Minimum curvature flow and martingale exit times

We study the following question: What is the largest deterministic amount of time $T_*$ that a suitably normalized martingale $X$ can be kept inside a convex body $K$ in $\mathbb{R}^d$? We show, in a viscosity framework, that $T_*$ equals the time it takes for the relative boundary of $K$ to reach $X(0)$ as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio--Soner codimension-$(d-1)$ mean curvature flow of the $1$-skeleton of $K$. Our results are obtained by a mix of probabilistic and analytic methods.


Introduction and main results
Let d ≥ 2 and let K ⊂ R d be a convex body, i.e. a nonempty compact convex set.If X = (X 1 , . . ., X d ) is a d-dimensional continuous martingale that starts inside K and whose quadratic variation satisfies tr X (t) = X 1 (t)+• • •+ X d (t) ≡ t, then X eventually leaves K. What is the maximal deterministic lower bound T * on the exit time, across all such martingales X?The answer is linked to the evolution of the (relative) boundary of K as it undergoes a geometric flow that we refer to as minimum curvature flow: T * is equal to the lifetime of this flow.The minimum curvature flow resembles the well-known mean curvature flow, in particular its version in codimension d − 1 introduced by Ambrosio and Soner (1996).Our goal is to develop the connection between the exit time problem and the minimum curvature flow in detail.
Our original motivation comes from a long-standing problem in mathematical finance, namely to characterize the worst-case time horizon for so-called relative arbitrage.In a suitably normalized setup, the answer turns out to be precisely T * , with K being the standard d-simplex.We do not discuss this connection further here; instead we provide full details in the companion paper Larsson and Ruf (2021).Let us however emphasize that this application motivates us to consider convex bodies K with nonsmooth boundary.
To give a precise description of our main results, let X denote the coordinate process on the Polish space Ω = C(R + , R d ) of all continuous trajectories in R d with the locally uniform topology.Thus X(t, ω) = ω(t) for all ω ∈ Ω and t ∈ R + .Write P(Ω) for the set of all probability measures on Ω with the topology of weak convergence.For each x ∈ R d , define P x = {P ∈ P(Ω) : X is a P-martingale and P(X(0) = x) = P(tr X (t) ≡ t) = 1} , where the martingale property is understood with respect to the (raw) filtration generated by X.We always take K ⊂ R d to be compact, but not necessarily convex unless explicitly stated.The first exit time from K is and we are interested in computing the value function v(x) = sup P∈Px Pess inf τ K .
(1.2) This is the largest deterministic almost sure lower bound on the exit time τ K across all martingale laws P ∈ P x .
Our first result states that the value function solves a PDE with (degenerate) elliptic nonlinearity F (p, M ) = inf − 1 2 tr(aM ) : a 0, tr(a) = 1, ap = 0 , (1.3) where a ranges through all symmetric matrices of appropriate size, and a 0 refers to the positive semidefinite order.The theorem uses the notion of viscosity solution, which is reviewed in Section 3 where also the proof is given.
Theorem 1.1.Let d ≥ 2 and suppose K is compact, but not necessarily convex.The value function v is an upper semicontinuous viscosity solution to the nonlinear equation in int(K) with zero boundary condition (in the viscosity sense).
The value function is always an upper semicontinuous viscosity solution.As our next result shows, it is actually the unique viscosity solution in this class, provided that K satisfies a certain additional condition.This condition holds for all strictly star-shaped compact sets, in particular for all convex bodies with nonempty interior.Our condition is however more general than that; see Example 4.2.We also show that uniqueness may fail for star-shaped but not strictly star-shaped domains; see Example 4.3.This answers a question of Kohn and Serfaty (2006, Section 1.8).The proof of the following uniqueness theorem is given in Section 4, and follows from a comparison principle proved there, Theorem 4.1.
Theorem 1.2.Let d ≥ 2 and suppose K is compact.Assume there exist invertible affine maps T λ on R d , parameterized by λ ∈ (0, 1), such that T λ (K) ⊂ int(K) and lim λ→1 T λ = I (the identity).Then the value function v is the unique upper semicontinuous viscosity solution to (1.4) in int(K) with zero boundary condition (in the viscosity sense).
Remark 1.3.We point out that this uniqueness result is designed to handle the non-smooth convex domains that arise in the financial applications of interest.There are however other natural domains that are not covered by this result, such a various non-convex domains with smooth boundary.Proving comparison theorems (and hence uniqueness results) for such domains is an interesting problem which we do not consider here; see however Soner (1986a,b); Barles et al. (1999); Barles and Da Lio (2004).
Theorem 1.2 characterizes the value function even in cases where it is not continuous.In fact, we will give examples showing that the value function may be discontinuous even when K is a convex body.
Before describing this and related results, we briefly discuss links to the existing literature and the connection to geometric flows.
Our results tie in with a well established literature on stochastic representations of geometric PDEs, initiated by Buckdahn et al. (2001) and Soner and Touzi (2002aTouzi ( ,b, 2003)).In particular, Soner and Touzi introduced the notion of stochastic target problem and based their analysis on an associated dynamic programming principle; see also Bouchard and Vu (2010a).Part of our analysis can be cast in the language of stochastic target problems, and this connection is described further in Remark 2.6.
The control problem (1.2) is formulated over an infinite time horizon.As a result, our PDE is elliptic rather than parabolic, and, as explained next, the solution acquires the interpretation of arrival time of an evolving surface.This is reminiscent of the two-person deterministic game introduced by Spencer (1977) and linked to the positive curvature flow by Kohn and Serfaty (2006).In a similar spirit there are also the works of Peres et al. (2009) on the tug-of-war game and infinity Laplacian, and more recently Drenska and Kohn (2020) and Calder and Smart (2020).
The geometric meaning of (1.4) is most clearly conveyed by reasoning as in Section 1.2 of Kohn and Serfaty (2006).This is standard in the literature on geometric flows and paraphrased here for convenience.Let K be strictly convex with smooth boundary ∂K.
Suppose we are given a family {Γ t : t ≥ 0} of smooth convex surfaces with Γ 0 = ∂K, that evolve with normal velocity equal to (half) the smallest principal curvature at each point x ∈ Γ t .It is natural to call this minimum curvature flow, by analogy with mean curvature flow whose normal velocity is the average curvature.
Let u be the arrival time function: for each x ∈ K, u(x) is the time it takes the evolving front to reach x (we assume the front passes through each point in K exactly once.)Thus Γ t = {x : u(x) = t} is a level surface of u, and the gradient ∇u(x) is a normal vector at x.If ∇u(x) = 0, the minimal principal curvature of Γ t at x is the smallest value of as y ranges over all tangent unit vectors: |y| = 1 and y ∇u(x) = 0. 1 On the other hand, since u(x) is the arrival time, the speed of normal displacement at x is 1/|∇u(x)|.We therefore expect u to satisfy at least at points where ∇u = 0.It is not hard to check that this is precisely (1.4).In the planar case d = 2, Γ t has only one principle curvature direction, and (1.5) reduces to the well-known arrival time PDE for the mean curvature flow, Remark 1.4.Let us outline how the minimum curvature flow can be constructed rigorously using the level set method of Osher and Sethian (1988);Chen et al. (1991); Evans and Spruck (1991) and then linked to (1.2) and (1.4).Fix a time horizon T > max x∈K v(x) and consider the geometric parabolic equation with an initial condition U 0 (x) that is positive on int(K), negative on K c , and constant, say equal to −1, outside some large compact set.Chen et al. (1991, Theorems 6.7-6.8)yields existence and uniqueness of a continuous solution U (t, x) of the initial value problem.One now defines the evolving front of the minimum curvature flow at time t to be the boundary of the superlevel set, ∂{x : U (t, x) > 0}.The time u(x) = inf{t : U (t, x) < 0} at which the front passes through x ∈ K can then, under suitable conditions, be shown to be an upper semicontinuous viscosity solution of the elliptic equation (1.4).If uniqueness holds for this equation, for instance if Theorem 1.2 is applicable, it follows that u actually coincides with the value function v in (1.2).
