# Convex ordering of solutions to one-dimensional SDEs

Benjamin Jourdain 33footnotemark: 3 CERMICS, Ecole des Ponts, INRIA, Marne-la-
Vallée, France. E-mail<EMAIL_ADDRESS>Gilles Pagès Laboratoire de
Probabilités, Statistique et Modélisation, UMR 8001, Campus Pierre et Marie
Curie, Sorbonne Université case 158, 4, pl. Jussieu, F-75252 Paris Cedex 5,
France. E-mail<EMAIL_ADDRESS>research benefited
from the support of the “Chaire Risques Financiers”, Fondation du Risque

###### Abstract

In this paper, we are interested in the propagation of convexity by the strong
solution to a one-dimensional Brownian stochastic differential equation with
coefficients Lipschitz in the spatial variable uniformly in the time variable
and in the convex ordering between the solutions of two such equations. We
prove that while these properties hold without further assumptions for convex
functions of the processes at one instant only, an assumption almost amounting
to spatial convexity of the diffusion coefficient is needed for the extension
to convex functions at two instants. Under this spatial convexity of the
diffusion coefficients, the two properties even hold for convex functionals of
the whole path. For directionally convex functionals, the spatial convexity of
the diffusion coefficient is no longer needed. Our method of proof consists in
first establishing the results for time discretization schemes of Euler type
and then transferring them to their limiting Brownian diffusions. We thus
exhibit approximations which avoid convexity arbitrages by preserving
convexity propagation and comparison and can be computed by Monte Carlo
simulation.

##  1 Introduction

In this paper, we carry on in-depth investigations about various aspects of
propagation of convexity (and monotone convexity) by the flow of a one-
dimensional Brownian diffusion

$dX_{t}=b(t,X_{t})dt+\sigma(t,X_{t})dW_{t}$

where $b,\sigma:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ and $(W_{t})_{t\geq
0}$ is a one-dimensional Brownian motion as well as the comparison results
with respect to marginal or functional convex orderings between two such
diffusions. Studies on convex ordering(s) for stochastic processes go back to
more than fifty years with a seminal work by Hajek in [10] (for increasing
convex ordering), followed in the late 1990’s by El Karoui, Jeanblanc and
Shreve in [9], Martini in [16] and then by various authors, among them Schied
and Stadje in [23], Bergenthum and Rüschendorf in a series of papers [4, 5, 7]
(on which we will comment further in Remark 4.10) or, more recently, [19] (for
various stochastic dynamics), [15, 14] (for McKean-Vlasov processes) or [11]
(for Volterra processes). Except for [10], most of these papers were motivated
by financial applications to the pricing and hedging of derivative products in
a complete market where the underlying traded asset price is supposed to be
driven by/obey a local volatility model.

In Section 2, we state a theorem (Theorem 2.3) which combines the state of the
art about convexity propagation and functional convex ordering(s) for strong
solutions to $1$-dimensional Brownian SDEs combined with recent improvements
developed while tackling similar problems for other classes of stochastic
processes. This statement assumes convexity of at least one of the two
diffusion coefficients in the spatial variable. The objective of this paper is
to generalize this theorem essentially in two directions. The first one is to
analyze the role played in dimension one by the convexity assumption made on
the diffusion coefficient in the propagation of convexity and subsequently in
the comparison property. The second one is to extend the class of functionals
for which such type of propagation/comparison can be established (see further
on).

Our first objective is motivated by a result from [9]. In this paper, without
assuming spatial convexity of the diffusion coefficient $\sigma$, the authors
establish the propagation of convexity property (and a somewhat hidden
comparison result through their proof of the tracking error property) for
$1$-marginals of the diffusion – i.e. at the fixed time $T$ – in the sense
that for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$

$x\longmapsto{\mathbb{E}}\,f(X^{x}_{{}_{T}})\quad\mbox{ is convex},$

where $(X^{x}_{t})_{t\in[0,T]}$ denotes the solution to the above stochastic
differential equation started from $X_{0}=x$. In Section 3.1, we revisit this
result in Theorem 3.2 under weaker regularity assumptions on the coefficients
and establish the convexity of $x\mapsto{\mathbb{E}}\,f(X^{x}_{{}_{T}})$ under
standard Lipschitz assumptions only. We prove accordingly under the same
assumptions that the convex ordering property for the marginal distributions
at time $T$ holds as well.

To elucidate this paradox about the spatial convexity assumption needed on the
diffusion coefficient, we were led to introduce the notion of $m$-marginal
convex ordering that is comparison in convex order of the $m$-dimensional
marginals of the diffusions. By contrast with the purely marginal case $m=1$,
we state in Section 3.2 that, as soon as convexity propagation holds for
convex functions of the diffusion at $m$ instants with $m\geq 2$, then
$x\mapsto{\mathbb{E}}\,\sigma(t,X^{x}_{t})$ is necessarily convex for every
$t\\!\in[0,T]$. In particular, in the autonomous case $\sigma(t,x)=\sigma(x)$,
this implies that the function $\sigma$ itself is convex. No such spatial
convexity is assumed by Schied and Stadje [23] to extend the results in [9]
(convexity propagation and somewhat hidden comparison through robustness of
the hedging strategy) to payoffs depending on $m$ instants which are
directionally convex. There is an overlap between the classes of directionally
convex and convex functions but none contains the other, except in dimension
one where they coincide. In our second main result (Theorem 3.10 in Section
3.3), we check that propagation and comparison hold for directionally convex
functionals.

On our way through the paper, we always provide results for both “regular” and
increasing convex propagation and ordering, thus extending the seminal theorem
by Hajek [10].

Our method of proof for all these results always relies on the same paradigm:
we first establish our results for time discretization schemes of Euler type
and then transfer them using appropriate convergence theorems of these schemes
to their respective Brownian diffusions. Like in [14], but in a much more
general and demanding case, we will need to consider Euler schemes with
truncated noise to obtain convexity propagation in discrete time. The
truncation is chosen in such a way that the strong $L^{1}$-convergence to the
limiting diffusion is preserved. Schemes involving time integrals of the
coefficients converge without continuity of these coefficients in the time
variable. Standard schemes avoiding such integrals are also considered under
more stringent continuity assumptions. This is important in practice,
especially in Finance when dealing with derivative products (European
options), since they provide approximations which preserve convexity
propagation and comparison along convex functionals thus avoiding convexity
arbitrages and can be computed by Monte Carlo simulation. Also note that the
results obtained in discrete time have their own interest for applications.

The paper is organized as follows. Section 2 is devoted to convexity
propagation for convex functionals and functional convexity comparison. In
Section 3, we make a few definitions precise and state the main results of the
paper. In Theorem 3.2 devoted to the $1$-marginal case, we show that spatial
convexity of the diffusion coefficients can be relaxed. Then, in Proposition
3.3, we check that as soon as the $2$-marginals functions come in the game,
the assumptions to be made on the coefficients of the diffusions are nearly
the same as those needed for general functionals of the whole paths of the
processes. The third part of this section is devoted to the class of
directionally convex functionals which is not a subset of convex functionals
(except in the $1$-marginal case) and for which we establish a theorem similar
to Theorem 2.3 without assuming spatial convexity of the diffusion
coefficients. Section 4 is devoted to several variants of truncated Euler
schemes. They turn out to be the main tools used in our proofs since we first
establish our results in discrete time and then transfer them to the
continuous time setting by appropriate limit theorems. Section 5 is devoted to
the proofs of the main results stated in Section 3.

#### Notation

  * •

We denote by ${\cal L}(X)$ the law of a random vector or a stochastic process
$X$.

  * •

For $p\in[1,+\infty]$ (resp. $p\in(0,1)$), $\|X\|_{p}$ denotes the
$L^{p}({\mathbb{P}})$-norm (resp. pseudo-norm) of the random vector $X$
defined on $(\Omega,{\cal A},{\mathbb{P}})$.

  * •

${\cal C}([0,T],{\mathbb{R}})$ denotes the space of continuous functions from
$[0,T]$ to ${\mathbb{R}}$.

  * •

We denote the sup-norm of a function $f\in{\cal C}([0,T],{\mathbb{R}})$ by
$\|f\|_{\infty}=\sup_{t\in[0,T]}|f(t)|\leq+\infty$.

##  2 Functional convex ordering of one-dimensional diffusions

Before being more specific on the state of the art functional convex ordering
for one dimensional Brownian diffusions, let us briefly recall some
definitions about convex ordering of random vectors (in view of the proofs)
and stochastic processes (which are main objects of interest in our results).

######  Definition 2.1 (Convex orderings)

Let $U$, $V\\!\in L^{1}({\mathbb{P}})$ be two ${\mathbb{R}}^{m}$-valued random
vectors and let $X=(X_{t})_{t\in[0,T]}$ and $Y=(Y_{t})_{t\in[0,T]}$ be
pathwise continuous real-valued stochastic processes such that
${\mathbb{E}}\,\sup_{t\in[0,T]}|X_{t}|$ and
${\mathbb{E}}\,\sup_{t\in[0,T]}|Y_{t}|<+\infty$.

$(a)$ Convex ordering. We say that $U$ is dominated for the convex ordering by
$V$, denoted

$U\preceq_{cvx}V,$

if, for every convex function $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$,

${\mathbb{E}}\,f(U)\leq{\mathbb{E}}\,f(V)\in{\mathbb{R}}\cup\\{+\infty\\}.$
(2.1)

$(b)$ Monotone convex ordering. We say that $U$ is dominated for the non-
decreasing convex ordering by $V$, denoted

$U\preceq_{icv}V,$

if (2.1) holds for every convex function $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$
non-decreasing in each of its variables.

$(c)$ Functional convex ordering. We say that $X$ is dominated by $Y$ for the
functional convex ordering, denoted

$X\preceq_{cvx}Y$

if for every lower semi-continuous (l.s.c.) convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$

${\mathbb{E}}\,F(X)\leq{\mathbb{E}}\,F(Y)\\!\in{\mathbb{R}}\cup\\{+\infty\\}.$
(2.2)

$(d)$ Functional monotone convex ordering.We say that $X$ is dominated by $Y$
(defined as above) for the increasing functional convex ordering, denoted

$X\preceq_{icvx}Y,$

if (2.2) holds true for every l.s.c. convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$, non-decreasing for the natural partial
order on ${\cal C}([0,T],{\mathbb{R}})$ (i.e. for every $(x,y)\\!\in{\cal
C}([0,T],{\mathbb{R}})\times{\cal C}([0,T],{\mathbb{R}}_{+})$, $F(x)\leq
F(x+y)$).

Obviously, $U\preceq_{cvx}V\Longrightarrow U\preceq_{icvx}V$ and
$X\preceq_{cvx}Y\Longrightarrow X\preceq_{icvx}Y$.

These definitions only involve the distributions of $U$, $V$ and $X$, $Y$
respectively. So we could have defined equivalently these orderings on the set
of ${\cal P}_{1}({\mathbb{R}}^{m})$ and ${\cal
P}_{1}\big{(}C([0,T],{\mathbb{R}})\big{)}$ of distributions with finite first
moment.

Note that the expectations involved in the definitions make sense in
${\mathbb{R}}\cup\\{+\infty\\}$ since any convex function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ (resp. l.s.c. convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$) is bounded from below by an affine
function (resp. by a continuous affine functional according to Lemma 7.5 [1]).
The next proposition which can be proved by inf-convolution arguments (see the
proof of [11, Lemma 1.9]) states a characterization of convex ordering(s)
based on the sole Lipschitz convex functions/functionals.

######  Proposition 2.2 (Convex ordering characterization)

$(a)$ For every (non-decreasing in each of its variables) convex function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$, there exists a non-decreasing sequence of
(non-decreasing in each of their variables) Lipschitz continuous convex
functions having $f$ as a pointwise limit.

$(b)$ To establish (increasing) convex ordering it suffices to establish (2.1)
for (non-decreasing in each of their variables) Lipschitz continuous convex
functions $f$.

$(c)$ For every (non-decreasing) l.s.c. convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ the exists a non-decreasing sequence of
(non-decreasing) Lipschitz continuous convex functionals having $F$ as a
pointwise limit.

$(d)$ To establish (increasing) functional convex ordering it suffices to
establish (2.2) for (non-decreasing) Lipschitz continuous convex functionals
$F$.

Let us now deal with the functional convex ordering of two one-dimensional
diffusions

$X_{t}=X_{0}+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma(s,X_{s})dW_{s},\quad
t\\!\in[0,T],$ (2.3)

and

$Y_{t}=Y_{0}+\int_{0}^{t}\beta(s,Y_{s})ds+\int_{0}^{t}\vartheta(s,Y_{s})dW_{s},\quad
t\\!\in[0,T],$ (2.4)

where $b$, $\beta$, $\sigma$,
$\vartheta:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ are measurable functions,
the initial positions $(X_{0},Y_{0})$ and the independent Brownian motion $W$
are defined on a probability space $(\Omega,{\cal A},{\mathbb{P}})$. We also
assume that $b,\sigma,\beta,\vartheta$ satisfy the following hypothesis
written for a generic function $f:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$:

$\displaystyle{\rm Lip}(f):=\sup_{t\in[0,T],x\neq
y\in{\mathbb{R}}}\frac{|f(t,x)-f(t,y)|}{|x-y|}<+\infty\mbox{ and
}\sup_{t\in[0,T]}|f(t,0)|<+\infty,$ (2.5)

which ensures that both diffusions have a unique adapted strong solution.

Moreover (see e.g. [20, Proposition 7.2] for a proof), if $X_{0}$ and
$Y_{0}\\!\in L^{p}({\mathbb{P}})$ for some $p>0$, then there exists a real
constant $C=C_{b,\beta,\sigma,\vartheta,p,T}$ only depending $b$, $\beta$,
$\sigma$,$\vartheta$, $p$ and $T$ such that

$\Big{\|}\sup_{t\in[0,T]}|X_{t}\Big{\|}_{p}\leq C(1+\|X_{0}\|_{p})\quad\mbox{
and }\quad\Big{\|}\sup_{t\in[0,T]}|Y_{t}\Big{\|}_{p}\leq C(1+\|Y_{0}\|_{p}).$
(2.6)

######  Theorem 2.3 (Scaled martingale diffusions, see [18], [15], [11])

Assume that the functions $b$, $\beta$, $\sigma$,
$\vartheta:{\mathbb{R}}\times[0,T]\to{\mathbb{R}}$ satisfy (2.5) and that
$X_{0},Y_{0}\in L^{1}({\mathbb{P}})$.

$(a)$ Monotone convex ordering and convexity propagation. If

$X_{0}\preceq_{icv}Y_{0}\quad\mbox{and}\quad\left\\{\begin{array}[]{ll}(i)_{\sigma,b}&\sigma(t,.):{\mathbb{R}}\to{\mathbb{R}}_{+}\mbox{
and }b(t,.):{\mathbb{R}}\to{\mathbb{R}}\mbox{ are convex for every
$t\\!\in[0,T]$},\\\ \mbox{or}\\\
(i)_{\vartheta,\beta}&\vartheta(t,.):{\mathbb{R}}\to{\mathbb{R}}_{+}\mbox{ and
}\beta(t,.):{\mathbb{R}}\to{\mathbb{R}}\mbox{ are convex for every
$t\\!\in[0,T]$},\\\ \mbox{}\\\
(ii)&\sigma(t,\cdot)\;\leq\;\vartheta(t,\cdot)\mbox{ and
}b(t,.)\leq\beta(t,.)\mbox{ for every $t\\!\in[0,T]$,}\end{array}\right.$

then

$X\preceq_{icv}Y.$ (2.7)

When $(i)_{\sigma,b}$ (resp. $(i)_{\vartheta,\beta}$) holds true then, for
every l.s.c. convex $F:{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ non-
decreasing for the natural partial order on ${\cal C}([0,T],{\mathbb{R}})$,

${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}\,F(X^{x})\in{\mathbb{R}}\cup\\{+\infty\\}(\mbox{resp.
}{\mathbb{R}}\ni
y\mapsto{\mathbb{E}}\,F(X^{y})\in{\mathbb{R}}\cup\\{+\infty\\})\mbox{ is
convex and non-decreasing},$ (2.8)

where $X^{x}$ (resp. $Y^{y}$) denotes the solution of (2.3) (resp. (2.4))
starting from $X_{0}=x$ (resp. $Y_{0}=y$).

$(b)$ Convex ordering and convexity propagation for scaled martingale
diffusions. Assume that both SDEs share the same drift affine in space of the
form

$\forall\,t\\!\in[0,T],\;\forall\,x\\!\in{\mathbb{R}},\quad
b(t,x)=\beta(t,x)=\lambda(t)+\mu(t)x,$

where $\lambda,\,\mu:[0,T]\to{\mathbb{R}}$ are two bounded Borel functions.

If

$X_{0}\preceq_{cvx}Y_{0}\quad\mbox{and}\quad\left\\{\begin{array}[]{ll}(i)_{\sigma}&\sigma(t,.):{\mathbb{R}}\to{\mathbb{R}}_{+}\mbox{
is convex for every $t\\!\in[0,T]$},\\\ \mbox{or}\\\
(i)_{\vartheta}&\vartheta(t,.):{\mathbb{R}}\to{\mathbb{R}}_{+}\mbox{ is convex
for every $t\\!\in[0,T]$},\\\ \mbox{}\\\
(ii)&\sigma(t,\cdot)\;\leq\;\vartheta(t,\cdot)\mbox{ for every
$t\\!\in[0,T]$,}\end{array}\right.$

then,

$X\preceq_{cvx}Y.$ (2.9)

When, $(i)_{\sigma}$ (resp. $(i)_{\vartheta}$) holds true, then, for every
lower semi-continuous (l.s.c.) convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ on ${\cal C}([0,T],{\mathbb{R}})$,

${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}\,F(X^{x})\in{\mathbb{R}}\cup\\{+\infty\\}(\mbox{resp.
}{\mathbb{R}}\ni
y\mapsto{\mathbb{E}}\,F(X^{y})\in{\mathbb{R}}\cup\\{+\infty\\})\mbox{ is
convex}.$ (2.10)

We do not provide a detailed proof of this result in this paper since it
appears as a synthesis of several more or less recently published papers. It
combines the results established in [18] (for true martingale diffusions as
weak solutions of Brownian SDEs) and in [15, 14] (devoted to convex ordering
for scaled martingale McKean-Vlasov diffusions and increasing convex ordering
for one-dimensional McKean-Vlasov diffusions respectively). Moreover, it
benefits from various significant technical improvements developed in [11]
(devoted to convex ordering for strong solutions of stochastic Volterra
equations).

