A multidimensional singular stochastic control problem on a finite time horizon

A singular stochastic control problem in n dimensions with timedependent coefficients on a finite time horizon is considered. We show that the value function for this problem is a generalized solution of the corresponding HJB equation with locally bounded second derivatives with respect to the space variables and the first derivative with respect to time. Moreover, we prove that an optimal control exists and is unique. Singular stochastic control is a class of problems in which one is allowed to change the drift of a Markov process (usually a diffusion) at a price proportional to the variation of the control used. Admissible controls do not have to be absolutely continuous with respect to the Lebesgue measure and they may have jumps. This setup is natural for many problems of practical interest, including portfolio selection in finance, control of queueing networks and spacecraft control, to mention just a few examples. The reader is referred to Chapter VIII of [5] for more information and basic references. One-dimensional singular stochastic control problems are well understood by now, see, e.g., [2] and the references given there. In this case, if the running cost is convex, the optimal control makes the underlying process a reflected diffusion at the boundary of the so-called nonaction region C. In the case of a diffusion with time-independent coefficients and discounted cost on the infinite time horizon, C is just an interval and the value function enjoys 2010 Mathematics Subject Classification. Primary: 93E20; Secondary: 35Q93.

Singular stochastic control is a class of problems in which one is allowed to change the drift of a Markov process (usually a diffusion) at a price proportional to the variation of the control used.Admissible controls do not have to be absolutely continuous with respect to the Lebesgue measure and they may have jumps.This setup is natural for many problems of practical interest, including portfolio selection in finance, control of queueing networks and spacecraft control, to mention just a few examples.The reader is referred to Chapter VIII of [5] for more information and basic references.
One-dimensional singular stochastic control problems are well understood by now, see, e.g., [2] and the references given there.In this case, if the running cost is convex, the optimal control makes the underlying process a reflected diffusion at the boundary of the so-called nonaction region C.In the case of a diffusion with time-independent coefficients and discounted cost on the infinite time horizon, C is just an interval and the value function enjoys C 2 -regularity (smooth fit).Both C 2 -regularity of the value function and the characterization of the optimally controlled process have been extended to the case of singular control for the two-dimensional Brownian motion [14].In n ≥ 3 dimensions, except for "close to one-dimensional" cases of a single push direction [15,16] and the radially symmetric running cost [9], only partial results are known.For example, for optimal control of the Brownian motion on the infinite time horizon, regularity of the boundary of C away from some "corner points" was shown in [17] and a characterization of the optimal control as a solution of the corresponding modified Skorokhod problem was given in [8].
In this paper we consider a n-dimensional singular stochastic control problem on a finite time horizon in which state is governed by a linear stochastic differential equation with time-dependent coefficients, the running cost is convex and controls may act in any direction.We provide estimates for the corresponding value function.These estimates imply that the value function has locally bounded generalized derivatives of the second order with respect to the space variable and of the first order with respect to the time variable.These properties are needed to consider the value function as a solution of the corresponding parabolic Hamilton-Jacobi-Bellman (HJB) equation in some generalized sense and to show existence and uniqueness of an optimal control.
Similar results have been shown in Theorem 2.1 and Theorem 3.4 from [2] in the one-dimensional case with a single push direction.The corresponding results for a multidimensional singular stochastic control problem on the infinite time horizon with time-independent drift, covariance, cost (i.e., for the elliptic case) can be found in [11].Our article contains a generalization (or adjustment) of the approach of [2,11] to an n-dimensional parabolic problem.It turns out that while the main ideas from those papers may be applied in our case, a mathematically rigorous analysis of our problem is somewhat delicate and needs rather careful arguments.
Our motivation for pursuing this project is the hope that the results given here will allow for a characterization of the optimal policy in the parabolic case as a solution to the corresponding Skorokhod problem for a domain with time-dependent (moving) boundary, which would be an analog of the main theorem from [8].Indeed, the analysis of [8] used the results from [11] as the starting point, so it is plausible that their analogs will be useful in proving the corresponding result on a finite time horizon.Such a characterization would address a long-standing open problem on the structure of the optimal control in the case under consideration.We hope to address this issue in a subsequent paper.