1 Indeed, if γ : R → Γt is a smooth geodesic curve with unit speed such that γ(0) = x and γ (0) = y, where k is the curvature of γ at 0.
The link to the control problem (1.2) can be understood as follows.Proceeding informally, we assume a C 2 solution u of (1.4) with u = 0 on ∂K is given.By Itô's formula, under any law P ∈ P x , where a(t) is the derivative of the quadratic variation of X and satisfies tr(a(t)) ≡ 1.The discussion of minimum curvature flow suggests that optimally, X should fluctuate tangentially to the level surfaces of u, that is, a(t)∇u(X(t)) ≡ 0.Then, due to the definition (1.3) of F and since u solves (1.4), Combining (1.6) and (1.7) leads to showing that τ K ≤ u(x).If a(t) maximizes the left-hand side of (1.7), we have equality and expect that u coincides with the value function.Still heuristically, this happens when X fluctuates only along the minimal principle curvature directions of the level surfaces of u.This minimizes the speed at which X moves "outwards" toward ∂K, and maximizes the amount of time X spends in K.This discussion suggests that optimally, X lies on the evolving front of the time-reversed minimum curvature flow.More precisely, before exiting K, one expects that X satisfies v(X(t)) = v(x) − t under some optimal law P ∈ P x , x ∈ K. Theorem 1.7 below shows that this is true if K is a polytope and v sufficiently regular.It is however false in general, even if v is smooth; see Example 2.3.
In the case where K is not convex, we get a somewhat different flow.Similarly to the positive curvature flow of Kohn and Serfaty (2006), it is now the positive part of the minimum principal curvature that determines the speed of the flow.
We now return to our main results, and focus on the case where K is a convex body.Theorems 1.1 and 1.2 yield upper semicontinuity of the value function v and characterize it as a viscosity solution of (1.4) with zero boundary condition (in the viscosity sense).If K has empty interior we simply apply these results in the affine span of K.The following result is a combination of Proposition 5.1 and Lemma 5.3 in Section 5.
Theorem 1.5.Let d ≥ 2 and suppose K is a convex body.Then the value function v is quasi-concave, vanishes on all faces of K of dimension zero and one, and is strictly positive elsewhere in K.
In particular, if K is strictly convex, then all its boundary faces have dimension zero, and v vanishes everywhere on ∂K.Because of upper semicontinuity, this implies that it is continuous at ∂K.In fact, Theorem 1.6 below shows that v is continuous everywhere in this case.
However, many convex bodies K have boundary faces of higher dimension.In this case v does not vanish everywhere on ∂K.This includes the standard d-simplex appearing in our motivating financial application.Additionally, and more subtly, there are convex bodies for which the value function is actually discontinuous.This is because in dimension d ≥ 4, there are convex bodies that admit boundary points x n , all contained in 1-dimensional boundary faces, whose limit x = lim n x n lies in the relative interior of a 2-dimensional boundary face; see Example 5.4.For such points, v(x n ) = 0 but v(x) > 0, so continuity fails.This is in sharp contrast to the more familiar case of mean curvature flow, where the arrival time function is continuous for any convex initial surface; see Evans and Spruck (1991, Theorem 7.4) and Evans and Spruck (1992, Theorem 5.5).
We prove continuity under the following regularity condition on the geometry of K. We require that the k-skeletons, defined by (1.8) be closed for k = 1, . . ., d (but not for k = 0, thus the set of extreme points need not be closed.)This condition is a weakening of a notion from convex geometry called stability, which is equivalent to all the k-skeletons being closed, including the 0-skeleton; see e.g.Papadopoulou (1977) and Schneider (2014, p. 78).Actually the d-, (d − 1)-and (d − 2)skeletons of a convex body are always closed, so this does not have to be assumed separately; see Lemma 5.7.The upshot is the following result, which is applicable in a number of interesting situations.In particular, it covers all convex bodies in R 3 , all polytopes in arbitrary dimension, and all convex bodies whose boundary faces all have dimension zero or one.It is a rewording of Theorem 5.8 in Section 5, and is proved using probabilistic arguments based on the control formulation (1.2).
Theorem 1.6.Let d ≥ 2 and suppose K is a convex body with F k closed for 1 ≤ k ≤ d − 3. Then the value function v is continuous on K.
The fact that v vanishes only at the 1-skeleton F 1 (the extreme points and lines), but not elsewhere in K, suggests that (1.4) describes a geometric flow also of F 1 , not only of ∂K.This flow of F 1 is the codimension-(d − 1) mean curvature flow of Ambrosio and Soner (1996), although here the initial set F 1 need not be a one-dimensional curve.
To spell this out, for any symmetric matrix A and eigenvector p of A, let λ min (A, p) denote the smallest eigenvalue of A corresponding to an eigenvector orthogonal to p. Then (1.5) states that λ min − 1 2 P ∇u(x) ∇ 2 u(x)P ∇u(x) , ∇u(x) = 1, (1.9) where Modulo sign conventions and the factor 1/2, the left-hand side of (1.9) is precisely the operator used by Ambrosio and Soner (1996).In fact, the function where v is the value function in (1.2), solves their parabolic equation on K with initial condition V (0, x) = −v(x), whose zero set (in K) is the 1-skeleton F 1 .This suggests interpreting the minimum curvature flow of ∂K as a codimension-(d − 1) mean curvature flow of F 1 .This perspective is particularly compelling when K is a polytope: F 1 is then a finite union of closed line segments and thus one-dimensional, albeit with "branching".In this case, the one-dimensional initial contour instantly develops higher-dimensional features as it evolves under the flow, and eventually becomes a closed hypersurface.This is illustrated schematically in Figure 1, where K is the standard 3-simplex.
Figure 1: Schematic illustration of the minimum curvature flow of the 3-simplex, regarded as codimension-2 mean curvature flow of its 1-skeleton as initial contour.In the second and third panel, the 1-skeleton is still shown for reference.
Returning to the minimum curvature flow as a flow of surfaces starting from ∂K, we see that points inside two-and higher dimensional faces remain stationary for some period of time.This behavior is analogous to the behavior of mean curvature flow of non-convex contours; see Kohn and Serfaty (2006, Figure 4) for an illustration.We thank R. Kohn for pointing this out to us.A similar phenomenon occurs for the Gauss curvature flow; see Hamilton (1994); Chopp et al. (1999); Daskalopoulos and Lee (2004).
We do not have much information about the regularity of the value function v in general, beyond the continuity assertion in Theorem 1.6 and the counterexample in Example 5.4.An exception is the planar case d = 2, where we recover the standard mean curvature flow.In this case, for K strongly convex with smooth boundary, Kohn and Serfaty (2006) proved that v is C 3 (see also Huisken (1993) for an earlier proof that v is C 2 ).In general, let us assume that v is C 2 inside each face of K, with just one critical point.If in addition K has at most countably many faces, it is then possible to construct optimal solutions of (1.2) where the intuitive notion that X should fluctuate tangentially to, and remain on, the level surfaces of v becomes rigorous.
Theorem 1.7.Let d ≥ 2 and let K be a convex body with at most countably many faces.Assume the value function v lies in C 2 (K).Assume also that in each face F of dimension at least two, either v has no critical point, or v has one single critical point which additionally is a maximum.Then for every x ∈ K there is an optimal solution P ∈ P x under which v(X(t)) = v(x) − t for all t < τ K .In particular, t 0 ∇v(X(s)) dX(s) = 0, t < τ K , and X lies on the evolving front of the time-reversed minimum curvature flow in the sense that X(t) ∈ Γ v(x)−t , where Γ t = {x : v(x) = t}, until it leaves K.
