Frédéric Bernardin
email: frederic.bernardin@developpement-durable.gouv.fr

Mireille Bossy
email: mireille.bossy@sophia.inria.fr

Miguel Martinez
email: miguel.martinez@univ-mlv.fr

Denis Talay
email: denis.talay@sophia.inria.fr

ELECTRONIC COMMUNICATIONS in PROBABILITY ON MEAN NUMBERS OF PASSAGE TIMES IN SMALL BALLS OF DISCRETIZED ITÔ PROCESSES

Keywords: Diffusion processes, sojourn times, estimates, discrete times

The aim of this note is to prove estimates on mean values of the number of times that Itô processes observed at discrete times visit small balls in R d . Our technique, in the infinite horizon case, is inspired by Krylov's arguments in [2, Chap.2]. In the finite horizon case, motivated by an application in stochastic numerics, we discount the number of visits by a locally exploding coefficient, and our proof involves accurate properties of last passage times at 0 of one dimensional semimartingales.

Introduction

The present work is motivated by two convergence rate analyses concerning discretization schemes for diffusion processes whose generators have non smooth coefficients: Bernardin et al. [START_REF] Bernardin | Viscosity solutions of parabolic differential inclusions and weak convergence rate of discretizations of multi-valued stochastic differential equations[END_REF] study discretization schemes for stochastic differential equations with multivalued drift coefficients; Martinez and Talay [START_REF] Martinez | Time discretization of one dimensional Markov processes with generators under divergence form with discontinuous coefficients[END_REF] study discretization schemes for diffusion processes whose infinitesimal generators are of divergence form with discontinuous coefficients. In both cases, to have reasonable accuracies, a scheme needs to mimic the local behaviour of the exact solution in small neighborhoods of the discontinuities of the coefficients, in the sense that the expectations of the total times spent by the scheme (considered as a piecewise constant process) and the exact solu-tion in these neighborhoods need to be close to each other. The objective of this paper is to provide two estimates for the expectations of such total times spent by fairly general Itô processes observed at discrete times; these estimates concern in particular the discretization schemes studied in the above references (see section 7 for more detailed explanations). To this end, we carefully modify the technique developed by Krylov [START_REF] Krylov | Controlled diffusion processes[END_REF] to prove inequalities of the type

E ∞ 0 e -λt f (X t )d t ≤ C f L d
for positive Borel functions f and R d valued continuous controlled diffusion processes (X t ) under some strong ellipticity condition (see also Krylov and Lipster [3] for extensions in the degenerate case). The difficulties here come from the facts that time is discrete and, in the finite horizon case, we discount by a locally exploding function of time the number of times that the discrete time process visits a given small ball in R d . Another difficulty comes from the fact that, because of the convergence rate analyses which motivate this work, we aim to get accurate estimates in terms of the time discretization step. Our estimates seem interesting in their own right: they contribute to show the robustness of Krylov's estimates. Let (Ω, , ( t ) t≥0 , P) be a filtered probability space satisfying the usual conditions. Let (W t ) t≥0 be a m-dimensional standard Brownian motion on this space. Consider two progressively measurable processes (b t ) t≥0 and (σ t ) t≥0 taking values respectively in R d and in the space d,m (R) of real d × m dimensional matrices. For X 0 in R d , consider the Itô process

X t = X 0 + t 0 b s ds + t 0 σ s dW s . ( 1
)
We assume the following hypotheses:

Assumption 1.1. There exists a positive number K ≥ 1 such that, P-a.s.,

∀t ≥ 0, b t ≤ K, ( 2
)
and

∀ 0 ≤ s ≤ t, 1 K 2 t s ψ(u)du ≤ t s ψ(u) σ u σ * u du ≤ K 2 t s ψ(u)du (3)
for all positive locally integrable map ψ : R + → R + .

Notice that (3) is satisfied when (σ t ) is a bounded continuous process; our slightly more general framework is motivated by the reduction of our study, without loss of generality, to one dimensional Itô processes: see section 2. The aim of this note is to prove the following two results. Our first result concerns discounted expectations of the total number of times that an Itô process observed at discrete times visits small neighborhoods of an arbitrary point in R d .

Theorem 1.1 (Infinite horizon). Let (X t ) be as in [START_REF] Bernardin | Viscosity solutions of parabolic differential inclusions and weak convergence rate of discretizations of multi-valued stochastic differential equations[END_REF]. Suppose that the hypotheses 1.1 are satisfied. Let h > 0. There exists a constant C, depending only on K, such that, for all ξ ∈ R d and 0 < ǫ < 1/2, there exists h 0 > 0 (depending on ǫ) satisfying

∀ h ≤ h 0 , h ∞ p=0 e -ph P X ph -ξ ≤ δ h ≤ Cδ h , ( 4
)
where

δ h := h 1/2-ǫ .
Our second result concerns expectations of the number of times that an Itô process, observed at discrete times between 0 and a finite horizon T , visits small neighborhoods of an arbitrary point ξ in R d . In this setting, the expectations are discounted with a function f (t) which may tend to infinity when t tends to T . Such a discount factor induces technical difficulties which might seem artificial to the reader. However, this framework is necessary to analyze the convergence rate of simulation methods for Markov processes whose generators are of divergence form with discontinuous coefficients: see section 7 and Martinez and Talay [START_REF] Martinez | Time discretization of one dimensional Markov processes with generators under divergence form with discontinuous coefficients[END_REF]. More precisely, we consider functions f satisfying the following hypotheses:

Assumption 1.2.
1. f is positive and increasing,

2. f belongs to C 1 ([0, T ); R + ), 3. f α is integrable on [0, T ) for all 1 ≤ α < 2, 4. there exists 1 < β < 1 + η, where η := 1 4K 4 , such that T 0 f 2β-1 (v) f ′ (v) (T -v) 1+η v η d v < +∞. (5)
Remember that we have supposed K ≥ 1. An example of a suitable function f is the function t → 1 T -t (for which one can choose β = 1 + 1 8K 4 ). Theorem 1.2 (Finite horizon). Let (X t ) be as in [START_REF] Bernardin | Viscosity solutions of parabolic differential inclusions and weak convergence rate of discretizations of multi-valued stochastic differential equations[END_REF]. Suppose that the hypotheses 1.1 are satisfied. Let f be a function on [0, T ) satisfying the hypotheses 1.2. There exists a constant C, depending only on β, K and T , such that, for all ξ ∈ R d and 0 < ǫ < 1/2, there exists h 0 > 0 (depending on ǫ) satisfying

∀ h ≤ h 0 , h N h p=0 f (ph)P X ph -ξ ≤ δ h ≤ Cδ h , (6)
where N h := ⌊T /h⌋ -1 and where δ h := h 1/2-ǫ .

Remark 1.1. Since all the norms in R d are equivalent, without loss of generality we may choose

Y = sup 1≤i≤d Y i , Y i denoting the i th component of Y .
Remark 1.2. Changing X 0 into X 0 -ξ allows us to limit ourselves to the case ξ = 0 without loss of generality, which we actually do in the sequel.