Existence results for multidimensional singular control problems closely related to our work may be found in [1,3,6].Apparently, in spite of their considerable generality, none of them contains our existence result as a special case.Indeed, in these papers optimal weak solutions to the corresponding SDEs are constructed, while we are concerned about finding an optimal strong solution, i.e., for the given (as opposed to some) filtration and underlying Brownian motion.Moreover, the problem considered in [1] is elliptic and the allowable control directions lie in a cone, the opening of which cannot be too wide.In [3,6] the time horizon is finite, but the problem considered in [3] has the final cost instead of the running cost, while in [6] the drift of the controlled diffusion is bounded, which excludes its linear dependence on the state.
The structure of this paper is as follows.In Section 1 we pose the singular stochastic control problem, give definitions and prove lemmas needed in further considerations.In Section 2 we prove estimates for the value function.In Section 3 we consider the Bellman's dynamic programming principle (DPP) and the HJB equation related to this problem.Section 4 contains proofs of existence and uniqueness of an optimal control.1. Notation, assumptions and lemmas.Let M n×n denote the set of matrices of dimension n × n with the operator norm, i.e. ||A|| = sup{|Ax| : x ∈ R n , |x| = 1}.Let T > 0 be a fixed number representing our time horizon.For a function u = u(x, t) : R n × [0, T ] → R we denote the gradient and the Hessian of u with respect to the space variables (i.e., x i ) by Du and D 2 u, respectively.
Let (W t , t ≥ 0) be a standard n-dimensional Brownian motion defined on a complete probability space (Ω, F, P ).Let (F t , t ≥ 0) be the augmentation of the filtration generated by W (see [7], p. 89).Denote by V the set of controls v which are left-continuous, adapted to the filtration (F t , t ≥ 0) random processes acting from [0, T ] into R n , with P -a.s.bounded variation and s.t.v(0) = 0 P -a.s.We note that these processes are also progressively measurable (see [7], Th. 1.1.13).As it is customary in singular stochastic control theory (see, e.g., [8]), we write v(t) = t 0 γ(s)dξ(s), where |γ(t)| = 1 for every t ∈ [0, T ] and ξ is nondecreasing and left-continuous.In other words, ξ(t) is the total variation of v on the time interval [0, t] and γ(t) is the Radon-Nikodym derivative of the vector-valued measure induced by v on [0, T ] with respect to its total variation ξ.
Consider the state process described by the stochastic integral equation and a, σ : [0, T ] → M n×n stand for the drift and the covariance terms.Note that (y xt (s)) s∈[t,T ] is a random process adapted to To each control v ∈ V, we associate a cost given by the payoff functional where f, α and c are respectively the running cost, the discount factor and the instantaneous cost per unit of "fuel".
Our purpose is to characterize the optimal cost, the so-called value function It is often convenient to consider the following penalized problem associated with (3): where > 0 and V is the set of all controls v ∈ V which are Lipschitz continuous and | dv dt (t)| ≤ 1 for almost every t ∈ [0, T ] almost surely.Definition 1.1.We say that the finite time horizon stochastic control problem has the dynamic programming property in the weak sense if for every Let us assume the following: such that for all t, t ∈ [0, T ], x, x ∈ R n and λ ∈ (0, 1) we have The last assumption implies strict convexity of the function f with respect to x.
Let us denote by c max and α max the maximum of the function c, α, respectively.Moreover, by a max , σ max , β max and b max we denote the maximum over t ∈ [0, T ] of the norms of the matrices a(t), σ(t), β(t) and the vector b(t) respectively, where β(t) = σ(t)σ T (t).Now we give lemmas needed for the proofs of the Theorems 2.1 and 2.2.The first one is well known.Lemma 1.2.For all x, y ≥ 0 we have Lemma 1.3 (See [10], Corollary 2.5.12).Consider an n-dimensional process described by a stochastic integral equation We assume that there exists a constant C such that for all x ∈ R n and t ≥ 0 Then for every q > 0 there exists a constant C 11 > 0 depending only on q, C such that for all t ≥ 0 Remark 1.4.For the process y xt defined by (1) with v ≡ 0 the assumption (10) holds.Indeed, σ is Lipschitz continuous, independent of x and defined on a finite time interval [0, T ], so it is bounded.We conclude the same about a, b, so |g(x, where Proof.In view of (1) we have Taking the derivative d/ds of both sides, we get the differential equation Using the Gronwall's inequality (see [4], p. 625), we get the second part of (12).