Remark. The above theorem also has a weak solution version in which instead of
assuming that $b$, $\beta$, $\sigma$, $\vartheta$ satisfy (2.5) and
$X_{0},Y_{0}\in L^{1}({\mathbb{P}})$ one supposes that $b$, $\beta$, $\sigma$,
$\vartheta$ are continuous on $[0,T]\times{\mathbb{R}}$ with at most affine
growth in space uniformly in time and $X_{0},Y_{0}\in L^{p}({\mathbb{P}})$ for
some $p>1$.

##  3 Main results

In order to state our main results, let us first introduce the $m$-marginal
convex ordering between two stochastic processes.

######  Definition 3.1 ($m$-marginal convex ordering)

Let $m\\!\in{\mathbb{N}}$. A real-valued stochastic process
$X=(X_{t})_{t\in[0,T]}$ such that $\forall
t\in[0,T],\;{\mathbb{E}}\,|X_{t}|<\infty$ is dominated by a real-valued
process $Y=(Y_{t})_{t\in[0,T]}$ such that $\forall
t\in[0,T],\;{\mathbb{E}}\,|Y_{t}|<\infty$ for the $m$-marginal convex ordering
if, for every $t_{1},\,t_{2},\cdots,\,t_{m}\\!\in[0,T]$ with $t_{1}\leq
t_{2}\leq\cdots\leq t_{m}$,

$(X_{t_{1}},\ldots,X_{t_{m}})\preceq_{cvx}(Y_{t_{1}},\ldots,Y_{t_{m}}).$

###  3.1 $1$-marginal convex ordering and convexity propagation without
convexity of the diffusion coefficient

For $1$-marginal convex ordering and convexity propagation, no spatial
convexity property of the diffusion coefficients is needed.

###### Theorem 3.2

[1-marginal convex orderings without convexity of the diffusion coefficients]
Assume that the functions $b$, $\beta$, $\sigma$,
$\vartheta:{\mathbb{R}}\times[0,T]\to{\mathbb{R}}$ satisfy (2.5) and that
$X_{0},Y_{0}\in L^{1}({\mathbb{P}})$.

$(a)$ Non-decreasing convex ordering. Assume that either
$\forall\,t\\!\in[0,T]$, $b(t,\cdot)$ is convex or $\forall\,t\\!\in[0,T]$,
$\beta(t,\cdot)$ is convex. We then have

$\Big{(}b\leq\beta\mbox{ and }0\leq\sigma\leq\vartheta\mbox{ on
}[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{icv}Y_{0}\Big{)}\Longrightarrow
X_{{}_{T}}\preceq_{icv}Y_{{}_{T}}.$

Moreover, when $\forall\,t\\!\in[0,T]$, $b(t,\cdot)$ is convex (resp.
$\forall\,t\\!\in[0,T]$, $\beta(t,\cdot)$ is convex), then for every non-
decreasing convex function $f:{\mathbb{R}}\to{\mathbb{R}}$,

$x\mapsto{\mathbb{E}}\,f(X^{x}_{T})\in{\mathbb{R}}\cup\\{+\infty\\}\left(\mbox{resp.
}y\mapsto{\mathbb{E}}\,f(Y^{y}_{T})\in{\mathbb{R}}\cup\\{+\infty\\}\right)\;\mbox{
is non-decreasing and convex.}$

$(b)$ Convex ordering. Let $b(t,x)=\beta(t,x)=\lambda(t)+\mu(t)x$,
$(t,x)\\!\in[0,T]\times{\mathbb{R}}$ be a time dependent affine function in
space where $\lambda$ and $\mu$ are bounded Borel functions. Then,

$\Big{(}0\leq\sigma\leq\vartheta\mbox{ on }[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{cvx}Y_{0}\Big{)}\Longrightarrow
X_{{}_{T}}\preceq_{cvx}Y_{{}_{T}}.$

Moreover, for every convex function $f:{\mathbb{R}}\to{\mathbb{R}}$,

$x\mapsto{\mathbb{E}}\,f(X^{x}_{T})\in{\mathbb{R}}\cup\\{+\infty\\}\left(\mbox{resp.
}y\mapsto{\mathbb{E}}\,f(Y^{y}_{T})\right)\in{\mathbb{R}}\cup\\{+\infty\\}\;\mbox{
is convex.}$

Remarks and comments. $\bullet$ From convex ordering to propagation of
convexity and back. The (non-decreasing) convexity of
$x\mapsto{\mathbb{E}}\,f(X^{x}_{T})$ for $f:{\mathbb{R}}\to{\mathbb{R}}$ (non-
decreasing) convex and the (increasing) convex ordering are closely
intertwined. For the choice $(\beta,\vartheta)=(b,\sigma)$ and
$X_{0}\sim\delta_{\alpha
x+(1-\alpha)y}\leq_{cvx}Y_{0}=\varepsilon\delta_{x}+(1-\varepsilon)\delta_{y}$
where $x,y\in{\mathbb{R}}$, $\alpha\in[0,1]$ and $\varepsilon$ independent of
$W$ follows the Bernoulli distribution with parameter $\alpha$, the
(increasing) convex ordering implies that ${\mathbb{E}}\,f(X^{\alpha
x+(1-\alpha)y}_{T})\leq\alpha{\mathbb{E}}\,f(X^{x}_{T})+(1-\alpha){\mathbb{E}}\,f(X^{y}_{T})$
for $f:{\mathbb{E}}\to{\mathbb{R}}$ (non-decreasing) convex while for the
choice $X_{0}=x\leq y=Y_{0}$ so that $X_{0}\leq_{icv}Y_{0}$, the increasing
convex ordering implies that
${\mathbb{E}}\,f(X^{x}_{T})\leq{\mathbb{E}}\,f(X^{y}_{T})$ for
$f:{\mathbb{E}}\to{\mathbb{R}}$ non-decreasing and convex. Conversely, we will
rely on Euler discretizations of the SDEs to prove Theorem 3.2 and at the
level of these Euler schemes, the propagation of convexity turns out to be a
key step to prove the convex ordering (see the proof of Proposition 4.6
below).

$\bullet$ Theorem 3.2 as an extension of Hajek’s theorem. As for the non-
decreasing convex ordering, the above comparison result extends a theorem by
Hajek [10] in which one of the two diffusion coefficients is supposed convex.
More precisely [10, Theorem 4.1] states that when

  * •

$b,\sigma$ are measurable functions satisfying (2.5),

  * •

$\beta(t,x)$ and $\vartheta(t,x)$ do not depend on the time variable and are
Lipschitz convex in the spatial variable,

  * •

$0\leq\sigma(t,x)\leq\vartheta(x)$ and $b(t,x)\leq\beta(x)$ for
$(t,x)\in[0,T]\times{\mathbb{R}}$,

  * •

$X_{0}=x_{0}\leq y_{0}=Y_{0}$,

then

$X_{T}\preceq_{icv}Y_{T}.$

$\bullet$ El Karoui, Jeanblanc and Shreve provide in [9] a direct proof of the
convexity of $x\mapsto{\mathbb{E}}\,f(X^{x}_{{}_{T}})$ for $f$ Lipschitz
convex using stochastic calculus when $b(t,x)=r(t)x$,
$\sigma(t,x)=x\gamma(t,x)$ with $\gamma:[0,T]\times(0,+\infty)\to{\mathbb{R}}$
continuous, bounded from above and such that $\partial_{2}\sigma(t,x)$ is
continuous in $(t,x)$ and Lipschitz continuous and bounded in
$x\in(0,+\infty)$ uniformly in $t\\!\in[0,T]$. According to Schied and Stadje
[23] who relie on arguments based on partial differential equations, the
convexity of $x\mapsto{\mathbb{E}}\,f(X^{x}_{{}_{T}})$ still holds for $f$
convex with polynomial growth when the drift vanishes and
$\sigma(t,x)=x\gamma(t,x)$ is Lipschitz in $x$ uniformly in $t\in[0,T]$ with
$\gamma:[0,T]\times(0,+\infty)\to(0,+\infty)$ continuously differentiable,
bounded and bounded away from $0$. The regularity assumptions in the above
theorem are weaker. For the reader’s convenience, we extend in Appendix A the
stochastic calculus argument in [9] to prove the (non-decreasing) convexity of
$x\mapsto{\mathbb{E}}\,f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ when $0\leq
t_{1}\leq\cdots\leq t_{m}\leq T$ and $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ is
directionally convex (and non-decreasing in each of its arguments). The case
$m=1$ and $t_{1}=T$ correspond to the framework of the current section.

###  3.2 A necessary condition for $2$-marginal convex ordering

Let us now explain why some stronger assumption on the diffusion coefficients
is needed to get convex order at the level of two-dimensional marginals. An
analogous result in discrete time is discussed in Remark 4.10.

######  Proposition 3.3 ($2$-marginal convex ordering)

$(a)$ Assume that $\sigma,\vartheta$ satisfy (2.5) and are continuous on
$[0,T]\times{\mathbb{R}}$, that $b(t,x)=\beta(t,x)=\lambda(t)+\mu(t)x$ with
continuous functions $\lambda,\mu:[0,T]\to{\mathbb{R}}$ and that
$X_{0},Y_{0}\in L^{1}({\mathbb{P}})$. If the solution $X=(X_{t})_{t\in[0,T]}$
of (2.3) is dominated by the solution $Y=(Y_{t})_{t\in[0,T]}$ of (2.4) for the
2-marginal convex ordering, then
${\mathbb{E}}\,|\sigma(t,X_{t})|\leq{\mathbb{E}}\,|\vartheta(t,Y_{t})|$ for
each $t\in[0,T]$.

$(b)$ In particular, when $\vartheta=\sigma$, the above $2$-marginal convex
ordering for all $X_{0},Y_{0}\in L^{1}({\mathbb{P}})$ such that
$X_{0}\leq_{cvx}Y_{0}$ implies the convexity of
$x\mapsto{\mathbb{E}}|\sigma(s,X^{x}_{s})|$ for each $s\in[0,T]$.

$(c)$ In the special case when $\sigma$ is autonomous in the sense that
$\sigma(t,x)=\sigma(x)$, the above $2$-marginal convex ordering implies that
$|\sigma|:{\mathbb{R}}\to{\mathbb{R}}$ is a convex function.

In fact, as shown by Schied et Stadje [23] in the vector case, to establish a
functional convex ordering and propagation without such stronger assumption on
the diffusion coefficient, we have to switch to a different class of
functionals, namely directionally convex functionals, which induce the
directional convex ordering. This is the object of the next section.

###  3.3 Ordering and propagation for directionally convex functionals

Prior to the main result of this section (Theorem 3.10), we first recall the
definitions of directional convexity in both vector and functional settings
and their first properties illustrated by some examples. In particular, we
stress the fact that directional convexity is not a restriction of the notion
of convexity but a different class of functions/functionals (except in one
dimension) with an overlap but none being more general than the other. We
refer to Section 3.12 of [17] where this notion is introduced and investigated
for more details. As a second step, we define the resulting stochastic order,
first for random vectors and then for stochastic processes. Finally we state
our main result.

######  Definition 3.4 (Directional convexity)

$(a)$ A function $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ is said to be
directionally convex if it is Borel measurable and

$\forall(x,y,z)\in{\mathbb{R}}^{m}\times{\mathbb{R}}_{+}^{m}\times{\mathbb{R}}_{+}^{m},\quad
f(x+y+z)-f(x+y)-f(x+z)+f(x)\geq 0.$ (3.11)

$(b)$ A functional $F:{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ is said to
be directionally convex if it is measurable and

$\forall(x,y,z)\in{\cal C}([0,T],{\mathbb{R}})\times{\cal
C}([0,T],{\mathbb{R}}_{+})^{2},\quad F(x+y+z)-F(x+y)-F(x+z)+F(x)\geq 0.$
(3.12)

######  Remark 3.5 (Characterisations)

$\mathbf{P1.}$ A directionally convex function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ is measurable and mid-point convex in each
of its variables (the other variables being fixed) whence, according to [24],
convex in each of its variables. This implies that for each
$i\in\\{1,\cdots,m\\}$, the right-hand derivative
$\partial_{x_{i}+}f(x_{1},\ldots,x_{i},\ldots,x_{m})$ is non-decreasing in
each of its variables and therefore locally bounded. As a consequence, the
function $f$ is continuous.

$\mathbf{P2.}$ A twice continuously differentiable function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ is directionally convex if and only if for
all $i,j\in\\{1,\cdots,m\\}$, $\partial^{2}_{x_{i}x_{j}}f\geq 0$.

$\mathbf{P3.}$ When $d=1$, directional convexity amounts to regular convexity.

$\mathbf{P4.}$ When $g:{\mathbb{R}}\to{\mathbb{R}}$ is convex, then for each
$i\in\\{1,\cdots,m\\}$, ${\mathbb{R}}^{m}\ni(x_{1},\ldots,x_{m})\mapsto
g(x_{i})$ is directionally convex.

$\mathbf{P5.}$ A twice continuously Fréchet differentiable functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ is directionally convex if and only if,
for every $x\\!\in{\cal C}([0,T],{\mathbb{R}})$ and every $u,\,v\\!\in{\cal
C}([0,T],{\mathbb{R}}_{+})$,

$D^{2}F(x).(u,v)\geq 0.$

The following examples illustrate that the notions of convexity and
directional convexity have overlap but that none of the two is more general
than the other.

###### Example 3.6

$(a)$ The funtion $f:{\mathbb{R}}^{2}\to{\mathbb{R}}$ defined by
$f(u,v)=|u-v|$ is (clearly) convex but not directionally convex. Indeed, if we
set

$x=y=(1,0),\quad z=(1,\tfrac{5}{2})$

then
$f(x+y+z)-f(x+y)-f(x+z)+f(x)=|3-\frac{5}{2}|-|2-0|-|2-\frac{5}{2}|+|1-0|=-1<0$
so that (3.11) is clearly violated.

$(b)$ Let us consider the quadratic function
$f:{\mathbb{R}}^{2}\to{\mathbb{R}}$ defined by

$f(u,v)=au^{2}+2\,cuv+bv^{2}\quad\mbox{where $a,b>0$}.$

One checks that $f$ is convex in each of its variables and that

  * •

$f$ is convex if and only if $|c|\leq\sqrt{ab}$,

  * •

$f$ is directionally convex if and only if $c\geq 0$ owing to the above
Property P2.

Consequently, $f$ can be convex but not directionally convex (if
$-\sqrt{ab}<c<0$), directionally convex without being convex (if
$c>\sqrt{ab}$) and both convex and directionally convex if $0\leq
c\leq\sqrt{ab}$.

$(c)$ By the same arguments, this time based on Property $\mathbf{P5}$, the
functional $F:{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ defined for every
continuous function $x$ by

$F(x)=\int_{[0,T]^{2}}(ax(s)^{2}+2cx(s)x(t)+bx(t)^{2})\,ds\,dt$

is

  * •

convex if and only if $|c|\leq\sqrt{ab}$,

  * •

directionally convex if and only if $c\geq 0$.

with the same conclusions as in $(b)$.

$(d)$ Let $F:{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ be the functional
defined by

$F(x)=\Psi\left(\int_{0}^{T}\varphi(x(s))ds\right),\quad x\\!\in{\cal
C}([0,T],{\mathbb{R}}),$

where $\Psi,\varphi:{\mathbb{R}}\to{\mathbb{R}}$ are two convex functions.

$\bullet$ If $\Psi$ is non-decreasing then $F$ is convex,

$\bullet$ If both $\Psi$ and $\varphi$ are non-decreasing, then $F$ is
directionally convex.

Consequently $F$ is “more often” convex than directionally convex. The proof,
based on a direct checking of (3.12) is postponed to Appendix B.

The following lemma is the counterpart for directionally convex functions of
the properties of monotone approximation of convex functions by Lipschitz
convex ones stated in Proposition 2.2.

###### Lemma 3.7

Let $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ be directionally convex (non-
decreasing in each of its variables). Then there exists a non-decreasing
sequence $(f_{n})_{n\geq 1}$ of Lipschitz continuous directionally convex
(non-decreasing in each variable) functions having $f$ as a pointwise limit.
Moreover, for any $m$-dimensional random vector $U$ such that $U\in
L^{1}({\mathbb{P}})$, $f^{-}(U)$ is integrable and ${\mathbb{E}}\,f(U)$ makes
sense in ${\mathbb{R}}\cup\\{+\infty\\}$.

Proof. The directionally convex function $f$ being convex in each of its
variables, it admits a left-hand derivative
$\partial_{i-}f(u_{1},\cdots,u_{i},\cdots,u_{m})$ and a right-hand derivative
$\partial_{i+}f(u_{1},\cdots,u_{i},\cdots,u_{m})$ with respect to its $i$-th
coordinate for $i\in\\{1,\cdots,m\\}$. The directional convexity implies that
$\partial_{i-}f$ and $\partial_{i+}f$ are non-decreasing in each of their
variables and therefore locally bounded. As a consequence $f$ is locally
Lipschitz continuous. The construction in [17, Theorem 3.12.7] then provides a
non-decreasing sequence of Lipschitz continuous directionally convex functions
$(f_{n})_{n\geq 1}$ that converge pointwise to $f$ as $n\to+\infty$ : for
$n\geq 1$,

$\displaystyle f_{n}(u_{1},\cdots$
$\displaystyle,u_{m})=f(\psi_{n}(u_{1},\cdots,u_{m}))$
$\displaystyle+\sum_{i=1}^{m}\left(\mathbf{1}_{\\{u_{i}\leq-n\\}}\partial_{i-}f(\psi_{n}(u_{1},\cdots,u_{m}))(u_{i}+n)+\mathbf{1}_{\\{u_{i}\geq
n\\}}\partial_{i+}f(\psi_{n}(u_{1},\cdots,u_{m}))(u_{i}-n)\right),$

where $\psi_{n}(u_{1},\cdots,u_{m})=(-n\vee u_{1}\wedge n,\cdots,-n\vee
u_{m}\wedge n)$. When $f$ is moreover non-decreasing in each of its variables,
then so is $f_{n}$ since $\psi_{n}$ also is and $\partial_{i-}f$ and
$\partial_{i+}f$ are non-negative.