The meaning of C 2 (K) and the notion of a critical point is explained in Section 6, where also the proof is given.The basic idea is to observe that v satisfies (1.9) classically at non-critical points.In particular, by definition of eigenvalue, the matrix is singular at all such points, so is of rank at most d − 1.This can be used to construct a martingale law P ∈ P x under which H(X(t))d X (t) = 0.This turns out to imply ∇v(X(t)) dX(t) = 0 and then dv(X(t)) = −dt.This is essentially the desired conclusion.Some effort is needed to construct P, basically because the Moore-Penrose inverse H(x) + of H(x) is no longer continuous in x.Moreover, X is obtained by constructing martingales on each face of K separately and then "gluing" these martingales together.This introduces some technical hurdles, and explains why the proof is somewhat lengthy.As an illustration, and for later use, we give a simple example where the value function v is known explicitly and happens to be smooth on K; see also Stroock (1971) and Fernholz et al. (2018).
Example 1.8.Let d ≥ 2 and let K = {x ∈ R d : |x| ≤ r} be the centered closed ball of radius r > 0. In this case, v(x) = r 2 − |x| 2 for all x ∈ K. To see this, choose any x ∈ K and P ∈ P x .We have Evaluating at t = τ K ∧ n, taking expectations, and letting n → ∞, one obtains E[τ K ] = r 2 − |x| 2 .In particular, this shows that X escapes from any bounded set in finite time, P-a.s.Moreover, since of course Pess inf τ K ≤ E[τ K ], we get v(x) ≤ |x| 2 − r 2 .In fact, we have equality.Indeed, let P ∈ P x be the law under which X 3 , . . ., X d are constant and where W denotes a one-dimensional Brownian motion.Such a probability measure P always exists, even if x = 0; see Lemma 3.4.An application of Itô's formula now yields τ K = r 2 − |x| 2 , P x -a.s.We deduce that v(x) = r 2 − |x| 2 for all x ∈ K. Furthermore, it is straightforward to verify that v satisfies (1.4) with boundary condition v = 0 on ∂K.
The reasoning in Example 1.8 directly yields the following upper bound on v.
Lemma 1.9.If K is compact, x ∈ K, and , where r is the radius of the smallest ball containing K. In particular, τ K < ∞, P-a.s., and the value function defined in (1.2) satisfies v(x) ≤ r 2 for all x.
The rest of the paper is organized as follows.Section 2 develops a number of general properties of the value function, as well as illustrative examples.In particular, a dynamic programming principle is proved.In Section 3 we prove Theorem 1.1 that the value function is a viscosity solution.In Section 4 we prove Theorem 4.1, a comparison principle for viscosity solutions of (1.4), and use it to deduce Theorem 1.2.In Section 5 we focus on the case where K is a convex body, and establish in particular Theorem 5.8 on continuity of the value function.In Section 6 we prove Theorem 1.7.
We end with a technical remark regarding filtrations and stopping times.Whenever X is said to be a martingale, this is understood with respect to its own filtration F X = (F X t ) t≥0 where F X t = σ(X s , s ≤ t).In this case, X is also a martingale for the right-continuous filtration F X + consisting of the σ-algebras u>t F X u , and similarly for the filtrations obtained by augmenting F X and F X + with nullsets.In particular, results such as the stopping theorem are applicable with τ K in (1.1), which is an F X + -stopping time but not an F X -stopping time.

The value function and dynamic programming
The purpose of this section is to establish a number of properties of the value function, in particular a dynamic programming principle.Throughout this section, K is compact but not necessarily convex.
Proof.We claim that ω → τ K (ω) is upper semicontinuous on Ω.To see this, let ω n , ω satisfy This proves upper semicontinuity of τ K since ε > 0 can be chosen arbitrarily small.Next, for every λ > 0 the Portmanteau theorem yields that the map Proposition 2.2.(i) P x is weakly compact for every x ∈ R d ; (ii) v, given in (1.2), is upper semicontinuous and there is a measurable map x → P x from R d into P(Ω) such that P x lies in P x and is optimal for all x ∈ R d ; (iii) v satisfies the following dynamic programming principle: for every x ∈ R d and every Moreover, the supremum is attained by any optimal P ∈ P x .
Proof.(i): Consider any P ∈ P x .Fix s ≥ 0 and define so that Kolmogorov's continuity criterion (see Revuz and Yor (1999), Theorem I.2.1 and its proof) then gives, for any fixed T > 0 and α ∈ (0, 1 4 ), for some constant c = c(T, α) that does not depend on P ∈ P x .Since Hölder balls are relatively compact in C([0, T ], R d ) by the Arzelà-Ascoli theorem, it follows that P x is tight and hence relatively compact by Prokhorov's theorem.To see that P x is closed, note that the martingale property of both X and |X|2 − t (and hence the property tr X (t) ≡ t) carries over to weak limits of sequences in P x .
(ii): First observe that P x consists of the pushforwards (x + •) * P with P ∈ P 0 .Thus v(x) = sup P∈P 0 f (x, P), where f (x, P) = g((x + •) * P) and g(P) = Pess inf τ K .By Lemma 2.1, the function g is upper semicontinuous.Since f is the composition of g with the continuous function (x, P) → (x + •) * P from R d × P(Ω) to P(Ω), it is also upper semicontinuous.Moreover, P 0 is compact by (i).A suitable selection theorem, see e.g.Bertsekas and Shreve (1978, Proposition 7.33), yields upper semicontinuity of v as well as a measurable map (iii): Fix x ∈ R d and an F X -stopping time θ.We first first fix P ∈ P x and prove that To this end, consider the extended space Ω × Ω with coordinate process (X, Y )(t, ω, ω) = (ω(t), ω(t)) and define a law P on (Ω × Ω, F ⊗ F) by P (dω, dω) = P X(θ(ω),ω) (dω)P(dω), where we use the measurable map R d y → P y ∈ P from (ii).We now consider the process ; thus θ depends on the trajectory of X like θ depends on the trajectory of X.
Since θ is an F X -stopping time, and since X (t) and X(t) coincide for all t ≤ θ, it follows by Galmarino's test that θ (ω, ω) = θ(ω) for all (ω, ω); see Stroock and Varadhan (2006, Lemma 1.3.3).Consequently, for all bounded measurable maps F, G : Ω → R, we have Thanks to the definition of Q we have Since also P y is optimal for every y, we get (2.4) Combining the definition of v(x), (2.2), (2.3), and (2.4), we get In the last step we used that θ ∧ τ K and 1 θ≤τ K are F X θ -measurable (even though τ K is only an F X + -stopping time) and hence have the same law under P as under Q due to (2.2).This proves (2.1).
It remains to prove that for any optimal P ∈ P x .The proof uses the notion of conditional essential infimum.For a random variable Y and a sub-σ-algebra G ⊂ F, the conditional essential infimum of Y given G is defined as the largest G-measurable random variable P-a.s.dominated by Y , denoted by Pess inf{Y | G}.Moreover, if {F ω } ω∈Ω is a regular conditional distribution of Y given G, we have Pess inf{Y | G}(ω) = ess inf F ω for P-a.e. ω, where we set ess inf For further details, see Barron et al. (2003); Larsson (2018).Now, fix any optimal P ∈ P x .Then, using (2.3), we get Stroock and Varadhan (2006, Theorem 1.3.4).In particular, {F ω } ω∈Ω with One readily verifies that Q ω ∈ P X(θ,ω) for P-a.e. ω.Hence Q ω -ess inf τ K ≤ v(X(θ, ω)) for P-a.e. ω, and we deduce that v(x) ≤ θ ∧ τ K + v(X(θ))1 θ≤τ K , P-a.s.This yields (2.5), and completes the proof of the proposition.