Lemma 1.6.Suppose that for some x ∈ R n , t ∈ [0, T ], v ∈ V we have for a suitable constant C > 0 independent of x, t.Then Proof.Indeed, multiplying both sides of our assumption by e T t α(r)dr , we get Of course, the left-hand side is not smaller than E T t f (y xt (s), s)ds.Lemma 1.7 (Compare a statement in [17], p. 181).The function J xt (v) is convex with respect to (x, v), more precisely, for all Proof.First, we note that the set V is obviously convex.Let y v xt (s) be the solution of (1) corresponding to a control v. Denote v 0 = θv 1 + (1 − θ)v 2 and x 0 = θx 1 + (1 − θ)x 2 .In view of the definition of J xt (v), it suffices to prove two following inequalities (14) f y v 0 x 0 ,t (s), s ≤ θf (y where ξ 0 , ξ 1 , ξ 2 are the total variations of v 0 , v 1 , v 2 respectively.The latter inequality is a consequence of the fact that the variation of the sum of functions is not greater than the sum of their variations.So ξ 0 ≤ θξ 1 + (1 − θ)ξ 2 .Because ξ 0 , ξ 1 , ξ 2 are nondecreasing and ξ 0 (0) = ξ 1 (0) = ξ 2 (0) = 0 P -a.s., we conclude that (15) is true.
Lemma 1.8.Suppose that for some t for a suitable constant C > 0 independent of x, t .Then there exists a constant C 17 > 0 independent of x, t such that Proof.Indeed, multiplying both sides of our assumption by e T −t 0 α(t +r)dr and using the lower bound of c, we get Proof.We observe that using (8) we have Hence, in view of Lemma 1.6 and Lemma 1.9, we get where for a suitable constant C > 0 independent of x, t , t.Then there exists a constant C 20 > 0 independent of x, t , t such that for all s ∈ [0, T − t] we have we have where Recall that a, b, σ are Lipschitz continuous with the constant L. The process M s is a martingale with quadratic variation This, together with the Burkholder-Davis-Gundy inequalities (see [7], Theorem 3.3.28),implies the existence of a constant C p , depending only on p, such that Clearly, (23) By the Hölder's inequality, for q = p/(p − 1) we have By Lemma 1.9, the inequality (18) holds for every t ∈ [0, T ].Lemma 1.2 and the relations ( 18), ( 21)-(25) imply that the random variable is integrable and hence, by the Lebesgue dominated convergence theorem, the function From Lemma 1.2 and ( 18), ( 21)-(25) we also have, for each s ∈ [0, This, together with the Gronwall's inequality (see, e.g., [7], Problem 5.2.7), implies that for all s ∈ [0, T − t], We have obtained (20) with for a suitable constant C > 0 independent of x, t.Then there exists a constant C 26 > 0 independent of x, x , t such that for every Proof.From ( 6) and Lemma 1.2 we have Now using Lemma 1.5, Lemma 1.9 and Lemma 1.2 again, we get where Lemma 1.13.Suppose that for some for a suitable constant C > 0 independent of x, t .Then there exists a constant C 27 > 0 independent of x, t , t such that for every t Proof.Using ( 6) and Lemma 1.2, we have In view of Lemma 1.9, the Fubini's theorem and Lemma 1.11, we get where The next two definitions and lemma refer to mollification of a given function (see [4], p. 629-630).
where the constant C 28 is selected so that

Estimates for the value function.
Let the assumptions from Section 1 appearing immediately after Definition 1.1 hold.
Theorem 2.1.Let u be the value function defined by (3).Then for some positive constants C 29 , C 30 , C 31 , the same p > 1 as in the assumptions (6)-( 9) and every t ∈ [0, T ], x, x ∈ R n and λ ∈ (0, 1), the following estimates hold: Proof: Proof of (29).Nonnegativity of u is the consequence of nonnegativity of f and c.Next, taking the control v ≡ 0 and using (6), the Fubini's theorem, Lemma 1.3 and Lemma 1.2, we get where C 29 depends only on C 0 , T, C 11 , p, so (29) is proved.
Proof of (30).Now we note that Applying (7), we can estimate the last expression from above by Using Lemma 1.5, we have We use the Hölder's inequality with exponent p p−1 to estimate the last expression above by By virtue of (29) we can consider only those controls v for which for some arbitrary > 0. From (32), Lemma 1.6 and Lemma 1.12 we see that where In an analogous manner we get the same estimate for u(x, t)−u(x+x , t).