When $U\in L^{1}({\mathbb{P}})$, then, since $f_{1}$ is Lipschitz continuous,
$f_{1}(U)$ is integrable. Since $f_{1}\leq f$, $f^{-}\leq f_{1}^{-}$ and we
deduce that $f^{-}(U)$ is integrable. $\Box$

An extension of this result to l.s.c. directionally convex functionals remains
an open question to us. Moreover, we do not whether a l.s.c. directionally
convex functional is bounded from below by a continuous affine functional. In
particular, we do not know whether ${\mathbb{E}}\,F(X)$ makes sense in
${\mathbb{R}}\cup\\{+\infty\\}$ when $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ is l.s.c. directionally convex and
$X=(X_{t})_{t\in[0,T]}$ is a continuous real-valued process such that
${\mathbb{E}}\,\sup_{t\in[0,T]}|X_{t}|<+\infty$. That is why we restrict the
class of directionally convex functionals in the next definition of
directionally convex orderings.

######  Definition 3.8 (Directionally convex orderings)

$(a)$ Let $U$, $V\\!\in L^{1}({\mathbb{P}})$ be two ${\mathbb{R}}^{m}$-valued
random vectors.

$\rhd$ Directionally convex ordering on ${\mathbb{R}}^{m}$. We say that $U$ is
dominated for the directionally convex ordering by $V$, denoted

$U\preceq_{dircvx}V$ (3.13)

if (2.1) holds for every directionally convex function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$.

$\rhd$ Monotone directionally convex ordering on ${\mathbb{R}}^{m}$. We say
that $U$ is dominated for the non-decreasing directionally convex ordering by
$V$, denoted

$U\preceq_{diricv}V$

if (2.1) holds for every directionally convex function
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ non-decreasing in each of its variables.

$(b)$ Let $X=(X_{t})_{t\in[0,T]}$ and $Y=(Y_{t})_{t\in[0,T]}$ be two pathwise
continuous ${\mathbb{R}}$-valued stochastic processes satisfying
${\mathbb{E}}\,\sup_{t\in[0,T]}|X_{t}|^{p}+{\mathbb{E}}\,\sup_{t\in[0,T]}|Y_{t}|^{p}<+\infty$.

$\rhd$ Functional directionally convex ordering. We say that $X$ is dominated
by $Y$ for the functional directionally convex ordering with growth index $p$,
denoted

$X\preceq_{dircvx(p)}Y,$

if for every continuous directionally convex functional $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ with polynomial growth of order $p$
(w.r.t. the sup norm) (111i.e. $\exists C_{{}_{F}}<\infty$ such that for every
$x\\!\in{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$, $|F(x)|\leq
C_{{}_{F}}(1+\|x\|^{p}_{\sup})$.)

${\mathbb{E}}\,F(X)\leq{\mathbb{E}}\,F(Y).$ (3.14)

$\rhd$ Monotone functional directionally convex ordering We say that $X$ is
dominated by $Y$ for the functional increasing directionally convex ordering
with growth index $p$, denoted

$X\preceq_{diricv(p)}Y,$

if (3.14) holds for every non-decreasing continuous directionally convex
functional $F:{\cal C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ with polynomial
growth of order $p$.

###### Remark 3.9

Property P4 in Remark 3.5 implies that

$\displaystyle(U_{1},\cdots,U_{m})\leq_{dircvx}(V_{1},\cdots,V_{m})\Longrightarrow\forall
i\in\\{1,\cdots,m\\},\;U_{i}\leq_{cvx}V_{i},$ (3.15)
$\displaystyle(U_{1},\cdots,U_{m})\leq_{diricv}(V_{1},\cdots,V_{m})\Longrightarrow\forall
i\in\\{1,\cdots,m\\},\;U_{i}\leq_{icv}V_{i}.$ (3.16)

We are now ready to state our result concerning directionally convex
functional ordering.

###### Theorem 3.10

Assume that the functions $b$, $\beta$, $\sigma$,
$\vartheta:{\mathbb{R}}\times[0,T]\to{\mathbb{R}}$ satisfy (2.5) and that
$X_{0},Y_{0}\in L^{p}({\mathbb{P}})$ for some $p\in[1,+\infty)$.

$(a)$ Non-decreasing directionally convex functional ordering. Assume that
either $\forall\,t\\!\in[0,T]$, $b(t,\cdot)$ is convex or
$\forall\,t\\!\in[0,T]$, $\beta(t,\cdot)$ is convex. Then,

$\Big{(}b\leq\beta\mbox{ and }0\leq\sigma\leq\vartheta\mbox{ on
}[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{icv}Y_{0}\Big{)}\Longrightarrow X\preceq_{dircvx(p)}Y.$

Moreover, when $\forall\,t\\!\in[0,T]$, $b(t,\cdot)$ is convex, then for each
continuous non-decreasing directionally convex $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ with polynomial growth,
${\mathbb{R}}\ni x\mapsto{\mathbb{E}}F(X^{x})\in{\mathbb{R}}$ is convex and
non-decreasing.

$(b)$ Directionally convex functional ordering. Let
$b(t,x)=\beta(t,x)=\lambda(t)+\mu(t)x$, $(t,x)\\!\in[0,T]\times{\mathbb{R}}$
be a time dependent affine function in space where $\lambda$ and $\mu$ are
bounded Borel functions. Then,

$\Big{(}0\leq\sigma\leq\vartheta\mbox{ on }[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{cvx}Y_{0}\Big{)}\Longrightarrow X\preceq_{diricv(p)}Y.$

Moreover, for each continuous directionally convex $F:{\cal
C}([0,T],{\mathbb{R}})\to{\mathbb{R}}$ with polynomial growth,
${\mathbb{R}}\ni x\mapsto{\mathbb{E}}F(X^{x})\in{\mathbb{R}}$ is convex.

Remark. According to Schied and Stadje [23] who relie on arguments based on
partial differential equations,
$x\mapsto{\mathbb{E}}\,f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is convex for
$0\leq t_{1}\leq\cdots\leq t_{m}\leq T$ and
$f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ directionally convex with polynomial
growth when the drift vanishes and $\sigma(t,x)=x\gamma(t,x)$ is Lipschitz in
$x$ uniformly in $t\in[0,T]$ with
$\gamma:[0,T]\times(0,+\infty)\to(0,+\infty)$ continuously differentiable,
bounded and bounded away from $0$. The convexity of
$x\mapsto{\mathbb{E}}\,f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is derived in
Appendix A by a generalization of the stochastic calculus argument in [9]
which requires more regularity on the coefficients of the SDE.

##  4 Convergence and ordering of Euler schemes with truncated noise

Our proofs rely on the same “transfer” paradigm: prove the results in discrete
time on discretization schemes of Euler type and then transfer them by a
strong (or functional weak ) convergence theorem toward the diffusion. We will
extensively use for the questions under consideration in this paper some
specific truncated Euler schemes, especially when dealing with monotone convex
ordering and propagation. We always consider a taylored Euler scheme for which
the lightest assumptions are needed but, when it turns out not to be
simulable, we also give a result for a simulable Euler scheme under slightly
more stringent assumptions.

It turns out that we will need to truncate the Brownian increments in the
Euler scheme to ensure propagation of convexity by the transitions associated
with this scheme. That is why, in the next section, we address the convergence
of such Euler schemes with truncated noise.

###  4.1 Convergence of Euler schemes with truncated noise

The (continuous time) Euler scheme with step $\frac{T}{m},\;m\geq 1$ that we
use to discretize (2.3) is defined by freezing the spatial coefficients (and
not the time coefficients) using $\underline{t}=\frac{kT}{m}$ when
$t\\!\in[\frac{kT}{m},\frac{(k+1)T}{m})$, $k=0,\ldots,m-1$ and
$\underline{T}=T$ :

$\bar{X}_{t}^{m}=X_{0}+\int_{0}^{t}b(s,\bar{X}^{m}_{\underline{s}})ds+\int_{0}^{t}\sigma(s,\bar{X}^{m}_{\underline{s}})dW_{s}.$
(4.17)

If we denote by $t_{k}=\frac{kT}{m}$ the discretization times, we have
$\bar{X}^{m}_{0}=X_{0}$ and

$\bar{X}^{m}_{t_{k+1}}=\bar{X}^{m}_{t_{k}}+\int_{t_{k}}^{t_{k+1}}b(s,\bar{X}^{m}_{t_{k}})ds+\int_{t_{k}}^{t_{k+1}}\sigma(s,\bar{X}^{m}_{t_{k}})dW_{s},\;k=0,\ldots,m-1.$
(4.18)

Introducing the independent standard Gaussian random variables
$G_{k+1}=(W_{t_{k+1}}-W_{t_{k}})/\sqrt{T/m}$, $k=0,\ldots,m-1$, we can also
associate a truncated Euler scheme relying on the truncated Brownian
increments

$Z^{m}_{k+1}=G_{k+1}\mathbf{1}_{\\{|G_{k+1}|\leq{\mathbf{s}}_{m}\\}},\quad
k=0,\ldots,m-1$ (4.19)

i.e. $\widetilde{X}^{m}_{0}=X_{0}$ and

$\widetilde{X}^{m}_{t_{k+1}}=\widetilde{X}^{m}_{t_{k}}+\int_{t_{k}}^{t_{k+1}}b(s,\widetilde{X}^{m}_{t_{k}})ds+\sqrt{\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\widetilde{X}^{m}_{t_{k}})ds}\,Z^{m}_{k+1},\;k=0,\ldots,m-1.$
(4.20)

###### Proposition 4.1

Assume $b,\sigma$ satisfy (2.5) and let $p\in[1,+\infty)$. There exists a
finite constant $C_{p}$ such that for each initial random variable $X_{0}$ and
each $m\geq 1$, the truncated Euler scheme (4.20) satisfies

${\cal W}_{p}({\cal L}(X_{t_{0}},\ldots,X_{t_{m}}),{\cal
L}(\widetilde{X}^{m}_{t_{0}},\ldots,\widetilde{X}^{m}_{t_{m}}))\leq
C_{p}\left(m^{-\frac{1}{2}}+{\Big{(}\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}}\Big{)}^{\frac{1}{2p}}}\right)(1+\|X_{0}\|_{p}),$

where, in the definition of the Wasserstein distance ${\cal W}_{p}$,
${\mathbb{R}}^{m+1}$ is endowed with
$\displaystyle\|(x_{0},\ldots,x_{m})\|_{\infty}=\max_{k=0,\ldots,m}|x_{k}|$.

###### Remark 4.2

We deduce that, as $m\to+\infty$,

$\liminf_{m}\frac{{\mathbf{s}}_{m}}{\sqrt{\ln m}}>\sqrt{2}\Rightarrow{\cal
W}_{p}({\cal L}(X_{t_{0}},\ldots,X_{t_{m}}),{\cal
L}(\widetilde{X}^{m}_{t_{0}},\ldots,\widetilde{X}^{m}_{t_{m}}))\to 0.$

Proof of Proposition 4.1. To analyse the convergence of the Euler scheme with
truncated Brownian increments, it is convenient to introduce a third scheme
$(\hat{X}^{m}_{t_{k}})_{k=0,\ldots,m}$ inductively defined by
$\hat{X}^{m}_{0}=X_{0}$ and

$\hat{X}^{m}_{t_{k+1}}=\hat{X}^{m}_{t_{k}}+\int_{t_{k}}^{t_{k+1}}b(s,\hat{X}^{m}_{t_{k}})ds+\sqrt{\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\hat{X}^{m}_{t_{k}})ds}\,G_{k+1},\;k=0,\ldots,m-1.$

Since $\bar{X}^{m}_{t_{0}}=\hat{X}^{m}_{t_{0}}=X_{0}$ and the conditional law
of $\bar{X}^{m}_{t_{k+1}}$ given
$(\bar{X}^{m}_{t_{0}},\ldots,\bar{X}^{m}_{t_{k}})$ is Gaussian with
expectation $\int_{t_{k}}^{t_{k+1}}b(s,\bar{X}^{m}_{t_{k}})ds$ and variance
$\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\bar{X}^{m}_{t_{k}})ds$ and the same
statement holds with $\bar{X}^{m}$ replaced by $\hat{X}^{m}$, we have by
induction that ${\cal
L}(\bar{X}^{m}_{t_{0}},\bar{X}^{m}_{t_{1}},\ldots,\bar{X}^{m}_{t_{m}})={\cal
L}(\hat{X}^{m}_{t_{0}},\hat{X}^{m}_{t_{1}},\ldots,\hat{X}^{m}_{t_{m}})$.

By standard arguments, the fact that $b,\sigma$ satisfy (2.5) implies the
existence of a finite constant $\kappa^{(1)}_{b,\sigma,p,T}$ such that

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}|\Big{\|}_{p}+\Big{\|}\sup_{t\in[0,T]}|\bar{X}^{m}_{t}|\Big{\|}_{p}+\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}|\Big{\|}_{p}+\Big{\|}\max_{k=0,\ldots,m}|\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq\kappa^{(1)}_{b,\sigma,p,T}(1+\|X_{0}\|_{p})$
$\displaystyle\mbox{ and
}\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}_{p}\leq\frac{\kappa^{(1)}_{b,\sigma,p,T}}{\sqrt{m}}(1+\|X_{0}\|_{p}).$

Since

$\displaystyle{\cal W}_{p}({\cal L}(X_{t_{0}},\ldots,X_{t_{m}}),{\cal
L}(\widetilde{X}^{m}_{t_{0}},\ldots,\widetilde{X}^{m}_{t_{m}}))$
$\displaystyle\leq{\cal W}_{p}({\cal L}(X_{t_{0}},\ldots,X_{t_{m}}),{\cal
L}(\bar{X}^{m}_{t_{0}},\ldots,\bar{X}^{m}_{t_{m}}))$
$\displaystyle\phantom{\leq}+{\cal W}_{p}({\cal
L}(\hat{X}_{t_{0}},\ldots,\hat{X}_{t_{m}}),{\cal
L}(\widetilde{X}^{m}_{t_{0}},\ldots,\widetilde{X}^{m}_{t_{m}}))$
$\displaystyle\leq\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}_{p}+\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p},$

to conclude, it is enough to check that

$\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq
2\kappa^{(1)}_{b,\sigma,2p}{\Big{(}\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}}\Big{)}^{\frac{1}{2p}}}(1+\|X_{0}\|_{p}),$
(4.21)

which we now do. To compare $(\hat{X}^{m}_{t_{k}})_{k=0,\ldots,m}$ and
$(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}$ we simply note that

$(\hat{X}^{m}_{t_{k}})_{k=0,\ldots,m}=(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\quad\mbox{
on the event
}\quad\Omega_{m}=\bigcap_{k=1}^{m}\left\\{|G_{k}|\leq{\mathbf{s}}_{m}\right\\}.$

Then, when $\|X_{0}\|_{2p}<\infty$,

$\displaystyle\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}$
$\displaystyle=\Big{\|}\mathbf{1}_{\Omega^{c}_{m}}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}$
$\displaystyle\leq{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2p}}\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}|+\max_{k=0,\ldots,m}|\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{2p}$
$\displaystyle\leq{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2p}}\left(\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}|\Big{\|}_{2p}+\Big{\|}\max_{k=0,\ldots,m}|\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{2p}\right)$
$\displaystyle\leq
2\kappa^{(1)}_{b,\sigma,2p}{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2p}}\big{(}1+\|X_{0}\|_{2p}\big{)}.$

In particular, denoting by $\hat{X}^{m,x_{0}}_{t_{k}}$ and
$\widetilde{X}^{m,x_{0}}_{t_{k}}$ the regular and the truncated Euler schemes
both starting from the deterministic initial position $x_{0}\in{\mathbb{R}}$,
we have

${\mathbb{E}}\left[\max_{k=0,\ldots,m}\left|\hat{X}^{m,x_{0}}_{t_{k}}-\widetilde{X}^{m,x_{0}}_{t_{k}}\right|^{p}\right]\leq(2\kappa^{(1)}_{b,\sigma,2p})^{p}{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2}}\big{(}1+|x_{0}|\big{)}^{p}.$

Integrating this inequality in $x_{0}$ with respect to the distribution of
$X_{0}$ and using the independence of this random variable and the vector of
Brownian increments $(G_{1},\ldots,G_{m})$, we deduce that

$\Big{\|}\max_{k=0,\ldots,m}|\hat{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq
2\kappa^{(1)}_{b,\sigma,2p}{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2p}}\|1+|X_{0}|\|_{p}\leq
2\kappa^{(1)}_{b,\sigma,2p}{\mathbb{P}}(\Omega^{c}_{m})^{\frac{1}{2p}}(1+\|X_{0}\|_{p}).$

Since, by the elementary inequality
${\mathbb{P}}(|G_{1}|>z)=2{\mathbb{P}}(G_{1}>z)\leq\left(1\wedge\sqrt{\frac{2}{\pi
z^{2}}}\right)e^{-\frac{z^{2}}{2}}$, $z\geq 0$,

$\displaystyle{\mathbb{P}}(\Omega^{c}_{m})={\mathbb{P}}\left(\max_{k=1,\ldots,m}|G_{k}|>{\mathbf{s}}_{m}\right)\leq
m{\mathbb{P}}\left(|G_{1}|>{\mathbf{s}}_{m}\right)$
$\displaystyle\leq\sqrt{\frac{2}{\pi}}\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}},$

we deduce (4.21). $\Box$

To transfer our discrete time results on convex ordering to continuous time,
we use the ${\cal C}([0,T],{\mathbb{R}})$-valued interpolation operator
$i_{m}$ associated to the mesh $(t_{k}=\frac{kT}{m})_{k=0,\ldots,m}$ already
introduced in [19]

$i_{m}:{\mathbb{R}}^{m+1}\ni(x_{0},\ldots,x_{m})\longmapsto\Big{(}t\mapsto\sum_{k=1}^{m}\mathbf{1}_{[t_{k-1},t_{k})}(t)\left(\frac{t_{k}-t}{t_{k}-t_{k-1}}x_{k-1}+\frac{t-t_{k-1}}{t_{k}-t_{k-1}}x_{k}\right)+\mathbf{1}_{\\{t=t_{n}\\}}x_{m}\Big{)}\\!.$

###### Corollary 4.3

Let us assume that $b,\sigma$ satisfy (2.5) and that $\|X_{0}\|_{p}<+\infty$
for some $p\in[1,+\infty)$. Then the truncated Euler scheme (4.20)
$(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}$ satisfies

