Correction: Smirnov, S. A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options. Mathematics 2019, 7, 1246

If the compacts $K_t(·)$, $t=0,\dots ,N$ are convex and the payoff functions $g_t$, $t=1,\dots ,N$ are convex, then the Bellman–Isaacs functions $v_t^{*}$, $t=0,\dots ,N$, defined by (1), are also convex.

Let us prove the lemma by induction, using the Bellman–Isaacs Equations (1). For $s=N$, function $v_s^{*}$ is convex because $v_N^{*}(·)=g_N(·)$. Let $v_s$ be convex for $s=N,\dots ,t+1$; now, we show that $v_t$ is also convex. Indeed, the function $({\overline{x}}_t,h)\mapsto v_{t+1}^{*}({\overline{x}}_t,x_t+y)-hy$ is convex for any $y\in K_t(·)$, so the function is convex with respect to $({\overline{x}}_t,h)$. Let us denote Consider ${\overline{x}}_t^1$ and ${\overline{x}}_t^2$ of ${\left({\mathbb{R}}^n\right)}^{t-1}$ and their convex combination ${\overline{x}}_t=q_1{\overline{x}}_t^1+q_2{\overline{x}}_t^2$ with weights $q_1,q_2\ge 0$, $q_1+q_2=1$. For any given $\epsilon >0$, there are $h_1=h_1(\epsilon ,{\overline{x}}_t^1)\in D_t(·)$ and $h_2=h_2(\epsilon ,{\overline{x}}_t^2)\in D_t(·)$ such that ${\phi }_t({\overline{x}}_t^1,h_1)\le {\psi }_t\left({\overline{x}}_t^1\right)+\epsilon$ and ${\phi }_t({\overline{x}}_t^2,h_2)\le {\psi }_t\left({\overline{x}}_t^2\right)+\epsilon$. Due to the convexity of $D_t(·)$, we obtain $q_1{h}_1+q_2{h}_2\in D_t(·)$. The following inequalities can be written: As $\epsilon >0$ is arbitrary, we obtain Since the function $v_t^{*}(·)$ is convex as the maximum of convex functions. □

Let the (deterministic) price dynamics be of Markov type, representable in the multiplicative form: the recurrent relation (R1) applies to the (discounted) price at time t of one unit of risky asset $i\in \{1,\dots ,n\}$where ${\check{C}}_t$ are convex compact sets with non-void interior such that ${\check{C}}_t\subseteq {(0,\infty )}^n$. Suppose that there are no trading constraints, i.e., $D_t(·)\equiv {\mathbb{R}}^n$, and the NDAO condition (no arbitrage opportunities) is satisfied. If the payoff functions on the American option ${\overline{x}}_t\mapsto g_t\left({\overline{x}}_t\right),t=1,\dots ,N$ are convex, then the solutions of Bellman–Isaacs equations ${\overline{x}}_t\mapsto v_t^{*}\left({\overline{x}}_t\right),t=N,\dots ,1$ are also convex.

Note first, that for the considered model the NDAO condition is tantamount to $e\in {\check{C}}_t$, where $e=(1,\dots ,1)\in {\mathbb{R}}^n$. Denote by $\Lambda \left(m\right)$ the diagonal matrix with main diagonal entries equal to $m^1,\dots ,m^n$. Considering $X_t$ as the vector-column of n prices of risky assets, we can rewrite (R1) as follows: Using the representation (R2) of price dynamics, let us prove the assertion by induction. The convexity is immediate for $s=N$. By induction, suppose that $v_s^{*}$ are convex for $s=N,\dots ,t\ge 1$. The convexity of $v_{t-1}^{*}$ follows from the formula (R3), which is a direct consequence of Theorem 2 in [6]: where ${\mathcal{P}}^{*}\left(E\right)$ stands for the class of all the probability measures, concentrated on finite subsets of a (non-void) set E. The convexity is conserved when taking integral (in fact, convex combinations) and supremum in (R3), whence the required result. □

Let the compact-valued mappings$K_t(·)$be continuous, convex-valued mappings, $D_t(·)$be weakly continuous (That is, lower semicontinuous and closed (see the terminology in § 14 in Reference [30])), functions, and  $g_t(·)$ be convex (since the functions $g_t(·)$ are convex everywhere on ${\mathbb{R}}^n$, they are continuous (see, for example, Reference [29], Corollary 10.1.1).), $t=1,\dots ,N$. Suppose that the robust condition of no sure arbitrage with unbounded profit $RNDSAUP$  and one of two following conditions hold:

Let there be no trading constraints; the condition of no sure arbitrage $NDSA$ be fulfilled. Suppose that the functions$g_t(·)$are convex and$K_t(·)$are convex polyhedra, $t=1,\dots ,N$.

Assume that the functions $v_t^{*}(·)t=0,\dots ,N$ are convex everywhere on ${\left({\mathbb{R}}^n\right)}^t$. Let there be no trading constraints; the condition of no sure arbitrage $NDSA$ is fulfilled. Suppose that the functions $g_t(·)$ are convex and $K_t(·)$ are convex polyhedra, $t=1,\dots ,N$.