2 Reduction to one dimensional (X t )'s

In order to prove Theorem 1.1, it suffices to prove that, for all 1 ≤ i ≤ d and h ≤ h 0 ,

h ∞ p=0 e -ph P X i ph ≤ δ h ≤ Cδ h ,
or, similarly, for Theorem 1.2

h N h p=0 f (ph)P |X i ph | ≤ δ h ≤ Cδ h .
Set

M i t := m j=1 t 0 σ i j s dW j s .
The martingale representation theorem implies that there exist a process ( f i t ) t≥0 and a standard t -Brownian motion (B t ) t≥0 on an extension of the original probability space, such that

M i t = t 0 f i s d B s ,
and

∀t > 0, t 0 ( f i θ ) 2 dθ = m j=1 t 0 σ i j θ 2 dθ .
Therefore it suffices to prove Theorems 1.1 and 1.2 for all one dimensional Itô process

X t = t 0 b s ds + t 0 σ s d B s (7) such that, ∃ K ≥ 1, P-a.s., ∀ t ≥ 0, |b t | ≤ K, ( 8
)
and

∀ 0 ≤ s ≤ t, 1 K 2 t s ψ(u)du ≤ t s ψ(u)σ 2 u du ≤ K 2 t s ψ(u)du (9)
for all positive locally integrable map ψ : R + → R + .

An estimate for a localizing stopping time

In this section we prove a key proposition which is a variant of Theorem 4 in [2, Chap.2]: here we need to consider time indices and stopping times which take values in a mesh with step-size h. We carefully modify Krylov's arguments. We start by introducing two non-decreasing sequences of random times (θ h p ) p and (τ h p ) p taking values in hN as

   θ h 0 = τ h 0 = 0 θ h p+1 = inf{t ≥ τ h p , t ∈ hN, X t ≤ δ h }, p ≥ 0 τ h p = inf{t ≥ θ h p , t ∈ hN, X t -X θ h p ≥ 1}, p ≥ 1. ( 10
)
Our goal is then to prove the following equality :

E   h +∞ p=0 e -ph 1 [-δ h ,δ h ] (X ph )   = +∞ p=0 E   1 θ h p <+∞ τ h p θ h p e -η s 1 [-δ h ,δ h ] (X η s )ds    , (11)
where

η s := h ⌊s/h⌋ for all s ≥ 0. ( 12
)
It is easy to see that (11) holds if the sequence θ h p p takes an infinite value in a finite time, or converges to infinity when p tends to infinity, which is a consequence of Lemma 3.1 (see below).

For the proof of Lemma 3.1, we need the following Proposition 3.1. Let (X t ) be as in [START_REF] Talay | Probabilistic numerical methods for partial differential equations: elements of analysis[END_REF]. Suppose that conditions ( 8) and ( 9) hold true. Let θ : Ω → hN be a ( t )-stopping time. Then τ := inf{t ∈ hN, t ≥ θ , X t -X θ ≥ 1} is a ( t )-stopping time, and

E 1 τ<+∞ e -τ θ ≤ 1 ch(µ) e -θ 1 θ <+∞ a.s., ( 13
)
where µ > 0 and 1 -Kµ -(1/2)K 2 µ 2 = 0.
Proof. Set π(z) := cosh(µz) for all z in R and

L u π(z) := b u π ′ (z) + 1 2 σ 2 u π ′′ (z).
Let A ∈ θ . Suppose that we have proven that, for all t > 0,

e -t∧θ ≥ E e -τ π(X t∧τ -X t∧θ ) t∧θ . (14)
Then we would have

E 1 A 1 θ <+∞ e -θ = lim t→+∞ E 1 A 1 θ ≤t e -θ ∧t ≥ lim t→+∞ E 1 A 1 θ ≤t E e -τ π(X t∧τ -X t∧θ ) t∧θ = lim t→+∞ E 1 A 1 θ ≤t E e -τ π(X t∧τ -X t∧θ ) θ = lim t→+∞ E 1 A 1 θ ≤t e -τ π(X t∧τ -X t∧θ ) = lim t→+∞ E 1 A 1 θ <+∞ e -τ π(1) ≥ E 1 A cosh(µ)1 τ<+∞ e -τ ,
and thus the desired result would have been obtained. We now prove the inequality (14). Fix t ≥ 0 arbitrarily and set

B 1 := {t ≤ θ } ∩ {t ≤ τ} = {t ≤ θ } ,
and

C 1 := {t > θ } ∩ {t ≤ τ} = {t > θ } ∩ ∀ s ∈ [θ , t] ∩ hN, X s -X θ < 1 .
The sets B 1 and C 1 are t -measurable. Consequently the set {t ≤ τ} is t -measurable and thus τ is a ( t )-stopping time.

To show (14), we start with checking that, P-a.s, for all 0 ≤ s ≤ t and z ∈ R,

t s ψ(u) π(z) -L u π(z) du ≥ cosh(µz) 1 -Kµ - 1 2 K 2 µ 2 t s ψ(u)du = 0 (15)
for all positive locally bounded map ψ : R + → R + . Indeed, for all z ∈ R and u ≥ 0,

π(z) -L u π(z) = cosh(µz) -µ b u sinh(µz) - 1 2 µ 2 σ 2 u cosh(µz).
As sinh(µ |z|) ≤ cosh(µz), we get (15) by using ( 8) and ( 9). Now, we observe that sup θ ≤t e γX θ is integrable for all γ > 0 since by ( 9), (b t ) and (〈X 〉 t ) are bounded. This justifies the following equality deduced from the Itô's formula : for all y ∈ R,

e -t∧θ π(X t∧θ + y) = E e -t∧τ π(X t∧τ + y) t∧θ + E t∧τ t∧θ e -s (π(X s + y) -L s π(X s + y))ds t∧θ a.s..
Therefore, in view of (15), there exists a set y with P( y ) = 0 such that

e -t∧θ π(X t∧θ + y) ≥ E e -τ π(X t∧τ + y) t∧θ on Ω -y . ( 16
)
We now use the continuity of π in order to replace y by -X t∧θ . For all positive integer q, let k q ( y) be the unique integer ℓ such that ℓ/2 q ≤ y < (ℓ + 1)/2 q . Then

π(X t∧τ -X t∧θ ) = lim q→+∞ π X t∧τ - k q (X t∧θ ) 2 q .
Therefore, by Fatou's lemma and ( 16),

E e -τ π(X t∧τ -X t∧θ ) t∧θ ≤ lim inf q→+∞ E e -τ π X t∧τ + k q (-X t∧θ ) 2 q t∧θ ≤ lim inf q→+∞ e -t∧θ π X t∧θ + k q (-X t∧θ ) 2 q = e -t∧θ .
We end this section by the following Lemma 3.1. For all p ≥ 0,

E 1 θ h p <+∞ e -θ h p ≤ 1 cosh(µ) p , ( 17
)
where µ is defined as in Proposition 3.1.