${\cal W}_{p}\left({\cal L}(X),{\cal
L}\left(i_{m}\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\right)\right)\to
0\quad\mbox{ as }\quad m\to+\infty\mbox{ when
}{\liminf_{m}\frac{{\mathbf{s}}_{m}}{\sqrt{\ln m}}>\sqrt{2}}.$

Proof. For $(x_{k})_{k=0,\ldots,m}\in({\mathbb{R}}^{d})^{m+1}$, the function
$i_{m}((x_{k})_{k=0,\ldots,m})$ being continuous and piecewise affine with
affinity breaks at times $t_{k}$ where it is equal to $x_{k}$,

$\|i_{m}((x_{k})_{k=0,\ldots,m})\|_{\sup}=\max_{k=0,\ldots,m}|x_{k}|.$

For $(x_{k})_{k=0,\ldots,m},(y_{k})_{k=0,\ldots,m}\in{\mathbb{R}}^{m+1}$,
since
$i_{m}((x_{k})_{k=0,\ldots,m})-i_{m}((y_{k})_{k=0,\ldots,m})=i_{m}((x_{k})_{k=0,\ldots,m}-(y_{k})_{k=0,\ldots,m})$,
we deduce that $i_{m}$ is Lipschitz with constant $1$ from
${\mathbb{R}}^{m+1}$ to $\big{(}{\cal
C}([0,T],{\mathbb{R}}),\|\cdot\|_{\sup}\big{)}$ :

$\|i_{m}((x_{k})_{k=0,\ldots,m})-i_{m}((y_{k})_{k=0,\ldots,m})\|_{\sup}=\max_{k=0,\ldots,m}|x_{k}-y_{k}|.$

By integrating this equality in
$((x_{k})_{k=0,\ldots,m},(y_{k})_{k=0,\ldots,m})$ against a ${\cal W}_{p}$
optimal coupling between ${\cal L}(X_{t_{0}},\ldots,X_{t_{m}})$ and ${\cal
L}(\widetilde{X}^{m}_{t_{0}},\ldots,\widetilde{X}^{m}_{t_{m}})$ and using
Proposition 4.1, we deduce that

${\cal W}_{p}\left({\cal
L}\left(i_{m}\big{(}(X_{t_{k}})_{k=0,\ldots,m}\big{)}\right),{\cal
L}\left(i_{m}\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\right)\right)\leq
C_{p}\left(m^{-\frac{1}{2}}+{\Big{(}\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}}\Big{)}^{\frac{1}{2p}}}\right)(1+\|X_{0}\|_{p}),$

where the right-hand side goes to $0$ as $m\to+\infty$ when
$\liminf_{m}\frac{{\mathbf{s}}_{m}}{\sqrt{\ln m}}>\sqrt{2}$. With the triangle
inequality for ${\cal W}_{p}$ and the inequality ${\cal W}_{p}\left({\cal
L}(X),{\cal
L}\left(i_{m}\big{(}(X_{t_{k}})_{k=0,\ldots,m}\big{)}\right)\right)\leq\Big{\|}\big{\|}X-i_{m}\big{(}(X_{t_{k}})_{k=0,\ldots,m}\big{)}\big{\|}_{\sup}\Big{\|}_{p}$
we conclude that it is enough to check that

$\Big{\|}\big{\|}X-i_{m}\big{(}(X_{t_{k}})_{k=0,\ldots,m}\big{)}\big{\|}_{\sup}\Big{\|}_{p}\to
0\quad\mbox{as}\quad m\to+\infty$ (4.22)

to complete the proof. The uniform continuity modulus
$w(\xi,\delta)=\sup_{0\leq s\leq t\leq(s+\delta)\wedge T}|\xi(t)-\xi(s)|$ of a
function $\xi\\!\in{\cal C}([0,T],{\mathbb{R}})$ converges to $0$ as
$\delta\to 0$. One easily checks that

$\big{\|}X-i_{m}\big{(}(X_{t^{m}_{k}})_{k=0,\ldots,m}\big{)}\big{\|}_{\sup}\leq
w\big{(}X,\tfrac{T}{m}\big{)}\leq 2\big{\|}X\big{\|}_{\sup}.$

As, by standard estimations, $\big{\|}\|X\|_{\sup}\big{\|}_{p}\leq
C(1+\|X_{0}\|_{p})<+\infty$, (4.22) holds by dominated convergence. $\Box$

No continuity of the coefficients $b$, $\sigma$ in their time variable is
required for the law of the Euler scheme (4.20) with truncated Brownian
increments to converge to that of the solution of the SDE (2.3). The
counterpart is that one needs closed-form formulas for the integrals
$\int_{t_{k}}^{t_{k+1}}b(s,x)ds$ and $\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,x)ds$
to implement this scheme in practice. If such formulas are not available, one
should rather rely on the the standard Euler scheme where the time variable of
the coefficients is also discretized : $\widetilde{X}^{m}_{0}=X_{0}$ and

$\widetilde{X}^{m}_{t_{k+1}}=\widetilde{X}^{m}_{t_{k}}+\frac{T}{m}b(t_{k},\widetilde{X}^{m}_{t_{k}})+\sqrt{\frac{T}{m}}\sigma(t_{k},\widetilde{X}^{m}_{t_{k}})\,Z^{m}_{k+1},\;k=0,\ldots,m-1.$
(4.23)

The corresponding Euler scheme with non-truncated increments reads

$\bar{X}_{t}^{m}=X_{0}+\int_{0}^{t}b({\underline{s}},\bar{X}^{m}_{\underline{s}})ds+\int_{0}^{t}\sigma({\underline{s}},\bar{X}^{m}_{\underline{s}})dW_{s}$
(4.24)

and evolves inductively from one discretization time to the next according to

$\bar{X}^{m}_{t_{k+1}}=\bar{X}^{m}_{t_{k}}+\frac{T}{m}b(t_{k},\bar{X}^{m}_{t_{k}})+\sqrt{\frac{T}{m}}\sigma(t_{k},\bar{X}^{m}_{t_{k}})\,G_{k+1},\;k=0,\ldots,m-1.$
(4.25)

Under the assumption

$\displaystyle\exists\gamma\\!\in$ $\displaystyle(0,1],\;\exists C^{\rm
Hol}_{b,\sigma},{\rm
Lip}(b,\sigma)<\infty,\;\forall\,s,\,t\\!\in[0,T],\;\forall\,x,\,y\\!\in{\mathbb{R}},$
$\displaystyle|b(s,x)-b(t,y))|+|\sigma(s,x)-\sigma(t,y)|\leq C^{\rm
Hol}_{b,\sigma}(1+|x|+|y|)|t-s|^{\gamma}+{\rm Lip}(b,\sigma)|x-y|,$ (4.26)

where, in addition to (2.5) for $b,\sigma$, a Hölder continuity of the
coefficients in their time variable is supposed, the following standard error
estimation holds for the scheme with non-truncated Brownian increments:

$\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}_{p}\leq\kappa^{(2)}_{b,\sigma,p}m^{-(\gamma\wedge\frac{1}{2})}(1+\|X_{0}\|_{p}).$
(4.27)

The next result addresses the convergence of the version of the scheme with
truncated increments.

###### Proposition 4.4

Let us consider the truncated Euler scheme (4.23) and let $p\in[1,+\infty)$.

$(a)$ Under (4.26), there exists a finite constant $C_{p}$ such that for each
initial random variable $X_{0}$ and each $m\geq 1$, the truncated Euler scheme
(4.20) satisfies

$\Big{\|}\max_{k=0,\ldots,m}|X_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq
C_{p}\left(m^{-(\gamma\wedge\frac{1}{2})}+m^{\frac{1}{2p}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{4p}}\right)(1+\|X_{0}\|_{p}).$

$(b)$ Moreover, if the coefficients $b,\sigma$ satisfy mere continuity on
$[0,T]\times{\mathbb{R}}$ in addition to (2.5) and $X_{0}\in
L^{p}({\mathbb{P}})$, then

$\Big{\|}\max_{k=0,\ldots,m}|X_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\to
0\mbox{ as }m\to+\infty\;\mbox{ when
}\;\liminf_{m}\frac{{\mathbf{s}}_{m}}{\sqrt{\ln m}}>\sqrt{2}.$

Proof of Proposition 4.4. $(a)$ Under (2.5), the standard estimation

$\Big{\|}\sup_{t\in[0,T]}|X_{t}|\Big{\|}_{p}+\Big{\|}\sup_{t\in[0,T]}|\bar{X}^{m}_{t}|\Big{\|}_{p}+\Big{\|}\max_{k=0,\ldots,m}|\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq\kappa^{(1)}_{b,\sigma,p,T}(1+\|X_{0}\|_{p})$

holds. Reasoning like in the last step of the proof of Proposition 4.1, one
deduces that

$\Big{\|}\max_{k=0,\ldots,m}|\bar{X}^{m}_{t_{k}}-\widetilde{X}^{m}_{t_{k}}|\Big{\|}_{p}\leq
2\kappa^{(1)}_{b,\sigma,2p,T}{\Big{(}\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}}\Big{)}^{\frac{1}{2p}}}(1+\|X_{0}\|_{p}).$

Combining this inequality with (4.27) thanks to the triangle inequality, one
obtains the first statement.

$(b)$ Since as $m\to+\infty$, when
$\liminf_{m}\frac{{\mathbf{s}}_{m}}{\sqrt{\ln m}}>\sqrt{2}$,
$\frac{m}{{\mathbf{s}}_{m}}e^{-\frac{{\mathbf{s}}_{m}^{2}}{2}}\to 0$, to prove
the second statement, it is enough to check that
$\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}_{p}\to 0$. Let us
suppose temporarily that $p\geq 2$ and $X_{0}\in L^{2p}({\mathbb{P}})$. Thanks
to the spatial Lipschitz continuity of the coefficients, standard estimations
based on the Hölder and Burkholder-Davis-Gundy inequalities and Gronwall’s
lemma ensure the existence of a finite constant not depending on $m$ and
$X_{0}$ such that

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}^{p}_{p}$
$\displaystyle\leq
C\int_{0}^{T}\|b(s,X_{s})-b({\underline{s}},X_{\underline{s}})\|^{p}_{p}+\|\sigma(s,X_{s})-\sigma({\underline{s}},X_{\underline{s}})\|^{p}_{p}ds.$

With standard estimations on the moments of the increments of $X$ under (2.5),
we deduce that

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}^{p}_{p}\leq$
$\displaystyle
C\int_{0}^{T}\|b(s,X_{s})-b({\underline{s}},X_{s})\|^{p}_{p}+\|\sigma(s,X_{s})-\sigma({\underline{s}},X_{s})\|^{p}_{p}ds$
$\displaystyle+Cm^{-\frac{p}{2}}(1+\|X_{0}\|_{p})^{p}.$ (4.28)

Let now $K\in[1,+\infty)$ and $\varepsilon_{m,K}:=\sup_{0\leq s\leq
t\leq(s+\frac{T}{m})\wedge
T,x\in[-K,K]}(|b(s,x)-b(t,x)|+|\sigma(s,x)-\sigma(t,x)|)$. By uniform
continuity of the coefficients $b,\sigma$ on the compact $[0,T]\times[-K,K]$,
$\varepsilon_{m,K}\to 0$ as $m\to+\infty$. One has, using the affine growth of
the coefficients in space uniform in time and $K\geq 1$,

$\displaystyle\sup_{s\in[0,T]}\left(|b(s,X_{s})-b({\underline{s}},X_{s})|+|\sigma(s,X_{s})-\sigma({\underline{s}},X_{s})|\right)$
$\displaystyle\leq\varepsilon_{m,K}+C\sup_{t\in[0,T]}|X_{t}|{\mathbf{1}}_{\\{\sup_{t\in[0,T]}|X_{t}|\geq
K\\}}.$

Therefore, using the Cauchy-Schwarz inequality, then
$\Big{\|}\sup_{t\in[0,T]}|X_{t}|\Big{\|}_{q}\leq\kappa^{(1)}_{b,\sigma,q}(1+\|X_{0}\|_{q})$
for $q\in\\{2p,1\\}$ and the Markov inequality, we obtain

$\displaystyle\sup_{s\in[0,T]}\big{(}\|b(s,X_{s})-b({\underline{s}},X_{s})\|^{p}_{p}$
$\displaystyle+\|\sigma(s,X_{s})-\sigma({\underline{s}},X_{s})\|^{p}_{p}\big{)}$
$\displaystyle\leq
C\left(\varepsilon_{m,K}^{p}+{\mathbb{E}}\left[\sup_{t\in[0,T]}|X_{t}|^{p}{\mathbf{1}}_{\\{\sup_{t\in[0,T]}|X_{t}|\geq
K\\}}\right]\right)$ $\displaystyle\leq
C\left(\varepsilon_{m,K}^{p}+\Big{\|}\sup_{t\in[0,T]}|X_{t}|\Big{\|}_{2p}^{p}{\mathbb{P}}^{1/2}\left(\sup_{t\in[0,T]}|X_{t}|\geq
K\right)\right)$ $\displaystyle\leq
C\left(\varepsilon_{m,K}^{p}+(1+\|X_{0}\|_{2p})^{p}\sqrt{1\wedge\frac{1+\|X_{0}\|_{1}}{K}}\right).$
(4.29)

Plugging this estimation into (4.28), we obtain that when $p\geq 2$ and
$X_{0}\in L^{2p}({\mathbb{P}})$,

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}^{p}_{p}\leq
C\left(m^{-\frac{p}{2}}(1+\|X_{0}\|_{p})^{p}+\varepsilon^{p}_{m,K}+(1+\|X_{0}\|_{2p})^{p}\sqrt{1\wedge\frac{1+\|X_{0}\|_{1}}{K}}\right).$

Going to deterministic initial conditions $x_{0}\in{\mathbb{R}}$ and using
that for $p\in[1,2]$, $\|\cdot\|^{p}_{p}\leq\|\cdot\|^{p}_{2}$, we deduce with
obvious notations, that when $p\geq 1$,

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X^{x_{0}}_{t}-\bar{X}^{m,x_{0}}_{t}|\Big{\|}^{p}_{p}\leq
C\left(m^{-\frac{p}{2}}(1+|x_{0}|)^{p}+\varepsilon^{p}_{m,K}+(1+|x_{0}|)^{p}\left(1\wedge\frac{1+|x_{0}|}{K}\right)^{\frac{1}{2}\wedge\frac{p}{4}}\right).$

By Blagove$\check{\rm s}\check{\rm c}$enkii-Freidlin’s theorem [22, Theorem
13.1, Section V.12-13, p.136], there exists a measurable function
$F:{\mathbb{R}}\times C([0,T],{\mathbb{R}})\to C([0,T],{\mathbb{R}})$
continuous in its first variable such that $X^{x_{0}}=F(x_{0},W)$ and
$X=F(X_{0},W)$. Moreover, by the inductive definition of the Euler scheme,
there also exists a measurable function $F_{m}:{\mathbb{R}}\times
C([0,T],{\mathbb{R}})\to C([0,T],{\mathbb{R}})$ such that
$\bar{X}^{m,x_{0}}=F_{m}(x_{0},W)$ and $\bar{X}^{m}=F_{m}(X_{0},W)$. Therefore
integrating the last inequality in $x_{0}$ with respect to the law of $X_{0}$
and using the independence of $X_{0}$ and $W$, we deduce that when $p\geq 1$
and $X_{0}\in L^{p}({\mathbb{P}})$

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-\bar{X}^{m}_{t}|\Big{\|}_{p}\leq
C\left(m^{-\frac{1}{2}}(1+\|X_{0}\|_{p})+\varepsilon_{m,K}+{\mathbb{E}}^{1/p}\left[(1+|X_{0}|)^{p}\left(1\wedge\frac{1+|X_{0}|}{K}\right)^{\frac{1}{2}\wedge\frac{p}{4}}\right]\right).$

Since, when $X_{0}\in L^{p}({\mathbb{P}})$, the expectation in the right-hand
side goes to $0$ as $K\to+\infty$ by Lebesgue’s theorem, we can choose $K$ so
that the last term in the right-hand side is arbitrarily small and then the
sum of the two first terms goes to $0$ when $m\to+\infty$. The conclusion
follows. $\Box$

###  4.2 Ordering of the Euler schemes with truncated noise

In the next proposition, we focus on orderings of the truncated Euler schemes.
To deal at the same time with the schemes (4.20) and (4.23) we set
$(\widetilde{X}^{m}_{0},\widetilde{X}^{m}_{0})=(X_{0},Y_{0})$ and

$\displaystyle\widetilde{X}^{m}_{t_{k+1}}=\widetilde{X}^{m}_{t_{k}}+\frac{T}{m}\widetilde{b}^{m}_{k}(\widetilde{X}^{m}_{t_{k}})+\sqrt{\frac{T}{m}}\widetilde{\sigma}^{m}_{k}(\widetilde{X}^{m}_{t_{k}})\,Z^{m}_{k+1},$
$\displaystyle\widetilde{Y}^{m}_{t_{k+1}}=\widetilde{Y}^{m}_{t_{k}}+\frac{T}{m}\widetilde{\beta}^{m}_{k}(\widetilde{Y}^{m}_{t_{k}})ds+\sqrt{\frac{T}{m}}\widetilde{\vartheta}^{m}_{k}(\widetilde{Y}^{m}_{t_{k}})\,Z^{m}_{k+1},\;k=0,\ldots,m-1.$
(4.30)

where either

for $k=0,\ldots,m-1$,
$(\widetilde{b}^{m}_{k}(\cdot),\widetilde{\sigma}^{m}_{k}(\cdot),\widetilde{\beta}^{m}_{k}(\cdot),\widetilde{\varsigma}_{k}^{m}(\cdot))$
is equal to

$\left(\frac{m}{T}\int_{t_{k}}^{t_{k+1}}b(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\cdot)ds},\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\beta(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\vartheta^{2}(s,\cdot)ds}\right)$

(schemes (4.20))

or

for $k=0,\ldots,m-1$,
$(\widetilde{b}^{m}_{k}(\cdot),\widetilde{\sigma}^{m}_{k}(\cdot),\widetilde{\beta}^{m}_{k}(\cdot),\widetilde{\varsigma}_{k}^{m}(\cdot))=\left(b(t_{k},\cdot),\sigma(t_{k},\cdot),\beta(t_{k},\cdot),\vartheta(t_{k},\cdot)\right)$
(schemes (4.23)). For $x,y\in{\mathbb{R}}$, we also denote by
$(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}$ and
$(\widetilde{Y}^{m,y}_{t_{k}})_{k=0,\ldots,m}$ these truncated Euler schemes
started from $\widetilde{X}^{m}_{0}=x$ and $\widetilde{Y}^{m}_{0}=y$.