It is not true in general that, under an optimal law, X(t) is located on the t-level surface of the value function, even if the value function is smooth.The following example illustrates this.
The following result can be viewed as an assertion about propagation of continuity: if the value function is continuous on a certain set, then it is also continuous on a larger set.Upper semicontinuity, which holds in general due to Proposition 2.2(ii), plays an important role.A refined version of this result is crucial in Section 5, where K will be a convex body.
Proposition 2.4.Let K be compact, and assume v| ∂K is continuous.Then v| K is continuous.
Proof.Since v is upper semicontinuous by Proposition 2.2(ii), since ∂K is compact, and since v| ∂K is continuous by assumption, Lemma 2.5 below gives a modulus ω such that (2.7) Fix x, ȳ ∈ K and an optimal law P ∈ P x.Define the process Y = X − x + ȳ and the F Xstopping time θ = inf{t ≥ 0 : We now combine this with two applications of the dynamic programming principle of Proposition 2.2(iii).We get In the last inequality, the application of the dynamic programming principle uses that the law of Y lies in P ȳ, that F Y = F X , and that θ ≤ inf{t ≥ 0 : Y (t) / ∈ K}, P-a.s.Since x, ȳ ∈ K were arbitrary, we deduce that v| K is uniformly continuous with modulus ω.
The following lemma is elementary, but crucial for our results on propagation of continuity.This is what allows us to exploit the fact that the value function is always upper semicontinuous.
Lemma 2.5.Let C ⊂ R d be a compact set, and let f : R d → R be a function that is upper semicontinuous at every point in C. If the restriction f | C is continuous, then there exists a modulus ω such that The balls B(y, δ y /2), y ∈ C, cover C. By compactness, there is a finite subcover B(y i , r i ), i = 1, . . ., n, where r i = δ y i /2.Define δ = min{r 1 , . . ., r n }.Suppose x ∈ R d , y ∈ C, and |x − y| < δ.Then y ∈ B(y i , r i ) for some i ∈ {1, . . ., n}, and hence |x − y As mentioned in Section 1, some of the analysis in this paper can be cast in the language of stochastic target problems.We end this section with a remark detailing this connection.Since this is not used in the analysis to come, we do not give proofs.
Remark 2.6.For any t ∈ [0, ∞), the target reachability set when the target is K and the controlled state dynamics is described by P x is defined by This is a "time-to-maturity" version, in a weak formulation, of the definition in Soner and Touzi (2002a, Section 2.4).Clearly V (0) = K, and one can show that V (t) = ∅ for all t > diam(K) 2 /4.One expects the following representation of the value function v in (1.2) in terms of the target reachability set: This equality can be shown to hold if K is convex, but there are non-convex examples where it fails.In such cases, one can work with the obstacle version of the stochastic target problem, where the reachability set is defined by This problem is discussed briefly in Section 7 of Soner and Touzi (2002a) and further in Bouchard and Vu (2010b) (where the terminology "obstacle version" is introduced).It is straightforward to show that regardless of the geometry of K.A suitable weak-formulation version of the geometric dynamic programming principle in Theorem 7.1 of Soner and Touzi (2002a) or Theorem 2.1 of Bouchard and Vu (2010b) could then be used to derive characterizations of W (t), and hence v(x), in terms of PDEs.

The value function is a viscosity solution
In this section we prove Theorem 1.1, the viscosity solution property, assuming that d ≥ 2 and that K is compact but not necessarily convex.(We already know from Proposition 2.2(ii) that v is upper semicontinuous.)A bounded function u : where an upper (lower) star denotes upper (lower) semicontinuous envelope (restricting the function to K).We say that u has zero boundary condition (in the viscosity sense) if The function u is said to be a viscosity supersolution in int(K) with zero boundary condition if the same conditions hold with u * , F * , max, ≤ replaced by u * , F * , min, ≥.It is a viscosity solution in int(K) with zero boundary condition if it is both a viscosity sub-and supersolution in int(K) with zero boundary condition.
To prove Theorem 1.1, we must establish the sub-and supersolution properties.We carry out these tasks separately in the following two subsections.To do so, the following description of the semicontinuous envelopes of F will be needed.Proof.From the representation (1.5) we have F (p, M ) = − 1 2 sup{y M y : |y| = 1, y p = 0}.One checks that this is continuous on the set (R d \ {0}) × S d , and in particular equal to F * and F * there.Next, we claim that for all (p, M ).The first inequality follows because sup{y M y : |y| = 1} = λ 1 (M ).For the second inequality, use the spectral theorem to write for an orthonormal basis w 1 , . . ., w d of eigenvectors of M .Express p and y is this basis, If π 1 = 0 one can take η 1 = 1 and In either case, the second inequality of (3.1) holds.
For any fixed M , there is a sequence (p n , M n ) → (0, M ) with F * (0, M ) = lim n F (p n , M n ).Thus by (3.1) and since λ 1 (M ) is continuous in M , we get This shows that F * (0, M ) = F (0, M ).On the other hand, with w 1 an eigenvector of M with eigenvalue λ 1 (M ), we have /2 and thus, by (3.1) and the continuity of For later use, let us also record the following observations.We let | • | op denote the operator norm of a matrix.
and taking infimum on the right-hand side gives the assertion, still for p = 0. Consider now the case p = 0 and consider a sequence (p n , M n ) converging to (p, M ) with p n = 0 such that lim n F (p n , M n ) = F * (p, M ).Since for sufficiently large n we have F (p n , M n ) > 0, we get from the case just established that as desired.
Corollary 3.3.Let B be a d × d invertible matrix, viewed as a linear map.Define K = B −1 (K) and w = |(BB ) −1 | op w • B. If, on K, w is a lower semicontinuous viscosity supersolution of (1.4) with zero boundary condition, then so is w on K .
Proof.The statement follows from the definition of viscosity supersolution, in conjunction with Lemma 3.2.

Subsolution property
We now prove the subsolution property claimed in Theorem 1.1.Since v is upper semicontinuous and F is lower semicontinuous, we may drop the stars in the definition of subsolution.
Proof of the subsolution property.Fix x ∈ K.If x ∈ int(K) then v(x) > 0 by Example 1.8.If x ∈ ∂K and v(x) = 0 then the subsolution property holds for this point.Hence, without loss of generality, we may assume that v(x) > 0.
Note also that there exists some c > 0 such that for all (x, a) where λ 1 (M ) denotes the largest eigenvalue of a symmetric matrix M .The boundedness comes from the continuity of λ 1 .Furthermore, we have Fix any optimal P ∈ P x.We then have a predictable S d + -valued process (a(s)) s≥0 such that X (t) = t 0 a(s)ds and tr(a(t)) = 1, dt ⊗ dP-a.e.

Define the stopping time
Clearly θ ≤ τ K by definition of v(x) and P[X(θ) We can now define the predictable set Next, the dynamic programming principle of Proposition 2.2(iii) yields Using (3.6) and then (3.5), we get Combining this with Itô's formula, the definition of J, and (3.4), we get Now, define the process Due to (3.3) and the definition of J, we then have Consider now the exponential local martingale Z given by This is well-defined since ∇ϕ is bounded on the closure of B ε (x), which contains X(t) for t ∈ J.An application of Itô's formula shows that multiplying (3.7) by Z(t) gives a local martingale, and hence a supermartingale since it is nonnegative.Therefore, using that θ < ∞, P-a.s., and P[X(θ) ∈ K ∩ ∂B ε (x)] > 0 for the first inequality.This contradiction completes the proof of the subsolution property.