Proof. Apply Proposition 3.1 to the stopping times θ h p and τ h p . It comes

E 1 τ h p <+∞ e -τ h p θ h p ≤ 1 cosh(µ) e -θ h p 1 θ h p <+∞ P -a.s.,
from which

E 1 θ h p+1 <+∞ e -θ h p+1 ≤ E 1 τ h p <+∞ e -τ h p = E E 1 τ h p <+∞ e -τ h p θ h p ≤ 1 cosh(µ) E 1 θ h p <+∞ e -θ h p ≤ • • • ≤ 1 cosh(µ) p+1 .
In the next section we carefully estimate the right-hand side of (11).

Estimate for the localized problem in the infinite horizon setting

In this section we aim to prove the following lemma.

Lemma 4.1. Consider a one dimensional Itô process [START_REF] Talay | Probabilistic numerical methods for partial differential equations: elements of analysis[END_REF] satisfying ( 8) and ( 9). Let θ : Ω → hN be a t -stopping time such that X θ ∈ [-δ h , δ h ] almost surely. Let τ be the stopping time

τ := inf{t ≥ θ , t ∈ hN, X t -X θ ≥ 1}.
There exists a constant C, independent of θ and τ, such that, for all h small enough,

E 1 θ <+∞ τ θ e -η s 1 [-δ h ,δ h ] (X η s )ds ≤ Cδ h E 1 θ <+∞ e -θ . ( 18
)
Proof. To start with, we construct a function I whose second derivative is 1 [-2δ h ,2δ h ] in order to be able to apply Itô's formula. Let I : R → R be defined as

I(z) =      0 if z ≤ -2δ h , 1 2 z 2 + 2δ h z + 2δ 2 h if -2δ h ≤ z ≤ 2δ h , 4δ h z if 2δ h ≤ z. ( 19
)
The map I is of class C 1 (R) and of class C 2 (R \ ({-2δ h } ∪ {2δ h })). In addition, for all z ∈ R,

I ′′ (z) = 1 [-2δ h ,2δ h ] (z) (we set I ′′ (-2δ h ) = I ′′ (2δ h ) = 1)
. For h small enough, there exists a constant C independent of h such that sup

-1-δ h ≤z≤1+δ h |I(z)| ≤ Cδ h , sup z∈R I ′ (z) ≤ Cδ h . (20)
Set

q h := 1 θ <+∞ τ θ e -η s 1 [-δ h ,δ h ] (X η s )ds, (21)
and observe that, for all s ≥ 0,

{X η s ∈ [-δ h , δ h ]} = {X η s ∈ [-δ h , δ h ]} ∩ {X s ∈ [-2δ h , 2δ h ]} ∪ {X η s ∈ [-δ h , δ h ]} ∩ {X s / ∈ [-2δ h , 2δ h ]} , so that 1 [-δ h ,δ h ] (X η s ) ≤ 1 [-2δ h ,2δ h ] (X s ) + 1 [-δ h ,δ h ] (X η s )1 R\[-2δ h ,2δ h ] (X s ). (22)
From ( 21) and ( 22),

q h ≤ A h + B h , (23)
where

A h = 1 θ <+∞ τ θ e -η s 1 [-2δ h ,2δ h ] (X s )ds (24)
and

B h = 1 θ <+∞ τ θ e -η s 1 [-δ h ,δ h ] (X η s )1 R\[-2δ h ,2δ h ] (X s )ds. ( 25
)
An upper bound for A h . In view of (9), one has

A h ≤ 1 θ <+∞ K 2 e -h τ θ e -s 1 [-2δ h ,2δ h ] (X s )σ 2 s ds = 1 θ <+∞ K 2 e -h τ θ e -s I ′′ (X s )σ 2 s ds.
Notice that ( t 0 e -s I ′ (X s )σ s dW s ) is a uniformly square integrable martingale. Thus, the limit ∞ 0 e -s I ′ (X s )σ s dW s exists a.s. As I ′′ is continuous except at points -2δ h and 2δ h , an extended Itô's formula (see, e.g., Protter [5, Thm.71,Chap.IV]) implies

A h ≤ 1 θ <+∞ K 2 e -h e -τ I X τ -e -θ I X θ + τ θ e -s I(X s )ds - τ θ e -s I ′ (X s )b s ds - τ θ e -s I ′ (X s )σ s dW s . (26)
In view of (20) we have |I(X s )| ≤ Cδ h even on the random set {τ = +∞}. In addition,

E 1 τ<∞ τ 0 e -s I(X s )ds ≤ E 1 τ<∞ τ 0 e -s |I(X η s )|ds +E   1 τ<∞ +∞ 0 e -s |I(X s ) -I(X η s )|ds   .
The first expectation is bounded from above by Cδ h since, for all integer k such that θ ≤ kh ≤ τ-h, one has |X kh | ≤ 1 + δ h . The second one is obviously bounded from above by C h. Thus,

E A h ≤ Cδ h E 1 θ <+∞ e -θ . (27)
An upper bound for B h . Setting M s := s 0 σ u dW u , one has

B h ≤ 1 θ <+∞ τ θ e -η s 1 [δ h ,+∞) ( X s -X η s )ds = 1 θ <+∞ e -θ τ-θ 0 e -η s 1 [δ h ,+∞) ( X s+θ -X η s +θ )ds ≤ 1 θ <+∞ e -θ +∞ 0 e -η s 1 [δ h ,+∞) s+θ η s +θ b u du + M s+θ -M η s +θ ds ≤ 1 θ <+∞ e -θ +∞ 0 e -η s 1 [δ h -Kh,+∞) M s+θ -M η s +θ ds ≤ 1 θ <+∞ e -θ +∞ 0 e -η s 1 [δ h /2,+∞) M s+θ -M η s +θ ds,
where we have used (8) and h is assumed small enough. We now successively use the Cauchy-Schwarz, Jensen and Bernstein inequalities (see, e.g., Revuz and Yor [6, Ex.3.16,Chap.IV] for the Bernstein inequality for martingales):

EB h ≤ E 1 θ <+∞ e -θ 2 E +∞ 0 e -η s 1 [δ h /2,+∞) M s+θ -M η s +θ ds 2 ≤ E 1 θ <+∞ e -θ E +∞ 0 e -η s 1 [δ h /2,+∞) M s+θ -M η s +θ ds ≤ E 1 θ <+∞ e -θ +∞ 0 e -η s P sup η s +θ ≤u≤s+θ M u -M η s +θ ≥ δ h 2 ds ≤ C E 1 θ <+∞ e -θ +∞ 0 e -η s exp - 1 8K 2 δ 2 h h ds ≤ C E 1 θ <+∞ e -θ exp - 1 16K 2 h -2ǫ . ( 28
)
One then deduces (18) from ( 21), ( 23), ( 27) and (28).

End of the proof of Theorem 1.1

We apply Lemmas 4.1 and 3.1 and the equality (11) and deduce

hE   +∞ p=0 e -ph 1 [-δ h ,δ h ] (X ph )   = +∞ p=0 E   1 θ h p <+∞ τ h p θ h p e -η s 1 [-δ h ,δ h ] (X η s )ds    ≤ Cδ h +∞ p=0 E 1 θ h p <+∞ e -θ h p ≤ Cδ h +∞ p=0 1 cosh(µ) p 2
, which ends the proof of Theorem 1.1.

