogous question for elliptic boundary/whole-space problems, i.e. which operators force convex solutions from convex data or boundary geometry.
4. Integro-differential equations: equations with nonlocal jump terms corresponding to jump-diffusion processes, in the viscosity framework.
5. Boundary conditions: determine necessary and sufficient conditions on the boundary geometry and boundary data, together with the interior structural conditions, under which convexity is preserved for boundary conditions other than state constraints.
6. Partial convexity: classify the operators (and the geometric situations) under which the solution is convex in a subset $y$ of the variables when $u_0$ is convex in $y$ only — the case the authors' preliminary results indicate fails in general; the classification should identify the nontrivial exceptions.
7. Joint state–strike convexity: for $N=1$ and call-type initial data $u_0(x)=(x-K)^+$, characterize when the solution is jointly convex in $(x,K)$, for the linear equation $\partial_tu-a^{ij}\partial_{ij}u=0$ and the fully nonlinear equation $\partial_tu+F(x,D^2u)=0$ of the source.

Each of the seven settings is an independent problem rather than a sub-step of a single combined proof. A secondary component recorded by the same source (Introduction, remark (ii)) is the question whether Theorem 2.1 for the linear equation admits a purely probabilistic proof, rather than the analytic maximum-principle argument; this component is independent of the seven settings above.

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