Supersolution property
The following result is used in the proof.
Lemma 3.4.Let m ∈ N with m ≥ 2, and let S be a nonzero skew-symmetric m×m matrix and let x, x ∈ R m .Then there exists a weak solution to the SDE Here W denotes a one-dimensional Brownian motion.Suppose now that S(x− x) = 0, and select points x n ∈ R m with x n → x and S(x n − x) = 0.For each n, let Y n be a solution to the SDE with Y n (0) = x n .Since tr Y n (t) ≡ t, the law of Y n − x n lies in P 0 , which is compact by Proposition 2.2(i).Thus after passing to a subsequence, we have t for all t ≥ 0} is closed, and since Y n lies in C almost surely for all n, the Portmanteau lemma implies that Y does as well.In particular, we have where Lf (u, y) = 1 2 (y − x) S ∇ 2 f (y)S(y − x)/(|S(y − x)| 2 + u) is the operator associated to the given SDE.Note that this uses that |S(Y n (u)− x)| 2 = |S(x− x)| 2 +u.The expression inside the expectation on the left-hand side of (3.8) is a bounded continuous function of the trajectory of Y n .We may therefore pass to the limit and deduce that the corresponding equality holds for Y as well.It follows that Y solves the martingale problem problem associated with the given SDE.Equivalently, Y is a weak solution, as desired.
We now turn to the supersolution property claimed in Theorem 1.1.
Proof of the supersolution property.Fix x ∈ K.If x ∈ ∂K then there is nothing to prove since v is nonnegative.Hence, we may assume throughout the proof that x ∈ O, where we write O = int(K).
In particular, there exists a skew-symmetric d × d matrix S such that σ = S∇ϕ(x); for instance, (3.9) Fix any x ∈ B ε (x).Define and let P be the law under which X satisfies where W is a one-dimensional Brownian motion and e 1 is the first canonical unit vector (any other unit vector would also do).Note that P ∈ P x and θ ≤ τ K , and thus θ < ∞, P-a.s. by Lemma 1.9.Define δ = min Using first that v ≥ v * ≥ ϕ + δ on ∂B ε (x); then Itô's formula; and finally (3.9) along with the fact that ∇ϕ S ∇ϕ = 0 by skew-symmetry of S, we get Combining this with the dynamic programming principle of Proposition 2.2(iii) yields Since x ∈ B ε (x) was arbitrary, we may send x → x such that v(x) → v * (x) = ϕ(x), and deduce 0 ≥ δ.This contradiction proves the supersolution property when ∇ϕ(x) = 0.
Fix any x ∈ B ε (x), and let P be a law under which X satisfies where W is a one-dimensional Brownian motion.Such P exists by Lemma 3.4, and it is clear that P ∈ P x .Itô's formula, the identity M S = γ 2 S, and the skew-symmetry of S give Example 4.2.If K is strictly star-shaped about the origin, meaning that λK ⊂ int(K) for all λ ∈ (0, 1), then it clearly satisfies the assumption of Theorem 4.1.In particular, this is the case if K is convex with 0 ∈ int(K).Here is an example of a body that is not star-shaped but satisfies the assumptions of Theorem 4.1: Indeed, one can use the linear maps T λ (x, y) = (λx, λ 2 y).It is easily verified that K is not star-shaped.
If K is star-shaped but not strictly star-shaped, then uniqueness among upper semicontinuous viscosity solutions may fail, as the following example shows; see also Soner (1993, Example 8.2).
The proof of Theorem 4.1 relies on the following maximum principle, which holds for arbitrary compact sets K.The boundary conditions in its statement should be understood in the viscosity sense.
Theorem 4.4.Let d ≥ 2 and suppose K is compact.Let u (w) be an upper (lower) semicontinuous viscosity subsolution (supersolution) of (1.4).Then there exists x ∈ ∂K that achieves max K (u−w).Moreover, if u in addition satisfies the zero boundary condition, and if w is a lower semicontinuous viscosity supersolution of (1.4) with zero boundary condition on some compact set K such that K ⊂ int(K ), then u ≤ w on K.
Proof.We proceed in several steps.
1.It is enough to prove the two assertions with u replaced by δu, for each δ ∈ (0, 1).Indeed, if δu ≤ w for all δ ∈ (0, 1) and if w is nonnegative (see Lemma 4.5 below) then also u ≤ w, yielding the second assertion.For the first assertion, assume we have xδ ∈ ∂K that achieves max K (δu − w), for each δ ∈ (0, 1).Then there exists a sequence (δ n ) such that lim n δ n = 1 and lim n xδn = x for some x ∈ ∂K.Then for all x ∈ K we have Sending n to infinity and using upper semicontinuity of u − w then shows that x achieves max K (u − w) as required.Now, δu is a subsolution of the equation and if u satisfies the zero boundary condition, then so does δu.Thus, by writing u instead of δu, we may and do assume throughout the proof that u itself is a subsolution of (4.1), where δ ∈ (0, 1) is arbitrary but fixed.2. For the first assertion, for every ε > 0, define for (x, y) ∈ K × K, and let (x ε , y ε ) maximize Φ ε over K × K. Then we have By compactness, (x ε , y ε ) converges to some (x, ȳ) ∈ K × K as ε → 0 along a suitable subsequence; in the following, ε is always understood to belong to this subsequence.Since by upper semicontinuity of u and of −w.Hence x maximizes u − w over K. Thus, to show the first assertion it suffices to argue that (x ε , y ε ) ∈ int(K) × int(K) is not possible.This forces x ∈ ∂K as desired.
For the second assertion, we define Φ ε as above, but now on the larger set K × K .Let (x ε , y ε ) again denote the corresponding maximizers, which converge along a subsequence to some (x, ȳ) ∈ K × K .We again obtain x = ȳ, thus ȳ ∈ K ⊂ int(K ), and therefore y ε ∈ int(K ) for all sufficiently small ε.We will use this to argue that u cannot satisfy the viscosity inequality at x ε .This forces x ε ∈ ∂K and, due to the boundary condition, u(x ε ) ≤ 0. Together with nonnegativity of w (see Lemma 4.5 below) this yields Φ ε (x ε , y ε ) ≤ 0 and thus, thanks to (4.2), max K (u − w) ≤ 0. This is the second assertion.
3. Both assertions can now be argued by contradiction in the same manner: for any fixed small ε > 0, we assume that both u and w simultaneously satisfy the viscosity inequalities at x ε and y ε , respectively, and use this to derive a contradiction.(Indeed, to prove the first assertion we had to exclude that (x ε , y ε ) ∈ int(K) × int(K), while for the second assertion we had to exclude that y ε ∈ int(K) and that u satisfies the viscosity inequality at x ε .) 4. Let us work under the assumptions of Step 3. Define To simplify notation, write We now claim that p = 0. Suppose for contradiction that p = 0. Then x ε = y ε , ∇ y ζ(x ε , y ε ) = 0, and ∇ yy ζ(x ε , y ε ) = 0. Since y ε minimizes y → w(y) + ζ(x ε , y) over K (respectively, over K ), the supersolution inequality states that 0 = F * (0, 0) ≥ 1.This contradiction confirms that p = 0.
Ishii's lemma, see Crandall et al. (1992, Theorem 3.2), now gives M, N ∈ S d such that Pre-and post-multiplying (4.4) by vectors of the form (z, z) and using (4.3) shows that M N .Now we use the fact that (p, M ) lies in limiting superjet of the subsolution u at x ε , the ellipticity of F , Lemma 3.1, the fact that p = 0, and finally that (−p, N ) lies in the limiting subjet of the supersolution w at y ε to get This is the required contradiction, which concludes the proof.
We used the following observation in the previous proof.The boundary condition in its statement should be understood in the viscosity sense.