Proof of Theorem 1.2

In the finite horizon setting, we do not need to localize by means of an appropriate sequence of stopping times as in the preceding section. We may start from

E   h N h p=0 f (ph)1 [-δ h ,δ h ] (X ph )   = N h p=0 E (p+1)h ph f (η s )1 [-δ h ,δ h ] (X η s )ds . ( 29
)
In view of ( 22) we have

E   h N h p=0 f (ph)1 [-δ h ,δ h ] (X ph )   ≤ E T 0 f (η s )1 [-2δ h ,2δ h ] (X s )ds + N h p=0 E D p h =: A h (T ) + N h p=0 E D h p , ( 30
)
where for all p ∈ {0, . . . , N h }

D h p := (p+1)h ph f (η s )1 [-δ h ,δ h ] (X η s )1 R\[-2δ h ,2δ h ] (X s )ds. ( 31
)

An upper bound for A h (T ).

In view of (9), as the function f is increasing, one has

A h (T ) ≤ K 2 E T 0 f (s)1 [-2δ h ,2δ h ] (X s )σ 2 s ds .
Since we may have lim s↑T f (s) = +∞, the proof of Lemma 4.1 does not apply.

Set τ y := inf{s > 0 : X sy = 0}. The generalized occupation time formula (see, e.g., Revuz and Yor [6, Ex.1.15,Chap.VI]) implies that

E T 0 f (s)1 [-2δ h ,2δ h ] (X s )σ 2 s ds = R d y 1 [-2δ h ,2δ h ] ( y)E T 0 f (s)d L y s (X ) = R d y 1 [-2δ h ,2δ h ] ( y)E 1 τ y ≤T T τ y f (s)d L y s (X ) , (32)
where {L y s (X ) : s ∈ [0, T ], y ∈ R} denotes a bi-continuous càdlàg version of the local time of the process (X t ) t≥0 .

Our aim now is to bound from above

E 1 τ y ≤T T τ y f (s)d L y s (X ) uniformly in y in [-2δ h , 2δ h ].
It clearly suffices to show that there exists a constant C T depending only β, K and T , such that

E T 0 f (s)d L 0 s (Y ) ≤ C T , (33)
where the semimartingale Y is defined as Y := Xy. Because f is not square integrable over [0, T ), we arbitrarily choose γ > 0. Now f is square integrable on [0, T -γ] and the function t

→ t 0 f (s)d L 0 s (Y ) is the local time of the local semi- martingale { f (g t )Y t , t ∈ [0, T -γ]},
where g t := sup{s < t : Y s = 0} (see, e.g., Revuz and Yor [6, Sec.4,Chap.VI]). The Stieltjes measure induced by the function s → f (g s ) has support := {s ≥ 0 : Y s = 0}, and therefore

T -γ 0 |Y θ |d f (g θ ) = 0, so that Itô-Tanaka formula implies E T -γ 0 f (s)d L 0 s (Y ) = E | f (g T -γ )Y T -γ | -f (g 0 )|Y 0 | -E T -γ 0 f (g s )sgn((Y s ))d Y s . ( 34
)
Apply Theorem 4.2 of Chap.VI in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]. It comes:

E | f (g T -γ )Y T -γ | = E | f (0)Y 0 + T -γ 0 f (g s )d Y s | , from which E | f (g T -γ )Y T -γ | ≤| f (0)Y 0 | +E | T -γ 0 f (g s )σ s dW s | + E | T -γ 0 f (s)b s ds | ≤ C T + K 2 T 0 E f 2 (g s ) ds 1 2 . ( 35
)
Coming back to (34) we deduce

E T -γ 0 f (s)d L 0 s (Y ) ≤ C T + K 2 T 0 E f 2 (g s ) ds 1 2
.

Thus, as β > 1 by assumption, (33) will be proven if we can show that

T 0 Ẽ f 2β (g s ) ds < +∞, ( 36
)
where P is the probability measure defined by a Girsanov transformation removing the drift of

(Y t , 0 ≤ t ≤ T ). Set Λ(t) := 〈Y 〉 t . Observe that sup{θ ≤ s; Y θ = 0} = Λ -1 sup{θ ≤ Λ(s); Y Λ -1 (θ ) = 0} .
In addition, the Dubins-Schwarz theorem implies that there exists a P-Brownian Motion ( Bt ) t≥0 such that Y s = BΛ(s) . Therefore,

P g s ≥ u = P Λ -1 sup{t<Λ(s) : X Λ -1 (t) =0} ≥ u = P sup{t < Λ(s) : Bt = 0} ≥ Λ(u) = P inf{t > Λ(u) : Bt = 0} ≤ Λ(s) ≤ P inf{t > 0 : Y u + Bu+t -Bu = 0} ≤ K 2 (s -u) , from which P g s ≥ u ≤ Ẽ   K 2 (s-u) 0 1 2πθ 3 |Y u | exp - Y 2 u 2 θ dθ   ≤ C Ẽ exp - Y 2 u 4K 2 (s -u) . For 0 ≤ v < s set m v := - v 0 Y θ 2K 2 (s -θ ) d Y θ . Itô's formula applied to h(v, y) = y 2 s -v yields Y 2 v s -v = Y 2 0 s + v 0 2Y θ s -θ d Y θ + v 0 Y 2 θ (s -θ ) 2 dθ + v 0 1 s -θ d〈Y 〉 θ ,
and then

- Y 2 v 4K 2 (s -v) ≤ m v - 1 4K 2 v 0 Y 2 θ (s -θ ) 2 dθ - 1 4K 2 v 0 1 s -θ d〈Y 〉 θ , Therefore, using (9), Ẽ exp - Y 2 v 4K 2 (s -v) = Ẽ exp m v - 1 2 〈m〉 v + 1 2 〈m〉 v exp - v 0 Y 2 θ 4K 2 (s -θ ) 2 dθ - v 0 d〈Y 〉 θ 4K 2 (s -θ ) ≤ Ẽ exp m v - 1 2 〈m〉 v + 1 2 v 0 Y 2 θ 4K 4 (s -θ ) 2 d〈Y 〉 θ exp - v 0 Y 2 θ 4K 2 (s -θ ) 2 dθ - v 0 dθ 4K 4 (s -θ ) ≤ Ẽ exp m v - 1 2 〈m〉 v + v 0 Y 2 θ 8K 2 (s -θ ) 2 dθ - v 0 Y 2 θ 4K 2 (s -θ ) 2 dθ exp - v 0 dθ 4K 4 (s -θ ) ≤ Ẽ exp m v - 1 2 〈m〉 v exp - v 0 dθ 4K 4 (s -θ ) , from which Ẽ exp - Y 2 v 4K 2 (s -v) ≤ exp - 1 4K 4 v 0 dθ s -θ = s -v s 1 4K 4 .
As f is increasing, we deduce