Let

$\displaystyle\mathbf{a}_{\sigma}=\inf\big{\\{}a\geq
0:x\mapsto\sigma^{2}(t,x)+ax^{2}\mbox{ is convex for all }\;t\\!\in[0,T]\\},$
(4.31) $\displaystyle\mathbf{c}_{b}=\inf\big{\\{}c\geq 0:x\mapsto
b(t,x)+cx\mbox{ is non-decreasing for all }\;t\\!\in[0,T]\\},$ (4.32)

and let $\mathbf{a}_{\vartheta}$, $\mathbf{c}_{\beta}$ be defined likewise.
Let also $\mathbf{c}_{\sigma}:=\mathbf{a}_{\sigma}+({\rm Lip}(\sigma))^{2}$,
$\mathbf{m}_{b,\sigma}:=\left(\frac{2\mathbf{c}_{b}}{\sqrt{\mathbf{c}_{\sigma}+2\mathbf{c}_{b}}-\sqrt{\mathbf{c}_{\sigma}}}\right)^{2}T$
and let $\mathbf{c}_{\vartheta},\mathbf{m}_{\beta,\vartheta}$ be defined
likewise.

###### Remark 4.5

The finiteness of $\inf\big{\\{}a\geq 0:x\mapsto f(x)+ax^{2}\big{\\}}$ is
often known as the semi-convexity property of $f:{\mathbb{R}}\to{\mathbb{R}}$.
It has been seemingly introduced in Mathematical Finance in [8] and can also
be found in various papers on the pricing and hedging of American style
options like [13], [2] or [3]. A typical example of semi-convex function is a
differentiable function with Lipschitz differential.

###### Proposition 4.6

Assume that the functions $b$, $\beta$, $\sigma$,
$\vartheta:{\mathbb{R}}\times[0,T]\to{\mathbb{R}}$ satisfy (2.5). Also assume

$(\mathcal{X})\equiv\left\\{\begin{array}[]{ll}(i)&\mathbf{a}_{\sigma}<+\infty\\\
(ii)&\forall\,t\\!\in[0,T],\;b(t,\cdot)\mbox{
convex}\end{array}\right.\quad\mbox{ or
}\quad(\mathcal{Y})\equiv\left\\{\begin{array}[]{ll}(i)&\mathbf{a}_{\vartheta}<+\infty\\\
(ii)&\forall\,t\\!\in[0,T],\;\beta(t,\cdot)\mbox{ convex}.\end{array}\right.$

We consider the Euler schemes $\widetilde{X}^{m}$ and $\widetilde{Y}^{m}$
given in (4.30) starting respectively from $X_{0}$ and $Y_{0}\\!\in
L^{1}({\mathbb{P}})$, with $m\in{\mathbb{N}}$, $m\geq{\mathbf{m}}_{b,\sigma}$
and truncated noise at level $\mathbf{s}_{m}\in\left[0,\frac{\sqrt{m}}{2{\rm
Lip}(\sigma)\sqrt{T}}\right]$ if $(\mathcal{X})$ holds or with
$m\in{\mathbb{N}}$, $m\geq{\mathbf{m}}_{b,\sigma}$ and truncated noise at
level $\mathbf{s}_{m}\in\left[0,\frac{\sqrt{m}}{2{\rm
Lip}(\vartheta)\sqrt{T}}\right]$ if $(\mathcal{Y})$ holds.

$(a)$ Non-decreasing directional convex ordering. We have

$\displaystyle\Big{(}b\leq\beta\mbox{ and }0\leq\sigma\leq\vartheta$
$\displaystyle\mbox{ on }[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{icv}Y_{0}\Big{)}\Longrightarrow(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\leq_{diricv}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}.$

Moreover, when $(\mathcal{X})$ (resp. $(\mathcal{Y})$) holds, then for each
directionally convex function $f:{\mathbb{R}}^{m+1}\to{\mathbb{R}}$ non-
decreasing in each of its variables,

$x\mapsto{\mathbb{E}}\,f((\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m})\in{\mathbb{R}}\cup\\{+\infty\\}\left(\mbox{resp.
}y\mapsto{\mathbb{E}}\,f((\widetilde{Y}^{m,y}_{t_{k}})_{k=0,\ldots,m})\in{\mathbb{R}}\cup\\{+\infty\\}\right)$

is non-decreasing and convex.

$(b)$ Directional convex ordering. Let $b(t,x)=\beta(t,x)=\lambda(t)+\mu(t)x$,
$(t,x)\\!\in[0,T]\times{\mathbb{R}}$ be a time dependent affine function in
space where $\lambda$ and $\mu$ are bounded Borel functions. Then

$\displaystyle\Big{(}0\leq\sigma\leq\vartheta$ $\displaystyle\mbox{ on
}[0,T]\times{\mathbb{R}}\mbox{ and
}X_{0}\preceq_{cvx}Y_{0}\Big{)}\Longrightarrow(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\leq_{dircvx}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}.$

Moreover, when $(\mathcal{X})$ (resp. $(\mathcal{Y})$) holds, then for each
directionally convex function $f:{\mathbb{R}}^{m+1}\to{\mathbb{R}}$,

$x\mapsto{\mathbb{E}}\,f((\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m})\in{\mathbb{R}}\cup\\{+\infty\\}\left(\mbox{resp.
}y\mapsto{\mathbb{E}}\,f((\widetilde{Y}^{m,y}_{t_{k}})_{k=0,\ldots,m})\in{\mathbb{R}}\cup\\{+\infty\\}\right)$

is convex.

###### Remark 4.7

In view of Property P3 in Remark 3.5, $(b)$ (resp. $(a)$) implies that when
$(\mathcal{X})$ holds and $b$ is affine (resp. convex) in space, then
$x\mapsto{\mathbb{E}}g(\widetilde{X}^{m,x}_{T})$ is convex (resp. non-
decreasing and convex) when $g:{\mathbb{R}}\to{\mathbb{R}}$ is convex (resp.
non-decreasing and convex).

The proof of Proposition 4.6 relies on the next results which deals with
directional convexity propagation by a truncated Euler operator.

###### Proposition 4.8

Assume that

  * •

the function $\beta:{\mathbb{R}}\to{\mathbb{R}}$ is convex and such that

$c_{\beta}:=\inf\big{\\{}c\geq 0:\;{\mathbb{R}}\ni x\mapsto\beta(x)+cx\mbox{
is non-decreasing}\big{\\}}<+\infty.$ (4.33)

  * •

the function $\varsigma:{\mathbb{R}}\to{\mathbb{R}}_{+}$ is Lipschitz
continuous with constant ${\rm Lip}(\varsigma)$ and such that

$a_{\varsigma}:=\inf\big{\\{}a\geq 0:x\mapsto\varsigma^{2}(x)+ax^{2}\hbox{ is
convex}\big{\\}}<+\infty,$ (4.34)

and let $c_{\varsigma}:=a_{\varsigma}+({\rm Lip}(\varsigma))^{2}$.

For $h\in{\mathbb{R}}_{+}^{*}$, let
$\mathcal{E}^{h}:{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}$ be the
function defined by

$\mathcal{E}^{h}(x,z):=x+\sqrt{h}\,\varsigma(x)z+h\beta(x),$

and $Z^{s}={\mathbf{1}}_{\\{G\in[-s,s]\\}}G$ with $G\sim{\cal N}_{1}(0,1)$ and
$s\in\left[0,\frac{1}{2{\rm Lip}(\varsigma)\sqrt{h}}\right]$. Then, the
following holds:

$(i)$ when $h\leq\frac{1}{2c_{\beta}}$, the random function
$\mathcal{E}^{h}(\cdot,Z^{s})$ is non-decreasing when the codomain is endowed
with the stochastic order,

$(ii)$ Let
$h\leq\bar{h}_{\beta,\varsigma}:=\left(\frac{\sqrt{c_{\varsigma}+2c_{\beta}}-\sqrt{c_{\varsigma}}}{2c_{\beta}}\right)^{2}$,
$k\in{\mathbb{N}}$ and $f:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$.

If $f$ is Lipschitz continuous, directionally convex and non-decreasing in
each of its variables, then
${\mathbb{R}}^{k}\times{\mathbb{R}}\ni(y,x)\mapsto{\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}(x,Z^{s})\big{)}$
is Lipschitz continuous, directionally convex and non-decreasing in each of
its variables.

If $\beta$ is affine and $f$ is Lipschitz continuous and directionally convex,
then
${\mathbb{R}}^{k}\times{\mathbb{R}}\ni(y,x)\mapsto{\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}(x,Z^{s})\big{)}$
is Lipschitz continuous and directionally convex.

We use the conventions $\frac{1}{0}=+\infty$ in particular in the definition
of the interval where the threshold $s$ can take its values and
$\frac{\sqrt{c_{\varsigma}+2c_{\beta}}-\sqrt{c_{\varsigma}}}{2c_{\beta}}$ is
conventionally equal to $\lim_{x\to
0+}\frac{\sqrt{c_{\varsigma}+x}-\sqrt{c_{\varsigma}}}{x}=\frac{1}{2\sqrt{c_{\varsigma}}}$
when $c_{\beta}=0$.

###### Remark 4.9

When $Z$ is centered and such that ${\rm Var}(Z)\in(0,+\infty)$, a necessary
condition for $\mathcal{E}^{h}(\cdot,Z)$ to be convex when the codomain is
endowed with the convex order is

$\displaystyle\forall x,y\in{\mathbb{R}},\;\forall\alpha\in[0,1],\;$
$\displaystyle{\mathbb{E}}\,\mathcal{E}^{h}(\alpha
x+(1-\alpha)y,Z)=\alpha{\mathbb{E}}\,\mathcal{E}^{h}(x,Z)+(1-\alpha){\mathbb{E}}\,\mathcal{E}^{h}(y,Z)$
$\displaystyle\mbox{ and }{\mathbb{E}}\,[\mathcal{E}^{h}(\alpha
x+(1-\alpha)y,Z)^{2}]\leq\alpha{\mathbb{E}}\,[\mathcal{E}^{h}(x,Z)^{2}]+(1-\alpha){\mathbb{E}}\,[\mathcal{E}^{h}(y,Z)^{2}]$

which also writes $\beta(\alpha
x+(1-\alpha)y)=\alpha\beta(x)+(1-\alpha)\beta(y)$ and

$\varsigma^{2}(\alpha x+(1-\alpha)y)+\frac{(\alpha x+(1-\alpha)y)^{2}}{h{\rm
Var}(Z)}\leq\alpha\left(\varsigma^{2}(x)+\frac{x^{2}}{h{\rm
Var}(Z)}\right)+(1-\alpha)\left(\varsigma^{2}(y)+\frac{y^{2}}{h{\rm
Var}(Z)}\right)$

so that $\beta$ is affine and $a_{\varsigma}\leq\frac{1}{h{\rm Var}(Z)}$.

###### Remark 4.10

Let $Q(x,dy)$ denote the distribution of $x+\sqrt{h}\varsigma(x)Z^{s}$ with
$s\in\left[0,\frac{1}{2{\rm Lip}(\varsigma)\sqrt{h}}\right]$. When
$h\leq\frac{1}{4c_{\varsigma}}$, the Markov kernel $Q$ is such that when
$\varphi:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is directionally convex, then
$x\mapsto{\mathbb{E}}[\varphi(x,x+\sqrt{h}\varsigma(x)Z^{s})]=\int_{\mathbb{R}}\varphi(x,y)Q(x,dy)$
is convex by Proposition 4.8 applied with $k=0$ and $\beta\equiv 0$. But this
convexity property may fail when $\varphi$ is convex .

Indeed, restricted to the class of functions $\varphi(x,y)=f(y-x)$ with convex
$f:{\mathbb{R}}\to{\mathbb{R}}$ for which
${\mathbb{E}}[\varphi(x,x+\sqrt{h}\varsigma(x)Z^{s})]={\mathbb{E}}[f(\sqrt{h}\varsigma(x)Z^{s})]$,
the convexity preservation would imply that

$x\mapsto\,{{\mathbb{E}}\,f(\sqrt{h}\varsigma(x)Z^{s})}\quad\mbox{ is
convex.}$

Setting $f(x)=|x|$ (which is also the choice made in the proof of Proposition
3.3) implies that

$x\mapsto\sqrt{h}\,{|\varsigma(x)|\,{\mathbb{E}}\,|Z^{s}|}\quad\mbox{ is
convex.}$

When ${\mathbb{E}}\,|Z^{s}|>0$, this implies that $\varsigma=|\varsigma|$ is
convex.

As a consequence, the proof of [5, Proposition 3.1], restated in [6, Lemma 1]
and in [7, p54-55], is not valid when ${\cal F}$ is the class of convex
functions.

Proof of Proposition 4.6. For $k=0,\ldots,m-1$, and
$f_{k+1}:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$ Lipschitz continuous, we set

$\displaystyle{\mathbb{R}}^{k+1}\ni
x_{0:k}=(x_{0},\ldots,x_{k})\mapsto\widetilde{P}_{k}^{(b,\sigma),m}f_{k+1}(x_{0:k})$
$\displaystyle={\mathbb{E}}\,f_{k+1}\left(x_{0:k},x_{k}+\frac{T}{m}\widetilde{b}^{m}_{k}(x_{k})+\sqrt{\frac{T}{m}}\widetilde{\sigma}^{m}_{k}(x_{k})\,Z^{m}_{k+1}\right)$
$\displaystyle{\mathbb{R}}^{k+1}\ni
x_{0:k}\mapsto\widetilde{P}_{k}^{(\beta,\vartheta),m}f_{k+1}(x_{0:k})$
$\displaystyle={\mathbb{E}}\,f_{k+1}\left(x_{0:k},x_{k}+\frac{T}{m}\widetilde{\beta}^{m}_{k}(x_{k})+\sqrt{\frac{T}{m}}\widetilde{\vartheta}^{m}_{k}(x_{k})\,Z^{m}_{k+1}\right).$

Since $b,\sigma,\beta,\vartheta$ satisfy (2.5),
$\widetilde{P}_{k}^{(b,\sigma),m}f_{k+1}$ and
$\widetilde{P}_{k}^{(\beta,\vartheta),m}f_{k+1}$ are Lipschitz continuous.

Still for $k=0,\ldots,m-1$, we set
$\widetilde{P}_{k:m-1}^{(b,\sigma),m}=\widetilde{P}^{(b,\sigma),m}_{k}\circ\cdots\circ\widetilde{P}_{m-1}^{(b,\sigma),m}$
and
$\widetilde{P}_{k:m-1}^{(\beta,\vartheta),m}=\widetilde{P}^{(\beta,\vartheta),m}_{k}\circ\cdots\circ\widetilde{P}_{m-1}^{(\beta,\vartheta),m}$.
By the Markov property, for $f:{\mathbb{R}}^{m+1}\to{\mathbb{R}}$ Lipschitz
continuous, we have

$\displaystyle\forall x\in{\mathbb{R}},\;$
$\displaystyle{\mathbb{E}}\,f\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}=\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(x)\mbox{
and
}{\mathbb{E}}\,f\big{(}(\widetilde{Y}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}=\widetilde{P}^{(\beta,\vartheta),m}_{0:m-1}f(x),$
$\displaystyle{\mathbb{E}}\,f\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}={\mathbb{E}}\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(X_{0})\mbox{
and
}{\mathbb{E}}\,f\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}={\mathbb{E}}\widetilde{P}^{(\beta,\vartheta),m}_{0:m-1}f(Y_{0}).$

Let $f:{\mathbb{R}}^{m+1}\to{\mathbb{R}}$ be Lipschitz continuous, non-
decreasing in each of its variables and directionally convex. We assume
$(\mathcal{X})$, that $m\geq{\mathbf{m}}_{b,\sigma}$ and
$\mathbf{s}_{m}\in\left[0,\frac{\sqrt{m}}{2{\rm Lip}(\sigma)\sqrt{T}}\right]$.
Then for each $k=0,\ldots,m-1$,
$a_{\widetilde{\sigma}^{m}_{k}}\leq\mathbf{a}_{\sigma}$, ${\rm
Lip}(\widetilde{\sigma}^{m}_{k})\leq{\rm Lip}(\sigma)$ so that
$c_{\widetilde{\sigma}^{m}_{k}}\leq\mathbf{c}_{\sigma}$ and
$c_{\widetilde{b}^{m}_{k}}\leq\mathbf{c}_{b}$. Since
${\mathbb{R}}^{*}_{+}\times{\mathbb{R}}^{*}_{+}\ni(z,w)\mapsto\frac{\sqrt{z+w}-\sqrt{z}}{w}$
is non-increasing in each variable,
$\left(\frac{\sqrt{c_{\widetilde{\sigma}^{m}_{k}}+2c_{\widetilde{b}^{m}_{k}}}-\sqrt{c_{\widetilde{\sigma}^{m}_{k}}}}{2c_{\widetilde{b}^{m}_{k}}}\right)^{2}\geq\left(\frac{\sqrt{\mathbf{c}_{\sigma}+2\mathbf{c}_{b}}-\sqrt{\mathbf{c}_{\sigma}}}{2\mathbf{c}_{b}}\right)^{2}$
and
$h=\frac{T}{m}\leq\bar{h}_{\widetilde{b}^{m}_{k},\widetilde{\sigma}^{m}_{k}}$.
Combining Proposition 4.8 applied with
$(\beta,\varsigma)=(\widetilde{b}^{m}_{k},\widetilde{\sigma}^{m}_{k})$,
$h=\frac{T}{m}$, $s=\mathbf{s}_{m}\in\left[0,\frac{1}{2{\rm
Lip}(\widetilde{\sigma}^{m}_{k})\sqrt{h}}\right]$ with an obvious backward
inductive reasoning, we check that for all $k=0,\ldots,m-1$,
$\widetilde{P}_{k:m-1}^{(b,\sigma),m}f$ is Lipschiz continuous, non-decreasing
in each of its variables and directionally convex. In particular,
${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}\,f\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}=\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(x)$
is non-decreasing and convex. We now proceed by a backward induction on $k$ to
show that
$\widetilde{P}^{(b,\sigma),m}_{k:m-1}f\leq\widetilde{P}_{k:m-1}^{(\beta,\vartheta),m}f$
for $k=0\ldots,m$, where, by convention,
$\widetilde{P}^{(b,\sigma),m}_{m:m-1}f=\widetilde{P}^{(\beta,\vartheta),m}_{m:m-1}f=f$
so that the inequality holds for $k=m$. Let for
$f_{k+1}:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$ Lipschitz continuous and
$(x_{0:k},u,v)\in{\mathbb{R}}^{k+1}\times{\mathbb{R}}\times{\mathbb{R}}$,
$Q_{k}f_{k+1}(x_{0:k},u,v)={\mathbb{E}}\,f_{k+1}\left(x_{0:k},x_{k}+\frac{T}{m}u+v\sqrt{\frac{T}{m}}Z^{m}_{k+1}\right)$.