Lemma 4.5.If w is a lower semicontinuous viscosity supersolution of (1.4) with zero boundary condition on some compact K ⊂ R d with d ≥ 0 then w ≥ 0.
Proof of Theorem 4.1.We assume for simplicity that the T λ are linear, not just affine; we may then identify T λ with its d × d matrix.Recall that | • | op denotes the operator norm.By Corollary 3.3, the function w λ = |(T λ T λ ) −1 | op w •T λ is a lower semicontinuous viscosity supersolution of (1.4) with zero boundary condition on K = T −1 λ (K).By the properties of T λ , we have K ⊂ int(K ).Theorem 4.4 then yields u ≤ w λ on K for all λ ∈ (0, 1).We thus obtain u ≤ lim sup λ→1 w λ ≤ w * on K as desired.

Convex bodies
Our next goal is to prove continuity of the value function v when K ⊂ R d (d ≥ 2) is a convex body satisfying an additional assumption.We first record the following simple property of the value function.
Proposition 5.1.Let K be a convex body.Then v is quasi-concave.
Proof.We must prove that v has convex super-level sets.Pick two distinct points x, y ∈ K, and let L be the line passing through x and y.Fix any point z ∈ L, and let P be the law under which X is a standard Brownian motion along L starting at z. Then P ∈ P z , and with θ = inf{t ≥ 0 : X(t) ∈ {x, y}} the dynamic programming principle yields v(z) ≥ Pess inf{θ + v(X(θ))} ≥ v(x) ∧ v(y).This proves quasi-concavity.
Recall the following notions from convex geometry; see Rockafellar (1970); Schneider (2014) for more details.Let F be any subset of R d .The affine span of F is denoted by aff(F ), with dimension dim(F ).The relative interior ri(F ) is the interior of F in aff(F ), and the relative boundary is rbd A face is called a boundary face if it is nonempty and not all of K.The relative boundary rbd(K) is the union of all boundary faces.For every x ∈ K, there is a unique face of K whose relative interior contains x.We call this face F x .For each k = 0, . . ., d, the k-skeleton is defined as in (1.8), namely In particular, F 0 consists of all extreme points, F 1 consists of all extreme points and line segments, F d−1 is the boundary of K, and F d is K itself.For convenience we introduce the notation for the first exit time of X from a set F .This notation is consistent with (1.1).
Lemma 5.2.Let K be a convex body and consider a point x ∈ K.For every P ∈ P x, we have τ K = τ Fx , P-a.s.
and the statement is obvious.Otherwise, there exists a supporting halfspace If not, we iterate the procedure and fix another halfspace As above, we again obtain τ K 2 = τ K 1 = τ K .We proceed in the same way, but thanks to the reduction in dimension at most d times, until K k = F x for some k ∈ {1, . . ., d}.We then have τ Fx = τ K k = . . .= τ K 1 = τ K , which proves the statement.
Lemma 5.3.Let K be a convex body and consider a point x ∈ K. Then v(x) = 0 if and only if dim(F x ) ≤ 1.
Proof.Suppose dim(F x ) = 0, so that F x = {x} is a singleton.Then X leaves F x immediately under any P ∈ P x , that is, τ Fx = 0. Suppose instead that dim(F x ) = 1, so that F x is a line segment.Then under any P ∈ P x , X evolves like a one-dimensional Brownian motion along the line segment F x , at least until τ Fx .Thus Pess inf τ Fx = 0, since X reaches the endpoints of F x arbitrarily quickly with positive probability.By Lemma 5.2, we have Pess inf τ K = 0. Therefore, in either case, we deduce that v(x) = 0.For the converse direction, assume that dim(F x ) > 1.Then there exists a dim(F x )-dimensional closed ball We now discuss continuity of the value function v.It was shown in Proposition 2.2(ii) that the value function v is upper semicontinuous.Therefore, if v(x) = 0 at a point x ∈ K, then v must be continuous at x.Of course, many convex bodies K have boundary faces of dimension two or higher, in which case Lemma 5.3 shows that v will not be zero everywhere on the boundary.Still, even in such cases, one might hope that v remains continuous.Unfortunately, this is not true in general, as the following example shows.
In Example 5.4, continuity of v fails because F 1 is not closed.One might therefore hope that continuity can be proved if F 1 , . . ., F d are closed.(Requiring F 0 closed should be, and is, unnecessary because v is zero on all of F 1 .)This condition indeed turns out to imply continuity.The proof iterates over the k-skeletons, in each step making use of the following refined version of the argument in Proposition 2.4.The argument is probabilistic and rests on the dynamic programming principle.
Lemma 5.5.Let K be a convex body, fix k ∈ {1, . . ., d}, and assume v| cl(F k−1 ) is continuous.Then there is a modulus ω such that the following holds.
Proof.Since v is upper semicontinuous by Proposition 2.2(ii), since cl(F k−1 ) is compact, and since v| cl(F k−1 ) is continuous by assumption, Lemma 2.5 gives a modulus ω such that v(x) ≤ v(y) + ω(|x − y|) for all x ∈ R d and y ∈ cl(F k−1 ). (5.2) We now show that ω satisfies the claimed property.To this end, let x, ȳ, A, and Q be as in the statement of the lemma, and select an optimal law P ∈ P x.Lemma 5.2 asserts that X(t) ∈ F x for all t ≤ τ K , P-a.s.By modifying the behavior after τ K , which does not affect the optimality of P, we may therefore assume that X(t) ∈ A for all t ≥ 0, P-a.s. (5.3) Consider the affine isometry Φ : A → aff(F ȳ) given by Φ(x) = Q(x − x) + ȳ.Using this isometry, define Y = Φ(X) and θ = inf{t ≥ 0 : Note that P(θ < ∞) = 1 by Example 1.8.Due to (5.3), Y takes values in aff(F ȳ), and hence Y (θ) ∈ rbd(F ȳ) ⊂ F k−1 , P-a.s.Thus by (5.2) and monotonicity of ω we have, P-a.s., where c = diam(K).We now combine this with two applications of the dynamic programming principle of Proposition 2.2(iii).This is permissible because θ is P-a.s.equal to an F X -stopping time, despite not being an F X -stopping time itself in general.We get In the last inequality, the application of the dynamic programming principle uses that the law of Y lies in P ȳ due to the isometry property of Φ, that F Y = F X , and that θ ≤ inf{t ≥ 0 : Y (t) / ∈ K}, P-a.s.This completes the proof.
We now state the key propagation of continuity result, analogous to Proposition 2.4.Part of the proof is convenient to phrase in terms of convergence of affine subspaces.For affine subspaces A n and A of R d , we say that A n → A if dim(A n ) = dim(A) for all large n, there are points x n ∈ A n and x ∈ A such that x n → x, and A n − x n converges to A − x as elements of the Grassmannian Gr(dim(A), R d ) of dim(A)-dimensional linear subspaces of R d .In this case, there exist orthogonal d × d matrices Q n such that Q n → I and the map y → Q n (y − x) + x n maps A to A n for n sufficiently large.The Grassmannian is known to be compact.Therefore, whenever the affine subspaces A n contain points x n that converge to some limit, it is possible to select a convergent subsequence of the A n .
Proof.Since v is upper semicontinuous by Proposition 2.2(ii), it suffices to show that v| F k is lower semicontinuous.Since v| cl(F k−1 ) is continuous by assumption, this amounts to showing that (5.4) Let therefore x, x n , α be as in (5.4).Define r n = dist(x n , rbd(F xn )).This is the radius of the largest k-dimensional ball centered at x n and contained in F xn .We consider two separate cases.