Ẽ f 2β (g s ) = f 2β (s) f 2β (0) P f 2β (g s ) ≥ u du + f 2β (0) = f 2β (s) f 2β (0) P g s ≥ ( f 2β ) -1 (u) du + f 2β (0) = s 0 P g s ≥ v 2β f 2β-1 (v) f ′ (v)d v + f 2β (0) ≤ C T s 0 s -v s 1 4K 4 f 2β-1 (v) f ′ (v)d v + f 2β (0).
Therefore, using the Fubini-Tonelli theorem for positive functions and the hypothesis (5), we get

T 0 Ẽ f 2β (g s ) ds ≤ C T T 0 d v f 2β-1 (v) f ′ (v) T v s -v s 1 4K 4 ds + C T T f 2β (0) ≤ C T T 0 d v v 1 4K 4 f 2β-1 (v) f ′ (v)(T -v) 1+ 1 4K 4 + C T T f 2β (0) < +∞. (37)
In view of (32), (33), (36), and (37), we deduce that

A h (T ) ≤ C T δ h . ( 38
)
An upper bound for D h p . We proceed as in the proof of Lemma 4.1. We recall that M s := s 0 σ u dW u . The main modification is the following one: As

D h p ≤ h 0 f (ph)1 [δ h /2,+∞) M s+ph -M ph ds, we deduce ED p h ≤ E h 0 f (ph)1 sup 0≤u≤s |Mu+ph-Mph|≥δh/2 ds ≤ f (ph)E h 0 1 sup 0≤u≤s |Mu+ph-Mph|≥δh/2 ds ≤ C f (ph)exp - 1 16K 2 δ 2 h h ≤ C h f (ph)δ h . (39)
From ( 30), (38), and (39), we deduce that

hE   N h p=0 f (ph)1 [-δ h ,δ h ] (X ph )   ≤ C T δ h + Cδ h N h p=0 h f (ph) ≤ C T δ h + Cδ h T 0 f (s)ds ≤ C T δ h .
This ends the proof of Theorem 1.2.

Appendix

In this section, we shortly explain why our sharp estimates are useful in the convergence rate analysis of discretization schemes for stochastic differential equations with irregular coefficients. Consider a standard stochastic differential equation with smooth coefficients b and σ. Denote by P t the transition operator of the solution (X t ). Let (X h t ) be the Euler discretization scheme with step-size h. A now classical result (see Talay [START_REF] Talay | Probabilistic numerical methods for partial differential equations: elements of analysis[END_REF]) is that, for all smooth function f , or all measurable function f under an hypoellipticity condition,

∃C > 0, ∀n ≥ 1, E f (X h T ) -E f (X T ) = Ch + (h 2 ).
A preliminary step in the proof consists in getting the following expansion:

E f (X h T ) -E f (X T ) = h 2 N h p=0 EΨ ph, X h ph + N h p=0 R h p ; (40)
here we have set N h := ⌊T /h⌋ -1 and either by using the smoothness up to T of the function u(s, x) when f is smooth, or by using Malliavin calculus techniques when f is measurable only. In both cases, the smoothness of b and σ is necessary. When b or σ are locally discontinuous, the derivatives of u(s, x) may explode around their singularities. In order to prove that, however, the convergence rate of Euler type schemes is not worse than h 1/2-ε , one needs to modify (40) by considering the event A h p := [X h ph ∈ h ], where h is a neighborhood of the singularities whose size depends on h: to summarize long and technical calculations, one expands u((p + 1)h, X h (p+1)h )u(ph, X h ph ) only when A h (p+1) or A h p do not occur; if A h (p+1) and A h p occur, one needs to take into account sharp estimates on P(A h p ). In the case where the generator of (X t ) is a divergence form operator with a discontinuous coefficient, one even has to get sharp estimates on

R h N h := E f (X h T ) -E(P h f )(X h T -h );

for all p < n - 1 ,h s 3 )

 13 the function Ψ(ph, •) can be expressed in terms of the functions b, σ, u(ph, •) := P T -ph f (•) and their derivatives up to order 5; R h p is a sum of terms, each one of the form∂ α u(s 3 , X h s 3 ) + ϕ ♭ α (X s 3 )∂ α u(s 3 , X s 3 ))ds 3 ds 2 ds 1   ,with |α| ≤ 6; the functions ϕ ♮ α 's, ϕ ♯ α 's, ϕ ♭ α 's are products of functions which are partial derivatives up to order 3 of b and σ. A next step in the proof consists in proving that∃C > 0, ∀h < 1, max 0≤p≤N h -1 |R h p | ≤ Ch 3 ,

  comes from the explosion rate in time of the spatial derivatives of u(Tt, x) near the singularities of σ (see Martinez and Talay[START_REF] Martinez | Time discretization of one dimensional Markov processes with generators under divergence form with discontinuous coefficients[END_REF] for details).


Lesley Graham

AN UNPUBLISHED LETTER FROM ROBERT LOUIS STEVENSON TO VIOLET PAGET (VERNON LEE), 1885

Earlier this year a manuscript letter by Robert Louis Stevenson Lee's unflattering and one-sided treatment of him as a repulsive drunk in her account of the life of his wife in The Countess of Albany. [START_REF] Lee | The Countess of Albany[END_REF] Stevenson mentioned the prince in Kidnapped the following year: 'the Prince was a gracious, spirited boy, like the son of a race of polite kings, but not so wise as Solomon. I gathered, too, that while he was in the Cage, he was often drunk; so the fault that has since, by all accounts, made such a wreck of him, had even then begun to show itself.' [START_REF] Stevenson | Kidnapped[END_REF] He was also to write about the Prince several years later in the novel fragment The Young Chevalier (1892) [START_REF] Stevenson | Robert Louis Stevenson's "The Young Chevalier": Unimagined Space[END_REF] which paints a brief but psychologically nuanced portrait of "a boy at odds with life, a boy with a spark of the heroic, which he was now burning out and drowning down in futile reverie and solitary excess" (82-3); a portrait in line with the plea for indulgence expressed in this letter.

The bottom right hand corner of the first page of the letter is torn and the ends of four lines are consequently missing. In each case, our best guess as to the missing words or letters has been inserted between square brackets with a question mark. Skerryvore Bournemouth Oct 14. 1885 Dear Madam, I shall attend to the affair of Signora Santarelli 7 with my best diligence, which is a relative diligence. It is right, however, that I should explain to you how I stand. If the permission be granted in the case of the first series it will be of the grace of Messrs Chatto & Windus; and if in the case of the second, Signora Santarelli will have to divide her thanks between the authors and Messrs Longman. 8 So far the authors are concerned, it is already done; neither my wife nor I would dream of denying any invalid what may possibly prove to be an entertainment: we have both unfortunately too much reason to sympathise with the sick.

Your Prince 9 has arrived only this morning; and I have only read the introduction: if the rest be at all of a piece with it, you have sent me a great treat.

I believe we have two more common friends than you all[ude?] to 10 : Symonds 11 and Prince Charlie. I, who had mostly s[tolen?] the bright pages of Charle's [sic] Stuart's life, felt it as perhaps[s a?] defect in your very interesting "Countess of Albany", that yo[u had] failed to bring out the contrast. He was a bright boy; rather he was the bright boy of history, full of dash, full of endurance[,] full of a superficial 12 generosity, of blood more than of mind; he lived through great feats and dangers not unworthily. I should have liked perhaps, if you could not screw out a tear over so base a fall, that you had smiled a little sadly. We may all fall as low before we are done with it, and not have the picturesque and generous to look back upon. And indeed if you introduce your pretty countess to the bottle and keep her for months in Hebridean caves 13 with no other consolation, I suspect she would sink as low.