Assume that
$\widetilde{P}^{(b,\sigma),m}_{k+1:m-1}f\leq\widetilde{P}_{k+1:m-1}^{(\beta,\vartheta),m}f$
for some $k\in\\{0,\ldots,m-1\\}$. Then, since
$\widetilde{P}^{(b,\sigma)}_{k+1:m-1}f$ is non-decreasing in its last
variable, $\forall(x_{0:k},v)\in{\mathbb{R}}^{k+1}\times{\mathbb{R}}$,
${\mathbb{R}}\ni u\mapsto
Q_{k}\widetilde{P}^{(b,\sigma)}_{k+1:m-1}f(x_{0:k},u,v)$ is non-decreasing.
Moreover, since $\widetilde{P}^{(b,\sigma)}_{k+1:m-1}f$ is convex in its last
variable, $\forall(x_{0:k},u)\in{\mathbb{R}}^{k+1}\times{\mathbb{R}}$,
${\mathbb{R}}\ni v\mapsto
Q_{k}\widetilde{P}^{(b,\sigma)}_{k+1:m-1}f(x_{0:k},u,v)$ is convex and, by
Jensen’s inequality, minimal for $v=0$, so that ${\mathbb{R}}_{+}\ni v\mapsto
Q_{k}\widetilde{P}^{(b,\sigma)}_{k+1:m-1}f(x_{0:k},u,v)$ is non-decreasing.
Combined with
$\widetilde{b}^{m}_{k}(x_{k})\leq\widetilde{\beta}^{m}_{k}(x_{k})$ and
$0\leq\widetilde{\sigma}^{m}_{k}(x_{k})\leq\widetilde{\vartheta}^{m}_{k}(x_{k})$,
this ensures that for $x_{0:k}\in{\mathbb{R}}^{k+1}$,

$\displaystyle\widetilde{P}^{(b,\sigma),m}_{k:m-1}f(x_{0:k})=Q_{k}\widetilde{P}^{(b,\sigma),m}_{k+1:m-1}f\big{(}x_{0:k},\widetilde{b}^{m}_{k}(x_{k}),\widetilde{\sigma}^{m}_{k}(x_{k})\big{)}$
$\displaystyle\leq
Q_{k}\widetilde{P}^{(b,\sigma),m}_{k+1:m-1}f\big{(}x_{0:k},\widetilde{\beta}^{m}_{k}(x_{k}),\widetilde{\vartheta}^{m}_{k}(x_{k})\big{)}.$

With the positivity of $Q_{k}$ as a linear operator on functions and the
induction hypothesis, we deduce that

$\widetilde{P}^{(b,\sigma),m}_{k:m-1}f(x_{0:k})\leq
Q_{k}\widetilde{P}_{k+1:m-1}^{(\beta,\vartheta),m}f\big{(}x_{0:k},\widetilde{\beta}^{m}_{k}(x_{k}),\widetilde{\vartheta}^{m}_{k}(x_{k})\big{)}=\widetilde{P}^{(\beta,\vartheta),m}_{k:m-1}f(x_{0:k}),$

which concludes the proof by induction. The monotonicity and convexity of
$\widetilde{P}^{(b,\sigma),m}_{0:m-1}f$ combined with
$X_{0}\preceq_{icv}Y_{0}$ and the previous inequality for $k=0$ then imply
that

${\mathbb{E}}\,f\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}={\mathbb{E}}\,\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(X_{0})\leq{\mathbb{E}}\,\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(Y_{0})\leq{\mathbb{E}}\,\widetilde{P}^{(\beta,\vartheta),m}_{0:m-1}f(Y_{0})={\mathbb{E}}\,f\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}.$

When $\big{(}{\cal Y}\big{)}$ holds and $m\geq{\mathbf{m}}_{\beta,\vartheta}$,
$\mathbf{s}_{m}\in\left[0,\frac{\sqrt{m}}{2{\rm
Lip}(\vartheta)\sqrt{T}}\right]$, then for each $k=0,\ldots,m-1$,
$x_{0:k}\mapsto P^{(\beta,\vartheta),m}_{k:m-1}f(x_{0:k})$ is non-decreasing
and convex by Proposition 4.8 and backward induction and we use

$\displaystyle
Q_{k}\widetilde{P}^{(b,\sigma),m}_{k+1:m-1}f\big{(}x_{0:k},\widetilde{b}^{m}_{k}(x_{k}),\widetilde{\sigma}^{m}_{k}(x_{k})\big{)}$
$\displaystyle\leq
Q_{k}\widetilde{P}^{(\beta,\vartheta),m}_{k+1:m-1}f\big{(}x_{0:k},\widetilde{b}^{m}_{k}(x_{k}),\widetilde{\sigma}^{m}_{k}(x_{k})\big{)}$
$\displaystyle\leq
Q_{k}\widetilde{P}_{k+1:m-1}^{(\beta,\vartheta),m}f\big{(}x_{0:k},\widetilde{\beta}^{m}_{k}(x_{k}),\widetilde{\vartheta}^{m}_{k}(x_{k})\big{)}$

in the backward inductive proof of
$\widetilde{P}^{(b,\sigma),m}_{k:m-1}f\leq\widetilde{P}_{k:m-1}^{(\beta,\vartheta),m}f$
for $k=0,\ldots,m-1$. We also use
${\mathbb{E}}\,\widetilde{P}^{(b,\sigma),m}_{0:m-1}f(X_{0})\leq{\mathbb{E}}\,\widetilde{P}^{(\beta,\vartheta),m}_{0:m-1}f(X_{0})\leq{\mathbb{E}}\,\widetilde{P}^{(\beta,\vartheta),m}_{0:m-1}f(Y_{0})$
to conclude that

${\mathbb{E}}\,f\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\leq{\mathbb{E}}\,f\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}.$

With Lemma 3.7, we deduce that
$(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\leq_{diricv}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}$.

Let us now check that mere directional convexity and monotonicity in each
variable of $f$ (without Lipschitz continuity) is enough to ensure that
$x\mapsto{\mathbb{E}}\,f\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}$
is non-decreasing and convex. By Lemma 3.7, there exists a non-decreasing
sequence of Lipschitz continuous, non-decreasing in each variable and
directionally convex functions $(f_{n})_{n\geq 1}$ converging pointwise to
$f$. By Beppo-Levi’s theorem,
${\mathbb{E}}\,\big{[}(f_{n}-f_{1})\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{]}$
converges to
${\mathbb{E}}\,\big{[}(f-f_{1})\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{]}$
possibly equal to $+\infty$ as $n\to+\infty$. Since $f_{1}$ is Lipschitz
continuous,
${\mathbb{E}}\,\big{|}f_{1}\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{|}<\infty$
and we deduce that
$x\mapsto{\mathbb{E}}\,f_{n}\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}$
converges pointwise to
$x\mapsto{\mathbb{E}}\,f\big{(}(\widetilde{X}^{m,x}_{t_{k}})_{k=0,\ldots,m}\big{)}$
as $n\to+\infty$ so that the non-decreasing convex property of the former
functions is transferred to the latter.

The proof is analogous for the convex ordering using that the affine property
of the drift coefficients enables backward propagation of directional
convexity and their equality permits to freeze
$u=\widetilde{b}^{m}_{k}(x_{k})=\widetilde{\beta}^{m}_{k}(x_{k})$ in
$Q_{k}f_{k+1}(x_{0:k},u,v)$. $\Box$

Proof of Proposition 4.8. Step 1. (Preliminaries).

Let $\rho$ be a $C^{\infty}$ probability density on the real line with compact
support such that

$\int_{{\mathbb{R}}}u\rho(u)du=0\mbox{ and }\int u^{2}\rho(u)du=1.$ (4.35)

We associate to $\rho$ its sequence of mollifiers $\rho_{n}(x)=n\rho(nx)$,
$n\in{\mathbb{N}}$.

We set, for every $n\geq 1$,

$\varsigma_{n}(x)=\sqrt{\frac{1}{n}+\rho_{n}\star\varsigma^{2}(x)}\quad\mbox{
and }\quad\beta_{n}=\rho_{n}\star\beta.$

The convex function $\beta$ being continuous, the function $\beta_{n}$ is
infinitely differentiable, convex and such that
$x\mapsto\beta_{n}(x)+c_{\beta}x$ is non-decreasing, so that

$\beta^{\prime}_{n}\geq-c_{\beta}.$ (4.36)

Furthermore, $\beta_{n}$ converges pointwise to $\beta$ as $n\to+\infty$. By
continuity of $\varsigma^{2}$, $\rho_{n}\star\varsigma^{2}$ converges to
$\varsigma^{2}$ so that $\varsigma_{n}$ also converges pointwise to
$\varsigma$. As a consequence,

$\mathcal{E}^{h}_{n}(x,Z^{s}):=x+\sqrt{h}\,\varsigma_{n}(x)Z^{s}+h\beta_{n}(x)\to\mathcal{E}^{h}(x,Z^{s})\quad\mbox{
as }\quad n\to+\infty.$

Moreover

$\big{|}\mathcal{E}^{h}_{n}(x,Z^{s})-\mathcal{E}^{h}(x,Z^{s})\big{|}\leq\sqrt{h}|\varsigma_{n}(x)-\varsigma(x)||Z|+h|\beta_{n}(x)-\beta(x)|\to
0\mbox{ as }n\to+\infty.$

For the non-decreasing stochastic order, this ensures that the non-decreasing
property of $x\mapsto\mathcal{E}^{h}_{n}(x,Z^{s})$ that we are going to check
for each $n\in{\mathbb{N}}$ is transferred to
$x\mapsto\mathcal{E}^{h}(x,Z^{s})$. This also ensures that for a Lipschitz
function $f:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$ and
$(y,x)\in{\mathbb{R}}^{k}\times{\mathbb{R}}$, one has
${\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}\to{\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}(x,Z^{s})\big{)}$
as $n\to+\infty$. Therefore it is enough to prove that the conclusion holds
for ${\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}$ for each
$n\in{\mathbb{N}}$, which we are also going to do.

Since, by Jensen’s inequality
$\varsigma\star\rho_{n}\leq\sqrt{\varsigma^{2}\star\rho_{n}}\leq\varsigma_{n}$,
we obtain that for $x,x^{\prime}\in{\mathbb{R}}$,

$\displaystyle|\varsigma^{2}_{n}(x)-\varsigma^{2}_{n}(x^{\prime})|$
$\displaystyle\leq\int_{\mathbb{R}}|\varsigma(x-z)-\varsigma(x^{\prime}-z)|(\varsigma(x-z)+\varsigma(x^{\prime}-z))\rho_{n}(z)dz$
$\displaystyle\leq{\rm
Lip}(\varsigma)|x-x^{\prime}|\left(\varsigma\star\rho_{n}(x)+\varsigma\star\rho_{n}(x^{\prime})\right)\leq{\rm
Lip}(\varsigma)|x-x^{\prime}|(\varsigma_{n}(x)+\varsigma_{n}(x^{\prime})).$

Hence the function $\varsigma_{n}$ is Lipschitz continuous with constant ${\rm
Lip}(\varsigma)$. Moreover, since the Lipschitz function $\varsigma$ has at
most affine growth, the non-negative function $\rho_{n}\star\varsigma^{2}$ is
infinitely differentiable and so is $\varsigma_{n}$. The Lipschitz property of
$\varsigma_{n}$ implies that

$|\varsigma_{n}^{\prime}|\leq{\rm Lip}(\varsigma).$ (4.37)

Moreover, it is clear that $\varsigma(x)^{2}+a_{\varsigma}x^{2}$ is convex
since the infimum holds as a minimum. Hence its convolution by $\rho_{n}$ is
convex too and one checks that it is infinitely differentiable. On the other
hand, using (4.35), one has

$\int\big{[}\varsigma^{2}(x-y)+a_{\varsigma}(x-y)^{2}\big{]}\rho_{n}(y)dy=\rho_{n}\star\varsigma^{2}(x)+a_{\varsigma}x^{2}+\tfrac{a_{\varsigma}}{n^{2}}=\varsigma^{2}_{n}(x)+a_{\varsigma}x^{2}+\tfrac{a_{\varsigma}}{n^{2}}-\tfrac{1}{n},$

so that $\varsigma^{2}_{n}(x)+a_{\varsigma}x^{2}$ is convex and
$(\varsigma_{n}^{2})^{\prime\prime}\geq-2\,a_{\varsigma}$. As a consequence,

$\displaystyle\forall
x\in{\mathbb{R}},\;\varsigma_{n}\varsigma_{n}^{\prime\prime}(x)$
$\displaystyle=\frac{1}{2}(\varsigma_{n}^{2})^{\prime\prime}(x)-(\varsigma^{\prime}_{n}(x))^{2}\geq-
a_{\varsigma}-({\rm Lip}(\varsigma))^{2}=-c_{\varsigma}.$ (4.38)

Step 2 (Non-decreasing stochastic ordering). When $h\leq\frac{1}{2c_{\beta}}$,
then $h\beta_{n}^{\prime}(x)\geq-\frac{1}{2}$ and, with
$s\in\left[0,\frac{1}{2{\rm Lip}(\varsigma)\sqrt{h}}\right]$ and (4.37), we
deduce that

$\forall
x\in{\mathbb{R}},\;\partial_{x}\mathcal{E}^{h}_{n}(x,Z^{s})=1+\sqrt{h}\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x)\geq
1-\frac{1}{2}-\frac{1}{2}=0.$ (4.39)

Therefore, $x\mapsto\mathcal{E}^{h}_{n}(x,Z^{s})$ is non-decreasing for the
non-decreasing stochastic ordering and
${\mathbb{R}}^{k}\times{\mathbb{R}}\ni(y,x)\mapsto{\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}$
is non-decreasing in each of its variables when
$f:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$ is non-decreasing in each of its
variables and Lipschitz continuous.

Step 3 (Differentiation). Let $f:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$ be
continuously differentiable with bounded first order derivatives and
${\mathbb{R}}^{k}\times{\mathbb{R}}\ni(y,x)\mapsto
g(y,x)={\mathbb{E}}\,f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}$. One has

$\displaystyle\partial_{x}$ $\displaystyle
g(y,x)={\mathbb{E}}\,\partial_{k+1}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))+{\mathbb{E}}\left[\partial_{k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))(1+\sqrt{h}\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))\right].$
(4.40)

And when $f$ is twice continuously differentiable with bounded first and
second order derivatives, then for $i,j\in\\{1,\cdots,k\\}$,

$\displaystyle\partial_{y_{i}y_{j}}g(y,x)={\mathbb{E}}\,\partial_{ij}f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)},$
(4.41)
$\displaystyle\partial_{y_{i}x}g(y,x)={\mathbb{E}}\,\partial_{ik+1}f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}+{\mathbb{E}}\left[\partial_{ik+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))(1+\sqrt{h}\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))\right],$
(4.42) $\displaystyle\partial_{xx}$ $\displaystyle
g(y,x)={\mathbb{E}}\,\partial_{k+1k+1}f\big{(}y,x,\mathcal{E}^{h}_{n}(x,Z^{s})\big{)}$
$\displaystyle+2{\mathbb{E}}\left[\partial_{k+1k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))(1+\sqrt{h}\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))\right]$
$\displaystyle+{\mathbb{E}}\left[\partial_{k+2k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))(1+\sqrt{h}\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))^{2}+\partial_{k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))\sqrt{h}\varsigma_{n}^{\prime\prime}(x)Z^{s}\right]$
$\displaystyle+{\mathbb{E}}\left[\partial_{k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))h\beta_{n}^{\prime\prime}(x)\right].$
(4.43)

Step 4. For a Lipschitz continuous and directionally convex (non-decreasing in
each of its variables) test function $f:{\mathbb{R}}^{k+2}\to{\mathbb{R}}$,
the functions $f_{m}=\rho_{m}\star f$ are Lipschitz continuous and
directionally convex (non-decreasing in each variable), infinitely
differentiable with bounded derivatives and converge to $f$ uniformly as
$m\to+\infty$, so that
${\mathbb{E}}\,f_{m}(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))\to{\mathbb{E}}\,f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))$.
Therefore it is enough to consider twice continuously differentiable test
functions $f$ with bounded derivatives for which we may apply Step 3. When $f$
is non decreasing in each of its variables, then clearly so is $g$ in each of
its $k$ first variables. The function $g$ also is non-decreasing in its last
variable by (4.40) and (4.39). When $f$ is directionally convex, so that its
second order partial derivatives all are non-negative, then the first summands
in the right-hand sides of (4.41), (4.42) and (4.43) are non-negative and the
second summands in those of (4.42) and (4.43) are non-negative by (4.39). The
fourth summand in the right-hand side of (4.43) is non-negative when $\beta$
is convex and $f$ is non-decreasing in its last variable. It is equal to $0$
disregarding the monotonicity of $f$ when $\beta$ is affine since $\beta_{n}$
is affine too and $\beta_{n}^{\prime\prime}$ vanishes. Therefore, to conclude
that $g$ is directionally convex, it is enough to check that the third summand
in the right-hand side of (4.43) is non-negative. To do so, we use the
following equation based on an integration by parts for the second equality

$\displaystyle{\mathbb{E}}\left[\partial_{k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))Z^{s}\right]$
$\displaystyle=\int_{-s}^{s}\partial_{k+2}f(y,x,\mathcal{E}^{h}_{n}(x,z))ze^{-\frac{z^{2}}{2}}\frac{dz}{\sqrt{2\pi}}$
$\displaystyle=\int_{-s}^{s}\partial_{k+2k+2}f(y,x,\mathcal{E}^{h}_{n}(x,z))\sqrt{h}\varsigma_{n}(x)\left(e^{-\frac{z^{2}}{2}}-e^{-\frac{s^{2}}{2}}\right)\frac{dz}{\sqrt{2\pi}}$
$\displaystyle={\mathbb{E}}\left[\partial_{k+2k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))\sqrt{h}\,\varsigma_{n}(x){\mathbf{1}}_{\\{Z^{s}\neq
0\\}}\Big{(}1-e^{\frac{(Z^{s})^{2}-s^{2}}{2}}\Big{)}\right].$

We deduce that the third summand is equal to

$\displaystyle{\mathbb{E}}\bigg{[}\partial_{k+2k+2}f(y,x,\mathcal{E}^{h}_{n}(x,Z^{s}))\\!\\!\left(\\!\\!(1+\sqrt{h}\,\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))^{2}+h{\mathbf{1}}_{\\{Z^{s}\neq
0\\}}\Big{(}1-e^{\frac{(Z^{s})^{2}-s^{2}}{2}}\Big{)}\varsigma_{n}\varsigma_{n}^{\prime\prime}(x)\right)\\!\bigg{]}.$

When $c_{\beta}>0$,
$\frac{\sqrt{c_{\varsigma}+2c_{\beta}}-\sqrt{c_{\varsigma}}}{2c_{\beta}}$ is
the positive root of the quadratic function ${\mathbb{R}}\ni y\mapsto
c_{\beta}y^{2}+\sqrt{c_{\varsigma}}y-\frac{1}{2}$. As a consequence, when
$h\leq\bar{h}_{\beta,\vartheta}$,
$\frac{1}{2}-hc_{\beta}\geq\sqrt{hc_{\varsigma}}$ so that with
$s\in\left[0,\frac{1}{2{\rm Lip}(\varsigma)\sqrt{h}}\right]$, (4.36), (4.37)
and (4.38),

$\forall
x\in{\mathbb{R}},\;(1+\sqrt{h}\,\varsigma_{n}^{\prime}(x)Z^{s}+h\beta_{n}^{\prime}(x))^{2}+h{\mathbf{1}}_{\\{Z^{s}\neq
0\\}}\Big{(}1-e^{\frac{(Z^{s})^{2}-s^{2}}{2}}\Big{)}\varsigma_{n}\varsigma_{n}^{\prime\prime}(x)\geq\left(\frac{1}{2}-hc_{\beta}\right)^{2}-hc_{\varsigma}\geq
0.$

This concludes the proof. $\Box$

##  5 Proofs of the main results

The next result is the key to get rid at the level of SDEs of the spatial
semi-convexity of the square of one of the diffusion coefficients necessary
for the comparison at the level of the truncated Euler schemes.