Case 1: Suppose lim inf n→∞ r n = 0.After passing to a subsequence, we have r n → 0. Then there exist points y n ∈ rbd(F xn ) such that |x n − y n | → 0. Thus y n ∈ F k−1 and y n → x, so that x ∈ cl(F k−1 ) and v(y n ) → v(x).Moreover, applying Lemma 5.5 with x = y n , ȳ = x n , A = aff(F xn ), and Thus v(x) ≤ α, proving (5.4) in this case.
Case 2: Suppose instead there exists r > 0 such that r n ≥ r for all n.Then each F xn contains a k-dimensional ball B n of radius r centered at x n .After passing to a subsequence, we have aff(F xn ) → A for some k-dimensional affine subspace A. Thus there exist orthogonal d × d matrices Q n such that Q n → I and the affine isometry Φ n : x → Q n (x − x) + x n maps A to aff(F xn ) for each n.Now, let B ⊂ A be the k-dimensional ball of radius r centered at x.There is only one such ball, and we have B n = Φ n (B) for all n.For any x ∈ B we thus have Φ n (x) ∈ F xn ⊂ K and Φ n (x) → x.Since K is closed, it follows that B ⊂ K. Hence B ⊂ F x, so that A = aff(B) ⊂ aff(F x).On the other hand, dim(A) = k ≥ dim(F x), so in fact A = aff(F x).We now apply Lemma 5.5 with x, ȳ = x n , A = aff(F x), and with c = diam(K).Sending n → ∞ yields v(x) ≤ α and proves (5.4).
In view of Lemma 5.6, it is of interest to know whether the k-skeletons of a given convex body are closed.For some values of k, closedness is automatic.x admits an open neighborhood contained in ri(F ) ∪ int(K) ∪ K c , and therefore cannot lie in the closure of F d−2 .This contradiction finishes the proof.
n for all n ∈ N thanks to Example 1.8.Thus by upper semicontinuity, we have This yields continuity of v| cl(F 2 ) .Hence by Lemma 5.6, v| F 3 is continuous.Lemma 5.7 yields that F 3 , F 4 , F 5 = K are closed.Repeating the previous argument shows that v| K is continuous, even though F 2 is not closed.

Smooth value functions
The goal of this section is to prove Theorem 1.7.Let us first introduce some terminology.Let K be a convex body.We say that a function f lies in C 2 (K) if f | K is continuous, and the restriction f | ri(F ) to the relative interior of any face F of K lies in C 2 (ri(F )), understood in the usual sense of twice continuous differentiability on the dim(F )-dimensional open set ri(F ) ⊂ aff(F ).The gradient and Hessian computed relative to this set are then denoted are the projections of ∇f (x) and ∇ 2 f (x) onto aff(F − x), whenever the latter exist.A critical point of f in F is a point x ∈ ri(F ) where ∇ K f (x) = 0.
To prove Theorem 1.7, it is enough to prove the following.
Theorem 6.1.Let d ≥ 2 and let K be a convex body with at most countably many faces.Assume the value function v lies in C 2 (K).Assume also that in each face F of dimension at least two, either v has no critical point, or v has one single critical point which additionally is a maximum.Then for every x ∈ K there is an optimal solution P ∈ P x under which v(X(t)) = v(x) − t for all t < τ K .
Observe that the assumption that v lies in C 2 (K) immediately implies that v| aff(F ) is a classical solution of (1.4) in ri(F ) away from the critical point, for every face F of dimension at least two; just use v itself as test function in the definition of viscosity suband supersolutions.
The proof of Theorem 6.1 proceeds by first constructing solution laws P under which X behaves in the desired manner while inside any given face F of K. Then these laws are pasted together as X reaches ever lower-dimensional faces, until it leaves K. To implement this idea, for any face F of K with dim(F ) ≥ 2 and any point x ∈ ri(F ), we define where τ ri(F ) = inf{t ≥ 0 : X(t) / ∈ ri(F )} is the first time X leaves the relative interior of F .For points x ∈ K c ∪ F 1 , we somewhat arbitrarily set P * x = P x .
Let now the hypotheses of Theorem 6.1 be in force.Our first goal is to prove that P * x is nonempty for every x ∈ R d .This rests on the following construction of a martingale with increments in the kernel of a given location-dependent matrix.
is a martingale for each n, and using the uniform bound on the quadratic variations, we may pass to the limit to deduce that Y Y − Q is also martingale, and hence Q = Y .Furthermore, by Skorohod's representation theorem (see Billingsley (1999, Theorem 6.7)), we may assume that the (Y n , Y n ) and (Y, Y ) are defined on a common probability space (Ω , F , P ) and that, almost surely, We now verify (6.1).We first claim that To prove this, note that On the other hand, the left-hand side converges to t 0 H(Y (s))d Y (s).This yields (6.1) and completes the proof of the lemma.Remark 6.3.An examination of the proof of Lemma 6.2 shows that the process Y is of the form dY t = σ t dW t for some Brownian motion W , where σ t = a(Y t ) 1/2 for all t such that a is continuous at Y t .By properties of the Moore-Penrose inverse, a is continuous except on the boundaries of the sets {x ∈ O : rank H(x) = r}, r = 0, . . ., d − 1 and on ∂O.Thus if Y can be shown to spend zero time in these sets, it is a bona fide weak solution of dY t = a(Y t ) 1/2 dW t .Proposition 6.4.Continue to assume v ∈ C 2 (K).Then P * x is nonempty for every x ∈ R d .
Proof.If x ∈ K c ∪ F 1 , then P * x = P x and the statement is obvious.Below we prove the statement for x ∈ int(K); the case x ∈ ri(F ) for a face F with dim(F ) ≥ 2 is identical since all considerations are then restricted to aff(F ).So suppose x ∈ int(K) and, initially, also that x is not the maximizer of v over K; in particular x is not a critical point in K.
Since v lies in C 2 (K), it is a classical solution of (1.4) in int(K) away from the critical point.As explained in Section 1, an alternative form of this equation at non-critical points is (1.9).That is, where λ min (A, p) denotes the smallest eigenvalue of A corresponding to an eigenvector orthogonal to p, and Let O = {x ∈ int(K) : ∇v(x) = 0} be the set of non-critical points in int(K).Define and arbitrarily set H(x) = 0 for x / ∈ O.It is clear that H is locally bounded measurable and that H| O is continuous.Moreover, the equation ( 6.3) satisfied by v implies that H(x) is singular, i.e. rank H(x) ≤ d − 1, for all x ∈ O.We may thus apply Lemma 6.2 to obtain a martingale Y whose law we denote by P. Clearly P ∈ P x, and due to (6.1) we have where a(t) satisfies X = • 0 a(s)ds and tr(a(t)) = 1, and τ O = inf{t ≥ 0 : X(t) / ∈ O}.As a consequence, omitting the argument X(t) for readability, we have for t < τ O that 0 = ∇v Ha(t) = 1 2 ∇v P ∇v ∇ 2 vP ∇v a(t) + ∇v a(t) = ∇v a(t).
These computations are valid for t < τ O , so we deduce that v(X(t)) = v(x) − t for t < τ O .
In particular, X(t) will not attain a critical point before τ O , so in fact τ O = τ int(K) , the first exit time from int(K).Moreover, at the exit time, we have X(τ int(K) ) ∈ ∂K.This shows that P ∈ P * x , as desired.The case where x is a critical point still remains.In this case, we select points x n ∈ int(K) \ {x} with x n → x, and let P n ∈ P * xn .In particular, the laws Q n = (• − x n ) * P xn lie in P 0 , which is compact by Proposition 2.2(i).The Q n are thus subsequentially convergent toward some Q ∈ P 0 .Along this subsequence, the P n converge to P = (• + x) * Q ∈ P x.Lemma 6.5 below shows that the properties v(X(t)) = v(X(0)) − t for all t < τ int(K) and X(τ int(K) ) ∈ ∂K if τ int(K) < ∞ carry over to weak limits.This shows that P ∈ P * x , and completes the proof of the proposition.