I am a fault finder in grain 14 and you must not wonder if I sieze [sic] on the occasion of your letter to pick this quarrel which I have long been musing.

(I am amused at the way in which I have bracketed the living lion and the dead dog 15 , but I meant no disrespect to either, surely not to Symonds), With many thanks Believe me Yours truly 7 Signora Sofia Fortini-Santarelli, translator, wife of Cavaliere Emilio Santarelli of Florence who owned relics of the Young Pretender. She translated various works of Herbert Spencer, Ouida, and Symonds's The Renaissance in Italy. In her letter, Paget describes her as "a lady who has taken to translating for the pastime which her recuperation affords her in a maiming & incurable malady". 8 The references are to New Arabian Nights (1882) and More New Arabian Nights: The Dynamiter (1885). The latter was written in collaboration with Fanny Van de Grift Stevenson. 9 Vernon Lee, The Prince of the Hundred Soups: A Puppet Show in Narrative (1883), a harlequinade. 10 Paget had mentioned Henry James and John Sargent as friends they had in common.

11 John Addington Symonds (1840-1893), essayist, poet, and biographer best known for his cultural history of the Italian Renaissance. 12 Stevenson has deleted 'and not very wise'. 13 Charles hid out in some remote refuges in Benbecula and South Uist between April and June 1746. 14 In grain: short for dyed in grain i.e. ingrained (OED). 15 I.e. the two "acquaintances" they have in common: A. E. Symonds and Charles Edward Stuart.

  photographs of the two-page letter to their Twitter account.1 The letter does not

Posted on July

23rd 2019: https://twitter.com/The_PBS/status/1153667768287485952 2 The Letters of Robert Louis Stevenson, ed. Bradford A. Booth and Ernest Mehew (New Haven: Yale University Press: 1994-5). 3 GEN MSS 664 Series I, (5363-5364).


Tatenda Mpofu

Pitso Tsibolane
email: pitso.tsibolane@uct.ac.za

Richard Heeks
email: richard.heeks@manchester.ac.uk

Jean-Paul Van Belle
email: jean-paul.vanbelle@uct.ac.za

Risks and Risk-Mitigation Strategies of Gig Economy Workers in the Global South: The Case of Ride-Hailing in Cape Town

Keywords: gig economy, platform economy, ride-hailing, risk, South Africa

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Gig work -tasks done for money and managed via digital platforms that are marketplaces bringing together buyers and sellers [adapted from 12] -already employs up to 40 million people in the global South -some 1.5% of the workforce -and growth rates are rapid [START_REF] Heeks | How many platform workers are there in the global South?[END_REF]. Alongside high hopes around job creation and worker perceptions of benefits of such work, there have also been concerns that such work is precarious and risky [10,[START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF]. While there has been some research on gig work risks [START_REF] Christie | The health and safety risks for people who drive for work in the gig economy[END_REF][START_REF] De Stefano | The rise of the just-in-time workforce: On-demand work, crowdwork, and labor protection in the gig-economy[END_REF] little of it has focused on developing countries, and we found no research yet focused on risk-mitigation strategies. Yet such strategies are vital given the dangers with, for example, riders and drivers dying in the global South on a regular basis [START_REF] De Greef | One more way to die: delivering food in Cape Town's gig economy[END_REF][START_REF] Phillips | Murder of Uber drivers may be Brazil gang leader's 'revenge' for cancelled ride[END_REF].

To investigate these issues in more detail, the research reported here undertook an inductive, interpretive study of risks endured and risk-mitigation strategies adopted by ride-hailing drivers in Cape Town, South Africa. Following a short literature review, and an explanation of case methodology, the paper presents two sets of findings -on risks and on risk-mitigation strategies. We end with some recommendations and ideas for future research.

Literature Review

As noted above, the gig economy in the global South is already sizeable, providing work for tens of millions of workers with employment growth rates estimated at up to 30% per year [START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF]. There are many benefits brought by the growth in gig work: it seems likely to be increasing the quantum of employment in the global South; it is improving services and reducing costs for clients; and it can provide work for those excluded from other forms of employment [16]. But there have also been many concerns about the rapid growth in this form of employment; especially in comparison to traditional, formal-sector employment [START_REF] Graham | The Fairwork Foundation: strategies for improving platform work in a global context[END_REF]. In particular, that the mediation of digital platforms exacerbates labour market inequalities; notably those between capital (i.e. the platforms) and labour (i.e. the workers) [START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF]. These inequalities can be understood in various different ways: for example, inequality in capture of the financial benefits of gig work, or information inequality comparing the panopticon-style understanding of the platform compared to the porthole-style understanding of workers.

We could also see a potential inequality of risk, if the risks of work are being loaded by platforms onto workers, and if platforms are taking few responsibilities for mitigating risks or for responding when risks materialise [START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF]. This is something of an issue for digital gig work, defined as location-independent digitally-centred activity such as data entry, translation and web development via platforms such as Amazon Mechanical Turk, Upwork and Freelancer. Risks reported by global South digital gig workers include discrimination by clients who prefer to employ workers from the global North, over-work, or clients refusing to pay for work [START_REF] Graham | The Risks and Rewards of Online Gig Work at the Global Margins[END_REF][START_REF] Malik | Understanding the development implications of online outsourcing[END_REF].

Risks are much greater, though, for physical gig work, defined as location-bound physical activity such as taxi driving, food delivery and house cleaning via platforms such as Uber, Deliveroo, Rappi and GoJek. As noted above, at its most extreme, physical gig workers in developing countries are at risk of dying during their work. Other risks identified include "accidents, breakdowns, robbery, traffic tickets", lack of income and being stuck in traffic [START_REF] Gomez-Morantes | A multi-level perspective on digital platform implementation and impact: the case of EasyTaxi in Colombia[END_REF] and "fatigue, stress, hunger and sleep deprivation" [START_REF] Kashyap | Taxi drivers and taxidars: a case study of Uber and Ola in Delhi[END_REF]. These risks are undoubtedly greater within developing countries; impacted by a context of, for example, greater insecurity and crime in terms of personal risk; and a context of lack of unemployment benefits and other protections in terms of financial risk. One must take care when enumerating the risks of gig work in developing countries to ask what the comparison is -the alternative for many gig workers is not well-protected formal employment but equally-risky work in the informal sector or even black economy. However, this does not reduce the risks of gig work or the need to do something about them.

It is the "do something about them" that has been little researched to date: we were unable to identify any previous work focused on gig work risk-mitigation strategies. It is this knowledge gap that motivates the current study.