###### Proposition 5.1

Assume that $b,\sigma,\beta,\vartheta$ satisfy (2.5) and
$0\leq\sigma\leq\vartheta$. Then there is a sequence
$(\sigma_{n},\vartheta_{n})_{n\geq 1}$ of approximations of
$(\sigma,\vartheta)$ such that
$\sigma_{n},\vartheta_{n}:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ are
Lipschitz continuous in space uniformly in time and $n$,
$\mathbf{a}_{\sigma_{n}}\wedge\mathbf{a}_{\vartheta_{n}}>-\infty$ and
$0\leq\sigma_{n}\leq\vartheta_{n}$ for each $n\geq 1$, and when
$X_{0},Y_{0}\in L^{p}({\mathbb{P}})$ for some $p\in[1,+\infty)$ the solutions
$(X^{n},Y^{n})$ to

$X^{n}_{t}=X_{0}+\int_{0}^{t}b(s,X^{n}_{s})ds+\int_{0}^{t}\sigma_{n}(s,X^{n}_{s})dW_{s}\mbox{
and
}Y^{n}_{t}=X_{0}+\int_{0}^{t}\beta(s,Y^{n}_{s})ds+\int_{0}^{t}\vartheta_{n}(s,Y^{n}_{s})dW_{s},$

satisfy
$\lim_{n\to+\infty}\Big{\|}\sup_{t\in[0,T]}|X_{t}-X^{n}_{t}|\Big{\|}_{p}=\lim_{n\to+\infty}\Big{\|}\sup_{t\in[0,T]}|Y_{t}-Y^{n}_{t}|\Big{\|}_{p}=0$.

Proof of Proposition 5.1. Let $\rho$ be a $C^{\infty}$ probability density on
the real line compactly supported on $[-1,1]$. We associate to $\rho$ its
sequence of mollifiers $\rho_{n}(x)=n\rho(nx)$, $n\in{\mathbb{N}}$. For $n\geq
1$ and $t\in[0,T]$, we set

$\sigma_{n}(t,x)=\begin{cases}\rho_{n}\star\sigma(t,\cdot)(x)\mbox{ if
}x\in[-n,n]\\\
\rho_{n}\star\sigma(t,\cdot)(n)+\left(0\vee\partial_{x}\rho_{n}\star\sigma(t,\cdot)(n)\right)(x-n)\mbox{
if }x\geq n\\\
\rho_{n}\star\sigma(t,\cdot)(-n)+\left(0\wedge\partial_{x}\rho_{n}\star\sigma(t,\cdot)(-n)\right)(x+n)\mbox{
if }x\leq-n\end{cases},$

and

$\vartheta_{n}(t,x)=\begin{cases}\rho_{n}\star\vartheta(t,\cdot)(x)\mbox{ if
}x\in[-n,n]\\\
\rho_{n}\star\vartheta(t,\cdot)(n)+\left(0\vee\partial_{x}\rho_{n}\star\sigma(t,\cdot)(n)\vee\partial_{x}\rho_{n}\star\vartheta(t,\cdot)(n)\right)(x-n)\mbox{
if }x\geq n\\\
\rho_{n}\star\vartheta(t,\cdot)(-n)+\left(0\wedge\partial_{x}\rho_{n}\star\sigma(t,\cdot)(-n)\wedge\partial_{x}\rho_{n}\star\vartheta(t,\cdot)(-n)\right)(x+n)\mbox{
if }x\leq-n\end{cases}.$

The inequality $0\leq\sigma\leq\vartheta$ clearly implies that
$0\leq\sigma_{n}\leq\vartheta_{n}$ on $[0,T]\times[-n,n]$ and, with the above
choice of slopes, $0\leq\sigma_{n}\leq\vartheta_{n}$ on
$[0,T]\times{\mathbb{R}}$. For each $n\geq 1$ and $t\in[0,T]$, the function
$x\mapsto\rho_{n}\star\sigma(t,\cdot)(x)$ (resp.
$x\mapsto\rho_{n}\star\vartheta(t,\cdot)(x)$) is Lipschitz continuous with
constant ${\rm Lip}(\sigma)$ (resp. ${\rm Lip}(\vartheta)$) so that its
spatial derivative is bounded by ${\rm Lip}(\sigma)$ (resp. ${\rm
Lip}(\vartheta)$) and $x\mapsto\sigma_{n}(t,x)$ (resp.
$x\mapsto\vartheta_{n}(t,x)$) is Lipschitz continuous with constant ${\rm
Lip}(\sigma)$ (resp. ${\rm Lip}(\sigma)\vee{\rm Lip}(\vartheta)$). Moreover
since $\sigma$ satisfies (2.5) and $\rho_{n}$ and $\rho^{\prime\prime}_{n}$
are compactly supported and bounded,

$\inf_{(t,x)\in[0,T]\times[-n,n]}\partial^{2}_{xx}(\rho_{n}\star\sigma(t,\cdot))^{2}(x)\geq
2\inf_{(t,x)\in[0,T]\times[-n,n]}(\rho_{n}^{\prime\prime}\star\sigma(t,\cdot))(\rho_{n}\star\sigma(t,\cdot))(x)>-\infty.$

Since for each $t\in[0,T]$, the differences

$2\rho_{n}\star\sigma(t,\cdot)(n)\left(\partial_{x}\rho_{n}\star\sigma(t,\cdot)(n)\right)^{-}\quad\mbox{
and }\quad
2\rho_{n}\star\sigma(t,\cdot)(-n)\left(\partial_{x}\rho_{n}\star\sigma(t,\cdot)(-n)\right)^{+}$

between the right-hand slope and the left hand-slope of
$x\mapsto\sigma_{n}^{2}(t,x)$ at point $x=n$ and $x=-n$ respectively, is non-
negative and

$\forall
x\in(-\infty,-n)\cup(n,+\infty),\;\partial_{xx}\sigma^{2}_{n}(t,x)\in\left\\{2\left(\partial_{x}\rho_{n}\star\sigma(t,\cdot)(n)\right)^{2},0,2\left(\partial_{x}\rho_{n}\star\sigma(t,\cdot)(-n)\right)^{2}\right\\},$

we deduce that $\mathbf{a}_{\sigma_{n}}>-\infty$. We check in the same way
that $\mathbf{a}_{\vartheta_{n}}>-\infty$. Last, since $\rho$ is compactly
supported on $[-1,1]$,

$\sup_{(t,x)\in[0,T]\times[-n,n]}|\sigma_{n}(t,x)-\sigma(t,x)|\leq\frac{{\rm
Lip}(\sigma)}{n}\mbox{ and
}\sup_{(t,x)\in[0,T]\times[-n,n]}|\vartheta_{n}(t,x)-\vartheta(t,x)|\leq\frac{{\rm
Lip}(\vartheta)}{n}.$ (5.44)

Let us check that
$\lim_{n\to+\infty}\Big{\|}\sup_{t\in[0,T]}|X_{t}-X^{n}_{t}|\Big{\|}_{p}=0$
when $X_{0}\in L^{p}({\mathbb{P}})$, the convergence of $Y^{n}$ to $Y$
following the same argument. We have

$X_{t}-X^{n}_{t}=\int_{0}^{t}(b(s,X_{s})-b(s,X^{n}_{s}))ds+\int_{0}^{t}(\sigma_{n}(s,X_{s})-\sigma_{n}(s,X^{n}_{s}))dW_{s}+\int_{0}^{t}(\sigma(s,X_{s})-\sigma_{n}(s,X_{s}))dW_{s}.$

Thanks to the uniform in time and $n$ spatial Lipschitz continuity of the
coefficients $(b,\sigma_{n})$, standard estimations based on the Hölder and
Burkholder-Davis-Gundy inequalities and Gronwall’s lemma ensure for $p\geq 2$
the existence of a finite constant not depending on $m$ and $X_{0}\in
L^{p}({\mathbb{P}})$ such that

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-X^{n}_{t}|\Big{\|}^{p}_{p}$
$\displaystyle\leq
C\int_{0}^{T}\|\sigma(s,X_{s})-\sigma_{n}(s,X_{s})\|^{p}_{p}ds.$

One has, using (5.44) and the affine growth of the coefficients
$\sigma,\sigma_{n}$ in space uniform in time and $n\geq 1$,

$\displaystyle\sup_{t\in[0,T]}|\sigma(t,X_{t})-\sigma_{n}(t,X_{t})|\leq\frac{{\rm
Lip}(\sigma)}{n}+C\sup_{t\in[0,T]}|X_{t}|{\mathbf{1}}_{\\{\sup_{t\in[0,T]}|X_{t}|\geq
n\\}}.$

Reasoning like in the end of the proof of Proposition 4.4, we deduce that for
$p\geq 1$,

$\displaystyle\Big{\|}\sup_{t\in[0,T]}|X_{t}-X^{n}_{t}|\Big{\|}_{p}\leq
C\left(\frac{1}{n}+{\mathbb{E}}^{1/p}\left[(1+|X_{0}|)^{p}\left(1\wedge\frac{1+|X_{0}|}{n}\right)^{\frac{1}{2}\wedge\frac{p}{4}}\right]\right)$

where the right-hand side goes to $0$ as $n\to+\infty$ since $X_{0}\in
L^{p}({\mathbb{P}})$. $\Box$

Proof of Theorem 3.2. Step 1. Let us temporarily assume either $(\mathcal{X})$
or $(\mathcal{Y})$ in place of $\forall t\in[0,T]$, $b(t,\cdot)$ is convex or
$\forall t\in[0,T]$, $\beta(t,\cdot)$ is convex. We consider the truncated
Euler schemes (4.30) where for $k=0,\ldots,m-1$,
$(\widetilde{b}^{m}_{k}(\cdot),\widetilde{\sigma}^{m}_{k}(\cdot),\widetilde{\beta}^{m}_{k}(\cdot),\widetilde{\varsigma}_{k}^{m}(\cdot))$
is equal to

$\left(\frac{m}{T}\int_{t_{k}}^{t_{k+1}}b(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\cdot)ds},\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\beta(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\vartheta^{2}(s,\cdot)ds}\right),$

and with threshold $\mathbf{s}_{m}=\frac{\sqrt{m}}{2{\rm
Lip}(\sigma)\sqrt{T}}$ when $(\mathcal{X})$ holds or with threshold
$\mathbf{s}_{m}=\frac{\sqrt{m}}{2{\rm Lip}(\vartheta)\sqrt{T}}$ when
$(\mathcal{Y})$ holds. Since $b,\sigma,\beta,\vartheta$ satisfy (2.5) and
$X_{0}$, $Y_{0}\\!\in L^{1}({\mathbb{P}})$, it follows from Proposition 4.1
that

${\cal W}_{1}({\cal L}(X_{T}),{\cal L}(\widetilde{X}^{m}_{T}))+{\cal
W}_{1}({\cal L}(Y_{T}),{\cal L}(\widetilde{Y}^{m}_{T}))\rightarrow
0\quad\mbox{as}\quad m\to+\infty.$

To establish the (increasing) convex ordering, we may restrict ourselves to
Lipschitz (non-decreasing) convex functions $f:{\mathbb{R}}\to{\mathbb{R}}$
according to Proposition 2.2 . For such functions, by the dual formulation of
the Wasserstein distance ${\cal W}_{1}$ with index $1$,

$|{\mathbb{E}}\,f(\widetilde{X}^{m}_{{}_{T}})-{\mathbb{E}}\,f(X_{{}_{T}})|+|{\mathbb{E}}\,f(\widetilde{Y}^{m}_{{}_{T}})-{\mathbb{E}}\,f(Y_{{}_{T}})|\leq{\rm
Lip}(f)\left({\cal W}_{1}({\cal L}(X_{T}),{\cal
L}(\widetilde{X}^{m}_{T}))+{\cal W}_{1}({\cal L}(Y_{T}),{\cal
L}(\widetilde{Y}^{m}_{T}))\right)\to 0$

as $m\to+\infty$. So we can transfer to the diffusion processes both convex
orderings which hold for the truncated Euler schemes with $m$ large enough
according to Proposition 4.6. As for the (non-decreasing) convexity of
$x\mapsto{\mathbb{E}}\,f(X^{x}_{{}_{T}})$, it clearly holds for any (non-
decreasing) convex Lipschitz function $f$ owing to Remark 4.7 and the above
convergence. We can transfer the property to (non-decreasing) convex functions
$f$ by following the reasoning in the last but one paragraph in the proof of
Proposition 4.6.

Step 2. According to Step 1, the conclusion holds for $(X,Y)$ replaced by
$(X^{n},Y^{n})$ given in Proposition 5.1 and it is enough to consider
Lipschitz continuous test functions. We conclude since, by Proposition 5.1,
$\displaystyle\lim_{n\to+\infty}\Big{\|}\sup_{t\in[0,T]}|X_{t}-X^{n}_{t}|\Big{\|}_{1}=\lim_{n\to+\infty}\Big{\|}\sup_{t\in[0,T]}|Y_{t}-Y^{n}_{t}|\Big{\|}_{1}=0$.
$\Box$

Proof of Proposition 3.3. The second and third statements are straightforward
consequences of the first one for the choice $X_{0}=\alpha
x+(1-\alpha)y\leq_{cvx}\varepsilon x+(1-\varepsilon)y=Y_{0}$ where
$\varepsilon$ is distributed according to the Bernoulli law with parameter
$\alpha$ and independent from $W$, for arbitrary $x,y\in{\mathbb{R}}$ and
$\alpha\in[0,1]$.