Proof of (31).We observe that In view of ( 12) we can apply (9) to get we use the Hölder inequality with exponent p p−2 to get u(x + λx , t) By virtue of (29) we can consider only those controls v for which for some arbitrary > 0. From Lemma 1.6 and Lemma 1.2 we see that where C 31 = T 2/p C 0 (T + C 13 ) 1−2/p .We have proved the upper bound of (31).
To prove the lower bound of (31), it clearly suffices to prove convexity of u(x, t) with respect to the first variable.In view of the definition of u we know that for every > 0, Using Lemma 1.7, we get Because > 0 is arbitrary, we get convexity of u(x, t) with respect to the first variable.
Theorem 2.2.Let the assumptions of Theorem 2.1 be satisfied.Assume that the dynamic programming property in the weak sense holds (Definition 1.1).Then for some constant C 33 > 0 and every t, t ∈ [0, T ], x ∈ R n , we have Proof.We note that Let us denote the expectations of the first two integrals in the last expression by A and B, respectively.Because the last two integrals are nonnegative we get We can estimate B as follows: B ≤ E T −t 0 c(t + s)e − s 0 α(t+r)dr − c(t + s)e − s 0 α(t +r)dr dξ(s) .
Adding and subtracting c(t + s)e − s 0 α(t +r)dr under the absolute value sign and using the triangle inequality and positivity of α, we get Because |e x − e y | ≤ |x − y| for x, y ≤ 0 and c, α are Lipschitz continuous, we have By virtue of (29) we can consider only those controls v for which for some arbitrary > 0. Using Lemma 1.
Using the inequality |e x − e y | ≤ |x − y| for x, y ≤ 0 again, we get (36) By virtue of (29) we can consider only those controls v for which (37) E for some arbitrary > 0. Using (36) and Lemma 1.13, we get (38) , where C 38 = T LC 27 .
To estimate A 2 we use ( 7)-( 8) and we have that A 2 is less than or equal to Using the Hölder's inequality and the Fubini's theorem, we get From this together with (37), Lemma 1.13, Lemma 1.10 and Lemma 1.11 we have Hence, from Lemma 1.2, (39) where Furthermore, from Lemma 1.9 we get (40) In view of (34)-( 35) and ( 38)-(40) we get for t ≤ t, where To obtain a similar inequality for t < t we proceed as follows.Let y 0 xt (s) s∈[t,T ] be a solution of (1) with v ≡ 0. We can write the i-th coordinate of y 0 xt (s) as follows We need the following lemma.where C 47 = n 2 C 45 (a max + b max )2 p−1 .Now we show that B = 0. Indeed, From properties of the Itô's integrals (see [7], Section 3.2) the process Using Lemma 2.3 and Lemma 1.2, we have Using the Fubini's theorem and Lemma 1.3, we get Hence Z ij is a martingale and Z ij (t ) = 0. Now, using the conventional "multiplication rules" (see [7], p. 154), we know that dsds = 0, dsdW i s = 0, dW i s dW i s = ds, dW i s dW j s = 0 for i = j.
So in view of (42) we can write Hence, for t < t It is clear that (41) and ( 52) imply (33).
Now we give the proof of Lemma 2.3.
Proof: Proof of (44).The continuity of u(•, t ) is a consequence of (30).So in view of Lemma 1.16 we conclude that lim m→∞ u m (x) = u(x, t ).
Proof of (46).Let x ∈ R n and λ ∈ (0, 1).We have From (31) and nonnegativity of η m we conclude that ∂ 2 um(x) ∂x i ∂x j ≥ 0. On the other hand, using (31) and mimicking the proof of (45), we see that Thus, for all p > 1 we have Taking the limit as λ → 0 , we can conclude (46).Proof.Let x ∈ R n be arbitrary.Consider controls for which lim s→0+ v s = x.In view of ( 2) and (3) we have

Dynamic Programming Principle and HJB equation.
To consider the DPP and the HJB equation for our problem we will first prove the pointwise convergence of u to u if → 0 + .For this purpose we need an integral form of the Gronwall's inequality with locally finite measures.