Case and Methods

We chose to focus our study on ride-hailing in South Africa for two reasons. The gig economy in the country is relatively important. One estimate puts the number of physical gig workers at 30,000 [START_REF] Heeks | How many platform workers are there in the global South?[END_REF] with up 100,000 active digital gig workers [START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF][START_REF] Onkokame | What is the State of Microwork in Africa?[END_REF] While many of the latter will not be full-time, this still suggests gig work touches at least 1% of the workforce and, as noted above, growth rates are high. Second, this sector reflects wider patterns of the global South gig economy with risks and materialised dangers being a constant theme [START_REF] Burke | Violence erupts between taxi and Uber drivers in Johannesburg[END_REF][START_REF] De Greef | Driving for Uber when you can't afford a car[END_REF]. At the time of research, there were two main platforms operating in South Africa: Uber which has nearly three-quarters of the market with just over five million registered users and Bolt which has just over a quarter of the market with around two million users [START_REF] De Villiers | Taxify scraps ad that claims drivers earn R8000 a week[END_REF]. There was also a recent arrival -In-Driver -which was in process of trying to break into the market.

Because of the lack of prior work on risk mitigation and of understanding risks from a worker perspective, it was decided to follow a relatively inductive and interpretive research design. While sensitised to typical gig work risks and to some of the key issues to understand when identifying risk and mitigation strategies [START_REF] Rogers | Protection motivation theory[END_REF], this allowed drivers to represent their own lived experiences and for the researchers to then analyse and structure those experiences. The research complied with the University of Cape Town's prescribed code of ethics and obtained the necessary permission. Eighteen [START_REF] Malik | Understanding the development implications of online outsourcing[END_REF] semistructured, face-to-face interviews were conducted with a random / convenience sample of drivers, contacted through a combination of Uber/Bolt ride-booking and snowballing. The interview method was chosen because the objectives of this research centred on understanding experiences, opinions, attitudes, values, and processes relating to driver risk mitigation strategies. Data analysis followed a thematic analysis protocol: familiarisation with data, searching for and review of emergent themes (in this case, both risks and risk-mitigating strategies), definition and naming of themes [START_REF] Braun | Using thematic analysis in psychology[END_REF].

All drivers were men, reflecting the gender-skewed profile of ride-hailing in South Africa (and most of the global South). As summarised in Table 1, two-thirds were from Zimbabwe, three were from South Africa, and three from other African countries (DRC and Zambia); reflecting the heavy presence of immigrant workers as riders and drivers in South Africa's gig economy. Seven of the eighteen owned their vehicle; two were on a rent-to-own financing deal; the remaining nine were renting from an owner. The great majority (14 of the 18) said they were the main breadwinner for their household. Ten drivers worked only for Uber, three only for Bolt, and five worked for both platforms. Two drivers had registered with In-Driver but the novelty of the platform and lack of work meant it did not impinge on their responses. On average, they had worked for a ride-hailing platform for two years (the range was two months to four years).

✔ ✔ ✔ 2 ✔ ✔ ✔ 3 ✔ ✔ ✔ 4 ✔ ✔ ✔ 5 ✔ ✔ ✔ ✔ 6 ✔ ✔ ✔ 7 ✔ ✔ ✔ 8 ✔ ✔ ✔ ✔ 9 ✔ ✔ ✔ 10 ✔ ✔ ✔ 11 ✔ ✔ ✔ 12 ✔ ✔ ✔ 13 ✔ ✔ ✔ 14 ✔ ✔ ✔ 15 ✔ ✔ ? 16 ✔ ✔ ? 17 ✔ ✔ ✔ 18 ✔ ✔ ✔

Findings

Risks

We categorised the risks identified by drivers into three main themes/issues: income, safety and deactivation.

Income

Given their role as main breadwinners for their households, drivers felt a strong need to earn a certain level of income. This was exacerbated for many because they had other required payments: financed loans in the case of some of those who owned cars; weekly payments to owners of R2,000-R2,700 (c.US$130-180) in the case of those who were renting their vehicles. Meeting these multiple demands was jeopardised in various ways. Though not a risk but an integral part of ride-hailing, drivers resented the percentage of client payments taken by the platforms: 20% by Bolt plus a 5% booking fee; 25% by Uber plus a 3% booking fee at the time of writing [START_REF] Malinga | Uber, Bolt drivers plan shutdown of ride-hailing services[END_REF]: "we feel like Uber is ripping us off; 25% is too much. l feel like they are taking advantage of the situation in South Africa about unemployment and then now they just dictate it's going to be 25% … if you don't want the 25% to be taken what else are you going to do?" (D101 ) Dynamically, and because of their ability to control entry into the marketplace, platforms have increased the supply of cars/drivers on the platform over time. This has led to over-supply compared to demand, and drivers have thus faced falling income [START_REF]Market Inquiry into Land Based Public Passenger Transport: Metered Taxis and E-Hailing Services[END_REF] and concerns that platforms should stop adding cars in order to enable drivers to make enough income (D14). Further growth of supply and competition is a key part of income risk.

A central element of platforms' strategy has been to avoid regulation [16] and part of this has been to allow drivers to work without a taxi license, which the majority of drivers do: "You see I'm operating this car; it has no taxi operating license as you can see it only has one disc there for the vehicle" (D12). This therefore opens the drivers up to police impounding their vehicles and fining them. We estimate at least a 20% chance of being caught each year in Cape Town, with fines for repeat offences escalating up to R15,000 (just under US$1,000) and with drivers then unable to work while the police hold their car.

Safety

By far the main safety-related risk is the danger of robbery, including violent robbery:

"I was robbed. Someone requested when I was in Bellville and those guys were good looking; you can't even think that they can rob you. So their requesting was saying they are going to Bellville South, there were three; I drove them to Bellville South. So when we were almost close to the place of destination so they said don't mind that map we will direct you to the place we want to go, so they kept on saying turn right, turn left and go straight this and that until we got to a place which is a bit quiet and dull and the next thing they started pulling off their guns and they said I should with my phone give us the password so myself I had no choice I have to give them, bring that pocket and bring the next pocket pull it out and … then they took everything which I had." (D15) Drivers explained why they face these risks. When ride-hailing first launched in South Africa, it followed a global North model of payment via credit card but client demand led to cash payment options then being added, meaning criminals assume drivers carry cash. By definition, gig work also requires that they have a phone: "So they just want the money; they know every Uber driver has a smartphone and they think we carry money all the time." (D4) "Like I have a nice phone, I'm risking again because if you have a cheap phone it doesn't work here. Because this phone costs R1,800 and the thief will sell it for R500." (D14) Like many global South cities, Cape Town is heterogeneous in terms of its areas, with a number of high-crime locations that drivers may be asked to pick up from or drop off to, increasing the risk that they will be victims of crime:

"You see on Bolt, most of the Bolt clients use cash trips so and normally they go to those dangerous locations; Mitchells Plain, Hanover Park." (D4) The concern over cash-paying clients arises because the level of client identification is lower than for those paying via credit cards; making it possible for criminals to moreeasily create fake accounts. This was even more of a problem on Bolt because the platform provided few details: "I mean you get a request and you don't know whom you are picking … where they want a ride, or they want to rob you. You can't tell." (D5)