Let us prove the first statement. For $s\in[0,T)$ and $t=s+h$ where
$h=\frac{T}{m}$ with $m\in{\mathbb{N}}$ such that $m\geq\frac{T}{T-s}$ so that
$s+h\leq T$, the convex ordering $(X_{s},X_{t})\preceq_{cvx}(Y_{s},Y_{t})$
implies that ${\mathbb{E}}[|X_{s+h}-X_{s}|]\leq{\mathbb{E}}[|Y_{s+h}-Y_{s}|]$.
It is enough to check that
${\mathbb{E}}[|X_{s+h}-X_{s}|]\sim\sqrt{\frac{2h}{\pi}}{\mathbb{E}}[|\sigma(s,X_{s})|]$
and
${\mathbb{E}}[|Y_{s+h}-Y_{s}|]\sim\sqrt{\frac{2h}{\pi}}{\mathbb{E}}[|\vartheta(s,Y_{s})|]$
as $h\to 0$ to conclude, the case $s=T$ being recovered by continuity of
$[0,T]\ni
s\mapsto({\mathbb{E}}[|\sigma(s,X_{s})|],{\mathbb{E}}[|\vartheta(s,Y_{s})|])$.
Since
$\sqrt{\frac{2h}{\pi}}{\mathbb{E}}[|\sigma(s,X_{s})|]={\mathbb{E}}\left[\left|\int_{s}^{s+h}\sigma(s,X_{s})dW_{u}\right|\right]$,
using the triangle inequality and Jensen’s inequality, we get

$\displaystyle\left|{\mathbb{E}}\left[|X_{s+h}-X_{s}|-\sqrt{\frac{2h}{\pi}}|\sigma(s,X_{s})|\right]\right|$
$\displaystyle\leq{\mathbb{E}}\left[\left|\int_{s}^{s+h}b(u,X_{u})du\right|\right]$
$\displaystyle\hskip
71.13188pt+{\mathbb{E}}^{1/2}\left[\left|\int_{s}^{s+h}(\sigma(u,X_{u})-\sigma(s,X_{s}))dW_{u}\right|^{2}\right]$
$\displaystyle\leq
Ch(1+\|X_{0}\|_{1})+{\mathbb{E}}^{1/2}\left[\int_{s}^{s+h}(\sigma(u,X_{u})-\sigma(s,X_{s}))^{2}du\right].$

Using standard estimations on the moments of the increments of $X$ under
(2.5), we obtain that

$\displaystyle{\mathbb{E}}\left[\int_{s}^{s+h}(\sigma(u,X_{u})-\sigma(s,X_{s}))^{2}du\right]\leq
Ch^{2}(1+\|X_{0}\|_{2})^{2}+2{\mathbb{E}}\left[\int_{s}^{s+h}(\sigma(u,X_{s})-\sigma(s,X_{s}))^{2}du\right]$

and, when $X_{0}\in L^{4}({\mathbb{P}})$, reasoning like in the derivation of
(4.29), we check that

$\displaystyle\forall
u\in[s,s+h],\;\|\sigma(u,X_{s})-\sigma(s,X_{s})\|^{2}_{2}\leq
C\left(\varepsilon_{h,K}^{2}+(1+\|X_{0}\|_{4})^{2}\sqrt{1\wedge\frac{1+\|X_{0}\|_{1}}{K}}\right),$

where $K\geq 1$ and $\varepsilon_{h,K}:=\sup_{0\leq s\leq t\leq(s+h)\wedge
T,x\in[-K,K]}(|\sigma(s,x)-\sigma(t,x)|)$ goes to $0$ with $h$ by uniform
continuity of $\sigma$ on the compact set $[0,T]\times[-K,K]$. Combining the
three last inequalities with the subadditivity of the square root, we obtain
that for deterministic initial conditions $x_{0}$,

$\displaystyle\left|{\mathbb{E}}\left[|X^{x_{0}}_{s+h}-X^{x_{0}}_{s}|-\sqrt{\frac{2h}{\pi}}|\sigma(s,X^{x_{0}}_{s})|\right]\right|$
$\displaystyle\leq
C\sqrt{h}\left(\sqrt{h}(1+|x_{0}|)+\varepsilon_{h,K}+(1+|x_{0}|)\left(1\wedge\frac{1+|x_{0}|}{K}\right)^{\frac{1}{4}}\right).$

Using Blagove$\check{\rm s}\check{\rm c}$enkii-Freidlin’s theorem [22, Theorem
13.1, Section V.12-13, p.136], integrating with respect to the distribution of
$X_{0}$ and using the independence between $X_{0}$ and $W$, we deduce that

$\displaystyle\left|{\mathbb{E}}\left[|X_{s+h}-X_{s}|-\sqrt{\frac{2h}{\pi}}|\sigma(s,X_{s})|\right]\right|$
$\displaystyle\leq
C\sqrt{h}\left(\sqrt{h}(1+\|X_{0}\|_{1})+\varepsilon_{h,K}+{\mathbb{E}}\left[(1+|X_{0}|)\left(1\wedge\frac{1+|X_{0}|}{K}\right)^{\frac{1}{4}}\right]\right).$

For $X_{0}\in L^{1}({\mathbb{P}})$, by Lebesgue’s theorem, the last
expectation goes to $0$ as $K\to\infty$ so that we may choose $K$ such that it
is arbitrarily small, while for fixed $K$ the sum of the first two terms
between brackets in the right-hand side goes to $0$ with $h$. Therefore the
sum between these brackets goes to $0$ with $h$. The equivalent for
${\mathbb{E}}[|Y_{s+h}-Y_{s}|]$ is obtained in the same way. $\Box$

Proof of Theorem 3.10. The sequence $(X^{n})_{n\geq 1}$ (resp. $(Y^{n})_{n\geq
1}$) given by Proposition 5.1 converges in $L^{p}({\mathbb{P}})$ to $X$ (resp.
$Y$). Since $F$ is continuous, we deduce that $(F(X^{n}))_{n\geq 1}$ (resp.
$(F(Y^{n}))_{n\geq 1}$) converges in distribution to $F(X)$ (resp. $F(Y)$) and
since $F$ has polynomial growth with order $p$, this sequence is uniformly
integrable so that ${\mathbb{E}}\,F(X^{n})\to{\mathbb{E}}\,F(X)$ (resp.
${\mathbb{E}}\,F(Y^{n})\to{\mathbb{E}}\,F(Y)$) as $n\to+\infty$. Therefore we
can suppose that either $(\mathcal{X})$ or $(\mathcal{Y})$ holds. We consider
the truncated Euler schemes (4.30) where for $k=0,\ldots,m-1$,
$(\widetilde{b}^{m}_{k}(\cdot),\widetilde{\sigma}^{m}_{k}(\cdot),\widetilde{\beta}^{m}_{k}(\cdot),\widetilde{\varsigma}_{k}^{m}(\cdot))$
is equal to

$\left(\frac{m}{T}\int_{t_{k}}^{t_{k+1}}b(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\sigma^{2}(s,\cdot)ds},\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\beta(s,\cdot)ds,\sqrt{\frac{m}{T}\int_{t_{k}}^{t_{k+1}}\vartheta^{2}(s,\cdot)ds}\right),$

and with threshold $\mathbf{s}_{m}=\frac{\sqrt{m}}{2{\rm
Lip}(\sigma)\sqrt{T}}$ when $(\mathcal{X})$ holds or with threshold
$\mathbf{s}_{m}=\frac{\sqrt{m}}{2{\rm Lip}(\vartheta)\sqrt{T}}$ when
$(\mathcal{Y})$ holds. Since $b,\sigma,\beta,\vartheta$ satisfy (2.5) and
$X_{0}$, $Y_{0}\\!\in L^{p}({\mathbb{P}})$, it follows from Corollary 4.3
that, as $m\to+\infty$,

${\cal W}_{p}\left({\cal L}(X),{\cal
L}\left(i_{m}\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\right)\right)+{\cal
W}_{p}\left({\cal L}(Y),{\cal
L}\left(i_{m}\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\right)\right)\to
0,$

so that
${\mathbb{E}}\,F\big{(}i_{m}\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{)}\to{\mathbb{E}}\,F(X)$
and
${\mathbb{E}}\,F\big{(}i_{m}\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{)}\to{\mathbb{E}}\,F(Y)$.
Since the image of ${\mathbb{R}}_{+}^{m+1}$ by $i_{m}$ is included in
$C([0,T],{\mathbb{R}}_{+})$, $F\circ i_{m}$ is directionally convex on
${\mathbb{R}}^{m+1}$ and non-decreasing in each of its variables when $F$ is
non-decreasing. With Proposition 4.6, we deduce that for each $m\geq 1$,
${\mathbb{E}}\,F\big{(}i_{m}\big{(}(\widetilde{X}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{)}\leq{\mathbb{E}}\,F\big{(}i_{m}\big{(}(\widetilde{Y}^{m}_{t_{k}})_{k=0,\ldots,m}\big{)}\big{)}$.
By taking the limit $m\to+\infty$ in this inequality, we conclude that
${\mathbb{E}}\,F(X)\leq{\mathbb{E}}F(Y)$.

The (non-decreasing) convexity of ${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}\,F(X^{x})$ when $F$ is continuous, (non-decreasing)
directionally convex with polynomial growth is deduced by the argument in the
first point of the remark which follows Theorem 3.2. $\Box$

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##  Appendix A Generalization of the stochastic calculus approach by El
Karoui, Jeanblanc and Shreve in [9]

###### Proposition A.1

Let $b,\sigma:[0,T]\times{\mathbb{R}}\to{\mathbb{R}}$ satisfy (2.5). Assume
furthermore that:

$(i)$ the functions $b(t,\cdot)$ and $\sigma(t,\cdot)$ are continuously
differentiable with derivatives $\partial_{2}b(t,\cdot)$ and
$\partial_{2}\sigma(t,\cdot)$, for every $t\\!\in[0,T]$,

$(ii)$ $\partial_{2}b(t,\cdot)$ and $\partial_{2}\sigma(t,\cdot)$ are globally
$\delta$-Hölder, $\delta\in(0,1]$, uniformly in $t\in[0,T]$,

$(iii)$ $\sigma\partial_{2}\sigma(t,\cdot)$ is Lipschitz uniformly in
$t\in[0,T]$.

Let $X^{x}$ denote the solution of (2.3) starting from $x\\!\in{\mathbb{R}}$.
Let $f:{\mathbb{R}}^{m}\to{\mathbb{R}}$ be directionally convex (resp.
directionally convex and non-decreasing in each of its variables) and let
$0\leq t_{1}\leq t_{2}\leq\cdots\leq t_{m}\leq T$, $m\in{\mathbb{N}}$.

$(a)$ Then, ${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is convex (resp.
convex and non-decreasing) when $b(t,\cdot)$ is affine (resp. convex) for each
$t\in[0,T]$.

$(b)$ In the particular case $m=1$, and $t_{1}=T$, this amounts to convexity
(resp. non decreasing convexity) of ${\mathbb{R}}\ni
x\mapsto{\mathbb{E}}f(X^{x}_{T})$ when $f:{\mathbb{R}}\to{\mathbb{R}}$ is
convex (resp. non-decreasing and convex) and $b(t,\cdot)$ is affine (resp.
convex) for each $t\in[0,T]$.

Remark. Note that (2.5) implies that $\partial_{2}b$ and $\partial_{2}\sigma$
are bounded on $[0,T]\times{\mathbb{R}}$. On the other hand, condition $(iii)$
implies that $\sigma^{2}(t,\cdot)$ has a Lipschitz derivative and hence is
semi-convex.

Proof. Since directional and regular convexity coincide for a function from
${\mathbb{R}}$ to ${\mathbb{R}}$, it is sufficient to prove $(a)$. By Lemma
3.7, it is enough to deal with functions $f$ which are also Lipschitz
continuous. By spatial convolution with $n^{m}\rho(nx)$ where $\rho$ is a
compactly supported $C^{\infty}$ probability density on ${\mathbb{R}}^{m}$,
such a function may be approximated uniformly by functions which are moreover
$C^{\infty}$ and inherit its directional convexity and monotonicity
properties. Therefore, we may even suppose that $f$ is $C^{\infty}$ with
bounded first order derivatives.

By [12, Theorem 4.6.5] and its proof, there exists a modification of the
solution to the stochastic differential equation (2.3) such that for
$t\in[0,T]$, $x\mapsto X^{x}_{t}$ is a diffeomorphism of ${\mathbb{R}}$ with
derivative $\partial_{x}X^{x}_{t}$ solving the equation obtained by spatial
derivation of (2.3) :

$d\partial_{x}X^{x}_{t}=\partial_{2}b(t,X^{x}_{t})\partial_{x}X^{x}_{t}+\partial_{2}\sigma(t,X^{x}_{t})\partial_{x}X^{x}_{t}dW_{t},\;\partial_{x}X^{x}_{0}=1,$

so that

$\forall
t\in[0,T],\;\partial_{x}X^{x}_{t}=e^{\int_{0}^{t}\partial_{2}b(s,X^{x}_{s})ds}e^{\int_{0}^{t}\partial_{2}\sigma(s,X^{x})dW_{s}-\frac{1}{2}\int_{0}^{t}(\partial_{2}\sigma(s,X^{x}_{s}))^{2}ds}.$

As a consequence
$\partial_{x}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})=\sum_{k=1}^{m}\partial_{k}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})\partial_{x}X^{x}_{t_{k}}$.
The boundedness of the first order derivatives of $f$ and of $\partial_{2}b$,
$\partial_{2}\sigma$ implies that
$\sup_{x\in{\mathbb{R}}}{\mathbb{E}}\left[\left(\sum_{k=1}^{m}\partial_{k}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})\partial_{x}X^{x}_{t_{k}}\right)^{2}\right]<+\infty$
so that we may interchange the derivative and the expectation and obtain

$\partial_{x}{\mathbb{E}}\,f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})=\sum_{k=1}^{m}{\mathbb{E}}\Big{(}\partial_{k}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})\partial_{x}X^{x}_{t_{k}}\Big{)}.$

By boundedness of $\partial_{2}\sigma$,
${\mathbb{E}}\,e^{\int_{0}^{t}\partial_{2}\sigma(s,X^{x})dW_{s}-\frac{1}{2}\int_{0}^{t}(\partial_{2}\sigma(s,X^{x}_{s}))^{2}ds}=1$
for each $t\in[0,T]$. By Girsanov’s theorem, under the probability measure
${\mathbb{Q}}_{x,k}=e^{\int_{0}^{t_{k}}\partial_{2}\sigma(s,X^{x})dW_{s}-\frac{1}{2}\int_{0}^{t_{k}}(\partial_{2}\sigma(s,X^{x}_{s}))^{2}ds}\cdot{\mathbb{P}}$,
$\left(B^{x,k}_{t}=W_{t}-\int_{0}^{t}{\mathbf{1}}_{\\{s\leq
t_{k}\\}}\partial_{2}\sigma(s,X^{x}_{s})ds\right)_{t\geq 0}$ is a Brownian
motion and $X^{x,k}$ is solution to the SDE

$X^{x,k}_{t}=x+\int_{0}^{t}(b+{\mathbf{1}}_{\\{s\leq
t_{k}\\}}\sigma\partial_{2}\sigma)(s,X^{x,k}_{s})ds+\int_{0}^{t}\sigma(s,X^{x,k}_{s})dB^{x,k}_{s}.$

Since $b,\sigma$ and $b+\sigma\partial_{2}\sigma$ are Lipschitz continuous in
space uniformly in time, ${\cal L}_{{\mathbb{Q}}_{x,k}}(X^{x,k})={\cal
L}_{{\mathbb{P}}}(\tilde{X}^{x,k})$ where $\tilde{X}^{x,k}$ follows the same
SDE as $X^{x,k}$ but with the Brownian motion $B^{x,k}$ replaced by $W$ :

$\tilde{X}^{x,k}_{t}=x+\int_{0}^{t}(b+{\mathbf{1}}_{\\{s\leq
t_{k}\\}}\sigma\partial_{2}\sigma)(s,\tilde{X}^{x,k}_{s})ds+\int_{0}^{t}\sigma(s,\tilde{X}^{x,k}_{s})dW_{s}.$

Then

$\displaystyle\partial_{x}{\mathbb{E}}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$
$\displaystyle=\sum_{k=1}^{m}{\mathbb{E}}_{{\mathbb{Q}}_{x,k}}\Big{(}\partial_{k}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})e^{\int_{0}^{t_{k}}\partial_{2}b(s,X^{x}_{s})ds}\Big{)}$
$\displaystyle=\sum_{k=1}^{m}{\mathbb{E}}\Big{(}\partial_{k}f(\tilde{X}^{x,k}_{t_{1}},\cdots,\tilde{X}^{x,k}_{t_{m}})e^{\int_{0}^{t_{k}}\partial_{2}b(s,\tilde{X}^{x,k}_{s})ds}\Big{)}.$

At this stage, since $b,\sigma$ and $b+\sigma\partial_{2}\sigma$ are Lipschitz
continuous in space uniformly in time, by [21, Theorem 3.7 p.394],
$x\mapsto\tilde{X}^{x}$ is (a.s.) pathwise non-decreasing. For each
$k\in\\{1,\cdots,m\\}$, since $\partial_{k\ell}f\geq 0$ for all
$\ell\in\\{1,\cdots,m\\}$ by Property $\mathbf{P2}$ in Remark 3.5,
$x\mapsto\partial_{k}f(\tilde{X}^{x,k}_{t_{1}},\cdots,\tilde{X}^{x,k}_{t_{m}})$
is non-decreasing as well. When $b(t,\cdot)$ is affine and therefore
$\partial_{2}b(t,\cdot)$ constant for each $t\in[0,T]$, we deduce that
$x\mapsto\partial_{x}{\mathbb{E}}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is
non-decreasing and therefore
$x\mapsto{\mathbb{E}}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is convex. The
conclusion remains valid when $b(t,\cdot)$ is convex and therefore
$\partial_{2}b(t,\cdot)$ non-decreasing for each $t\\!\in[0,T]$ and $f$ is
non-decreasing in each of its variables so that $\partial_{k}f\geq 0$ for each
$k\in\\{1,\cdots,m\\}$. Finally, the pathwise non-decreasing property of
$x\mapsto X^{x}$ deduced from [21, Theorem 3.7 p.394] implies that
$x\mapsto{\mathbb{E}}f(X^{x}_{t_{1}},\cdots,X^{x}_{t_{m}})$ is non-decreasing
when $f$ is non decreasing in each of its variables. $\Box$

##  Appendix B Proof of Example 3.6$(d)$

The convexity of $\varphi$ implies that ${\cal C}([0,T],{\mathbb{R}})\ni
x\mapsto\int_{0}^{T}\varphi(x(s))ds$ is convex and the first statement follows
from the convexity of the composition of a non-decreasing convex function with
a convex function. To prove the second statement, let $x\in{\cal
C}([0,T],{\mathbb{R}})$ and $y,z\in{\cal C}([0,T],{\mathbb{R}}_{+})$. For
$s\in[0,T]$, we have $x(s)+y(s),x(s)+z(s)\in[x(s),x(s)+y(s)+z(s)]$ so that
$\varphi(x(s)+y(s)),\varphi(x(s)+z(s))\in[\varphi(x(s)),\varphi(x(s)+y(s)+z(s))]$
by monotonicity of $\varphi$ and
$\varphi(x(s)+y(s))+\varphi(x(s)+z(s))\leq\varphi(x(s))+\varphi(x(s)+y(s)+z(s))$
by convexity of this function. As a consequence,
$\int_{0}^{T}\varphi(x(s)+y(s))ds,\int_{0}^{T}\varphi(x(s)+z(s))ds\in\left[\int_{0}^{T}\varphi(x(s))ds,\int_{0}^{T}(\varphi(x(s)+y(s))+\varphi(x(s)+z(s))-\varphi(x(s)))\right]$
and

$\int_{0}^{T}\varphi(x(s)+y(s))ds+\int_{0}^{T}\varphi(x(s)+z(s))ds-\int_{0}^{T}\varphi(x(s))ds\leq\int_{0}^{T}\varphi(x(s)+y(s)+z(s))ds.$

With the convexity of $\Psi$ for the first inequality then its monotonicity
for the second, we conclude that

$\displaystyle F(x+y)+F(x+z)$ $\displaystyle\leq
F(x)+\Psi\left(\int_{0}^{T}(\varphi(x(s)+y(s))+\varphi(x(s)+z(s))-\varphi(x(s)))ds\right)$
$\displaystyle\leq F(x)+F(x+y+z).$