We now turn to the task of pasting solutions together as X reaches ever lower-dimensional faces of K.This uses a measurable selection of laws from P * x , which in turn requires suitable closedness properties of these sets.The following closedness result was already used in the proof of Proposition 6.4.Lemma 6.5.Let F be a face of K and write σ = τ ri(F ) for brevity.Then the set C F = {ω ∈ Ω : v(ω(t)) = v(ω(0)) − t ∀ t < σ(ω), and ω(σ(ω)) ∈ rbd(F ) if σ(ω) < ∞} is closed in Ω.As a consequence, {P ∈ P(Ω) : P(C F ) = 1} is closed in P(Ω).The same conclusion holds if v is only known to be continuous, not necessarily C 2 (K).
The following lemma produces the required measurable selection.This is actually the only step that uses that K has countably many faces.If the lemma could be established without assuming this, the assumption could be dropped from Theorem 6.1 (and Theorem 1.7).In fact, the current proof works for the more general situation where K has countably many faces of dimension two and higher, and arbitrarily many faces of dimension zero and one.Lemma 6.6.Assume K has countably many faces, and continue to assume v ∈ C 2 (K).Them there is a measurable map x → P x from R d to P(Ω) such that P x ∈ P * x for all x.
Proof.We apply the selection theorem of Kuratowski and Ryll-Nardzewski; see Aliprantis and Border (2006, Theorem 18.13).This requires that the set-valued map x → P * x be weakly measurable with nonempty closed values.By Proposition 6.4, P * x is nonempty for all x.For x ∈ K c ∪ F 1 , P * x = P x is closed (even compact) by Proposition 2.2(i).If F is a face of K with dim(F ) ≥ 2 and x ∈ ri(F ), then P * x = P x ∩ {P ∈ P(Ω) : P(C F ) = 1}, which is closed by Lemma 6.5.So P * x is closed for all x.We now argue weak measurability, initially for the map x → P x .We must show that for every open subset U ⊂ P(Ω), the set {x ∈ R d : P x ∩ U = ∅} is measurable; see Aliprantis and Border (2006, Definition 18.1).But since P x = (• + x) * P 0 , the condition P x ∩ U = ∅ means that there exists P ∈ P 0 such that (• + x) * P ∈ U .If this holds for some x ∈ R d , then it also holds for all y in a neighborhood of x since U is open and x → (• + x) * P is continuous.Thus {x ∈ R d : P x ∩ U = ∅} is actually open, and in particular measurable.So x → P x is weakly measurable.Furthermore, the set-valued map x → ϕ(x) specified by ϕ(x) = P(Ω) for x ∈ K c ∪ F 1 and ϕ(x) = {P ∈ P(Ω) : P(C F ) = 1} for x ∈ ri(F ) is constant on K c and on each face of K. Since K has countably many faces, we deduce that x → ϕ(x) is weakly measurable.By Aliprantis and Border (2006, Lemma 18.4(3)), it now follows that x → P * x = P x ∩ ϕ(x) is weakly measurable, as required.
for k = 2, . . ., d, and Thus Y first follows the dynamics of Y d while in the interior of K (a possibly empty time interval); then Y follows the dynamics of Y d−1 while inside the relative interior of a (d−1)dimensional face, and so on, until it reaches a face of dimension zero or one.From that point onwards, it follows a Brownian motion, scaled so that the quadratic variation has unit trace.Since the law of each Y k is chosen from the sets P * x , it is straightforward but somewhat tedious to make this intuitive description rigorous.One also finds that Y is a continuous martingale, starting at Y (0) = x and with tr Y (t) ≡ t, and (using that v| K is continuous) such that v(Y (t)) = v(x) − t for all t < τ K\F 1 = inf{t ≥ 0 : Y (t) / ∈ K \ F 1 }.Moreover, Y does not leave K before reaching F 1 , but then leaves K immediately since its dynamics switches to that of a scaled standard Brownian motion in d ≥ 2 dimensions.In particular, τ K\F 1 = τ K , and we have τ K = v(x) − v(Y (τ K )) = v(x), using also that v = 0 on F 1 .The law P of Y is therefore the required optimal law.

Furthermore
, we can select ε > 0 such that the closure of B ε (x) is contained in O and 2 ϕ S∇ϕ ≥ 0 on B ε (x).
Lemma 5.7.Let K be a convex body.Then F d , F d−1 , andF d−2 are closed.Proof.Both F d = K and F d−1 = ∂K are closed.To see that F d−2 is closed, assume for contradiction that there is a point x ∈ cl(F d−2 ) \ F d−2 .Then x lies in ∂K but not in F d−2, so must lie in the relative interior of a (d − 1)-dimensional boundary face F .But then Lemma 6.2.Let O ⊂ R d be open and let H : R d → S d be a locally bounded measurable map such that H| O is continuous and rank H(x) ≤ d − 1 for all x ∈ O.For every x ∈ O there exists a continuous martingale Y with Y (0) = x and tr Y (t) ≡ t such that t 0 H(Y (s))d Y (s) = 0, t < τ O , (6.1) Proof of Theorem 6.1.For k = 2, . . ., d, define U k = F k \ F k−1 .Equivalently, U k is the (possibly empty) union of the relative interiors of all k-dimensional faces of K. We work on the d-fold product Ω d = C(R + , R d ) d of the canonical path space, and let (W, Y 2 , . . ., Y d ) be the (R d ) d -valued coordinate process.Let x → P x ∈ P *x be the measurable map given by Lemma 6.6; we will use it to specify the law of Y 2 , . . ., Y d .Define random timesτ k−1 = inf{t ≥ 0 : Y k (t) / ∈ U k }.Given x ∈ K, let Y d have law P x.Next, if the law of (Y d , . . ., Y k ) has been specified for k ≥ 3, then specify the law of Y k−1 to be conditionally independent of (Y d , . . ., Y k ) given Y k (τ k−1 ), which is finite almost surely, with law Y k−1 ∼ P Y k (τ k−1 ) .That is, the regular conditional distribution of Y k−1 given Y k (τ k−1 ) = y is P y .This procedure specifies the law of Y 2 , . . ., Y d .Finally, let W have the law of an independent standard d-dimensional Brownian motion.Now, set τ d = 0 and define a process Y by Brownian motion.The law of Y n lies in P x for each n, so Proposition 2.2(i) shows that after passing to a subsequence, Y n ⇒ Y for some limiting process Y whose law again lies in P x.Since Y n (t) = t 0 a n (Y n (s))ds and the a n are uniformly bounded, after passing to a further subsequence we actually have 1, 0, ..., 0)Q , where the diagonal matrix contains r ones.It follows that a(x) 0 and tr(a(x)) = 1 for all x ∈ R d , and H(x)a(x) = 0 for all x ∈ O. Unless H has constant rank on O, a is not continuous on O. Consider therefore mollifications a n (x) = R d ϕ n (x − y)a(y)dy, where ϕ n (x) = n d ϕ(nx) for a positive mollifier ϕ supported on the centered unit ball.Then a n is continuous, positive semidefinite, and has unit trace.Thus there exist weak solutions Y n of the SDEs dY n(t) = a n (Y n (t)) 1/2 dW (t), Y n (0) = x,where the positive semidefinite square root is understood, and W is d-dimensional dy.Since a is bounded and the restriction H| O is continuous, arguing component by component, we see that the right-hand side converges to zero.This proves (6.2).Now pick t < τ O .Then Y (s) ∈ O for all s ≤ t.Since (Y n , Y n ) → (Y, Y ) locally uniformly, the bounded convergence theorem and (6.2) yield that t 0