Lemma 3.1 (see [18]).Let µ be a locally finite measure on the Borel σalgebra of [t, T ], where 0 ≤ t ≤ T .We consider a measurable function φ defined on [t, T ] such that T t |φ(r)|µ(dr) < ∞.We assume that there exists a Borel function ψ ≥ 0 on [t, T ] such that for all s ∈ [t, T ], Step 1.We show first that v ∈ L p (Ω × [0, T − t], P ⊗ µ Leb ), where µ Leb denotes the Lebesgue's measure.Since J xt (v) < ∞, we have Using (54) and properties of the normal distribution, we know that each term from the line above, maybe except for the last one, belongs to the space L p (Ω × [0, T − t]).But the last term belongs to this space, too.Indeed, Using the Hölder's inequality and (54), we can estimate the last expression above by where 1 p + 1 q = 1.Hence, from (55) we see that v ∈ L p (Ω × [0, T − t]).
Step 2. Now we define a sequence of bounded controls {v R , R > 0} such that v R is convergent to v in the space L p (Ω × [0, T − t]) and the total variation of v R is pointwise convergent to the total variation of v from below.Let Hence, from Lemma 1.2 and Step 1, and using the Lebesgue's dominated convergence theorem, we get The convergence in L p is proved.Moreover, if ξ(s), ξ R (s) denote the total variations on the interval [0, s] of the functions v, v R respectively, then for all s ∈ [0, T − t], Step 3. Let y v xt , y v R xt denote the state processes (see ( 1)) corresponding to the controls v, v R respectively.We want to show that {y v R xt } is convergent to y v xt in the space L p (Ω × [t, T ]).First we observe that for s ∈ [t, T ], where C 57 = e amaxT .So from Lemma 1.2 and the Hölder's inequality where 1 p + 1 q = 1 and C 58 = 2 p−1 (1 + C p 57 T p/q ).Finally, in view of Step 2 we have In view of (56), Using (56) again and the assumption that Hence, from the Lebesgue's dominated convergence theorem we get Using (7) and the Hölder's inequality, we have In view of Step 3, the second factor in the last expression goes to 0 if R → ∞.We must show that the first factor is bounded.Indeed, from (6) and Lemma 1.2 we can write Using (54) and Step 3 again, we conclude that the last expression is bounded uniformly in R. Hence , so we can consider only bounded controls.
Step 5. Consider v ∈ V such that ||v|| ∞ < R for some R > 0. We will construct a sequence of controls {v n , n ∈ N} convergent to v in L p (Ω × [0, T − t]) and such that v n ∈ V 1/(2nR) for all n.Besides we shall prove that the variation of v n is pointwise convergent to the variation of v from below.
Theorem 3.5 (The HJB equation).The value function u satisfies almost everywhere (a.e.) the following second-order differential equation: Proof.An application of the DPP for regular stochastic control problems yields for > 0 the following equation (see [5], Chapter IV.3): (66) In view of Theorems 2.1, 2.2, Remark 2.4, Theorem 3.2, Corollary 3.4 and the Arzela-Ascoli's theorem ( [7], Th. 2.4.9)we see that u → u uniformly on every compact set if → 0 + .Fix t ∈ [0, T ].From (31) and Remark 2.4 we see that D 2 u (•, t) are locally uniformly bounded for all > 0 in their domains, so using the Arzela-Ascoli's theorem from every sequence { m } m∈N convergent to 0, we can choose a subsequence {˜ m } m∈N such that But v must be equal to Du(•, t) in the distribution sense.Indeed, for any function φ ∈ C ∞ c (R n ) and any k = 1, . . ., n we have Letting m → ∞, we get From the Banach-Alaoglu theorem ( [12], Th. 3.15) we know that balls in the space L 2 are weakly compact.So for each sequence { m } m∈N convergent to 0, there exists a subsequence {˜ m } m∈N such that Au ˜ m v in L 2 ψ if m → ∞.We will show that v = Au in the distribution sense.Indeed, for any function φ belonging to the class C ∞ c (R n × [0, T ]), we have Letting m → ∞, we get which, together with (72) implies that

4.
Existence and uniqueness of the optimal control.The results of this section are analogous to Theorems 7 and 8 from [11].Fix (t, x) ∈ [0, T ) × R n (for t = T the only admissible control is v(0) = 0 a.s.).Let m t be the measure on [t, T ] × Ω, B([t, T ]) ⊗ F equal to the product of the Lebesgue's measure and P .Remark 4.1.If a process X is a modification of a process Y and both processes have left-continuous sample paths a.s., then the processes X, Y are indistinguishable (compare Problem 1.1.5,[7]).
Theorem 4.2.The optimal control v * ∈ V, if it exists, is unique up to the indistinguishability.

Proof. Suppose there
s) on some m t -nonzero set.This fact together with (75) and the definition of J xt imply that the inequality (74) is strict, so we get a contradiction.We conclude that v 1 , v 2 must be indistinguishable.
Proof.By the Hölder's inequality, the function g(s, ω) = s t a(r)z n (r, ω)dr satisfies so T is a bounded operator from L p (m t ) into L p (m t ).

Theorem 4.4.
There exists an optimal control v * ∈ V.

Proof
xt goes to 0 in L p (m t ) as k, m → ∞.Using Lemma 4.3 we conclude that {η k (• − t, •)} k∈N is a Cauchy sequence in L p (m t ) so it is convergent to a process v ∈ L p (m t ).Without loss of generality we may assume that v(0) ≡ 0.
Then P ( Ω) = 1 and µ Leb (S) = T − t.Let N be a countable subset of S, dense in [0, T −t], including 0 and let A N = s∈N A s .We have P (A N ) = 1.
Let ξ k (s) denote the total variation of η k on the interval [0, s].Because J xt (η k ) are uniformly bounded in k, there exists a constant C > 0 such that Eξ k (T − t) ≤ C for all k ∈ N. In view of the Fatou's lemma, E lim inf k→∞ ξ k (T − t) ≤ lim inf k→∞ Eξ k (T − t) ≤ C, so lim inf k→∞ ξ k (T − t) is finite a.s.
For ω ∈ A N ∩ Ω, the set A ω ∩ {s ∈ [0, T − t] : v(s, ω) = v * (s, ω)} is countable, so its Lebesgue's measure is equal to 0. Therefore v = v * m t -a.e.In particular, η k → v * in L p (m t ).Proceeding as in This together with the Fatou's lemma and the fact that P (A N ∩ Ω) = 1 yields Letting ||Π|| → 0, t m ↑ T − t so that each partition in the sequence is contained in the next one, by the monotone convergence theorem, we get (77).From (76) and (77) On the other hand, J xt (v * ) ≥ u(x, t) because v * ∈ V and hence J xt (v * ) = u(x, t) so v * is an optimal control.
[13] the proof of Theorem 3.5, to follow, for a similar argument, with u m replaced by u m ).This implies differentiability of u with respect to x in the classical sense (see, e.g., Theorem 7.17 in[13]), so Du is the classical gradient of u with respect to x at any point (x, t ) ∈ R n × [0, T ].Moreover, by (46) Du m are locally Lipschitz in x uniformly in m, so Du is also locally Lipschitz in x.Thus Theorems 2.1, 2.2 and their proofs imply that the value function u(x, t) has generalized derivatives of the first order with respect to t and of the second order with respect to x.
Remark 2.4.Theorems 2.1 and 2.2 are true for functions u (see (4)) instead of u.Indeed, in view of the proofs we see that the constants C 29 , C 30 , C 31 , C 33 do not depend on .Remark 2.5.It follows from (44)-(46) that for every t ∈ [0, T ] Du m (• ; t ) converges to Du(•, t ) the distributional gradient of u with respect to x almost uniformly as m → ∞ These generalized derivatives belongs to the space L ∞ loc (R n × [0, T ]) of all functions essentially bounded on every open bounded subset of the domain.Proposition 2.6.For all x ∈ R n and t ∈ [0, T ] we have u(x, t) ≤ (c max + C 29 )(1 + |x|).
Remark 2.7.The proof of Proposition 2.6 is not valid for u instead of u, because if a control v ∈ V , then it is continuous, so the conditionlim s→0+ v s = x is invalid for x = 0. Remark 2.8.The value function u(x, t) satisfies |Du(x, t)| ≤ c(t) for all (x, t) ∈ R n × [0, T ].Indeed, the gradient exists for all (x, t) ∈ R n × [0, T ] in view of Remark 2.5.From (53) we see that the first derivative of u(x, t) with respect to x in any direction is bounded by c(t).Hence, the norm of the gradient Du(x, t) is bounded by c(t), too.
The dynamic programming property in the weak sense holds (see Definition 1.1) and hence the value function satisfies (33).