A secondary safety risk arises from the danger of altercations with traditional meter taxi drivers who have been vehemently and sometimes violently opposed to the arrival of ride-hailing platforms in South Africa [START_REF] Mitchley | Uber and meter taxi feud claims life of driver[END_REF]:

"I also faced another that I was in Kuils River and I wanted to pick a rider … The moment the rider came to my car a taxi driver came to me he threatened me: 'Hey you Uber, who told you can pick up people here I am going to kill you' … I quickly picked up the rider and drove away … he was following me, but I just drove quickly off the place." (D12)

Deactivation

Very much related to concerns about earning enough income, drivers were concerned about the risk of deactivation from the platform which of course would mean they would be unable to work or earn. This could typically occur if their rating on the platform fell below a certain level and/or if a client made a complaint against them. This puts a lot of power into the hands of the client:

"I should exercise my rights as a driver; I can be an Uber driver and I'm still human, I have the right to express myself. But with this Uber app everyone who requests automatically becomes my boss or dictator or something. Someone comes into my car they start telling me can you do this, do that; they call it service but to us, on the other hand, it's a form of abuse and people take advantage of and tell us if you don't do this I will rate you and Uber will deactivate you." (D10) It also puts a lot of power and information in the hands of the platform, as the same driver related:

"Whenever a client is treating you badly, they know they have Uber at their back. Tomorrow Uber is going to support the client, of course they don't say that but that's the issue. We find our riders are becoming abusive to the drivers, they become rude, they don't treat us well, of course, we have no choice. For example, this is the only job that I have so now I'm forced to take what I don't want. I have an example one of the days they send me an email and they tell me that a rider complained; that a rider said I behaved in an inappropriate manner. Because that didn't go well with me I wrote an email back asking okay fine I got the email and they tell me that they can deactivate my account. I asked them if they could provide the details like which details were that, so that I take legal action if there's need. For me this was like defamation of character that client has taken advantage of me; because I'm not there and I have no room to defend myself they said whatever they liked about me, because now my job is at risk I don't know what you think about me also one thing that if tomorrow there's another complaint that means they are two and I might get deactivated. They told me because of their policy they don't provide details. These are things that we face. This was the end of the story there is nowhere I can go; I cannot face them if they can just block me and that's the end of my job. … My man, I feel like being oppressed; I have no more rights I can say ever since I joined Uber it's difficult for me to express my rights especially when I'm doing the job." (D10)

Risk-Mitigation Strategies

Before discussing risk-mitigation strategies, we must first note the number of drivers who do not adopt such strategies. In particular, the pressure to earn income -exacerbated by the platform's management of the market to create an excess of supply (drivers) over demand (riders) -has led drivers to engage in high-risk behaviours. Some simply hope or believe that nothing bad will happen to them; others turn to higher powers: "Like I said its God who protects me" (D1). But this is not enough: as an example, drivers still go to Nyanga at night: a district notorious as the "murder capital" of South Africa [START_REF] Mlamla | Crime stats[END_REF]. Predictably, there have then been repeated incidents of drivers murdered during robberies and hijackings [START_REF] Daniels | Cops offer to escort e-hailing drivers to Nyanga after murder[END_REF][START_REF] Nwokocha | Tragedy as second Nigerian taxi driver Joseph Ajouna murdered in South Africa within a week[END_REF]. Of the risk-mitigation strategies reported by the drivers, we identified three different themes/types: platform-initiated, driver-initiated, and driver group-initiated.

Platform-Initiated Risk Mitigation

Platforms may mandate certain risk-mitigation strategies. For example, "You can't drive on Uber without insurance" (D4): there is a check to ensure that all cars registered on the platform have vehicle insurance to avoid major financial loss in case of an accident.

Platforms may also provide risk mitigations themselves. Both Uber and Bolt provide a panic button as part of their app: interviewees were able to demonstrate its presence but were somewhat dubious about its value: "That one [panic button] doesn't work because in a robbery that's [phone] the first thing they will grab before you do anything in the car" (D4). Uber, alone, provides two other mitigations against loss of income. "If this car gets impounded by the traffic cops, Uber, they pay the amount to get back the vehicle but Bolt doesn't pay the amount" (D12). Uber will also pay the driver to cover the costs of riders who run off at the end of a journey without paying.

Individual Driver-Initiated Risk Mitigation

By far the largest number of strategies identified were the responsibility of individual drivers.

Income Mitigation

To mitigate the risk of not achieving the required level of income, drivers work very long hours: "It's not easy; the thing is like if you're working for someone, you need to pay the R2,500 a week [to the vehicle owner]. For you to pay that, to get that R2,500 a week you must be working; so you need R2,500 to pay the car first, money for your petrol, and money for yourself, so you must be on the road for long. That's why you see so many of the guys don't even go home, that's why they stay on the road." (D8) The average working day for drivers was just over 12 hours. Given they typically work six days a week, that means on average more than 72 hours per week; 60% longer than the maximum 45-hour working week set out in the country's Basic Conditions of Employment Act [START_REF] Rsa | Basic Conditions of Employment Act 1997, Republic of South Africa[END_REF]. Another platform-mandated risk mitigation is the imposition of a 12-hour maximum working day. However, workers get around this by registering on, and working for, more than one platform: hence D5 reported working up to 17 hours per day. Workers also register on multiple platforms in order to avoid the risk of underearning; taking rides on one platform if business is slack on the other.

There are also very situation-specific strategies. On Bolt, which does not compensate for rider non-payment, drivers reported "we sometimes take their [rider's] belongings so that we can recoup the money we used for our petrol" (D9) if the rider says at the end of the journey that they cannot pay. The intention is that the items are just held temporarily while the rider goes to get money for payment, to reduce the risk of nonpayment.

Safety Mitigation

To mitigate risks to personal safety, drivers undertake rider screening. Some of this is platform-assisted: drivers will look at the rider's rating on the platform and may refuse those with low ratings:

"You know we have this rating. When I see a 3 that person is in trouble and when I see 4 I don't doubt until 4.9. If it's 3, I doubt especially their attitude, 5 stars they are probably new, I don't trust them also." (D17) They may also refuse those seeking to travel to or be picked up from locations with high-crime reputations; and they may refuse those who are seeking to pay cash, particularly when travelling at night. In addition, on arrival at the pick-up, drivers assess the rider(s): and may drive off if they perceive the rider is drunk, or if the riders are a group of men. Again, these rider-refusal behaviours are more likely at night. Of course this all comes at a cost: not just loss of income but also algorithmic downgrading due to turning down rides; but those costs are seen to be outweighed by the benefits of risk mitigation.

Drivers also undertake various location-specific risk-mitigation strategies if they are driving in an area they consider to be dangerous. If picking someone up from such a location, they may avoid stopping until the rider is ready to quickly jump into the car:

"For me when I gets to the dangerous places, I don't stop my car when I get to the pick-up point. I just patrol there like someone patrolling while I inform the rider." (D11) When picking up someone to go to these areas, they may agree only to take the client to a safe place: "If he or she is going to Gugulethu … I don't mind going there but at this time I don't feel safe so I give the client options … either I take him halfway down when I get to this point or I drop you near the police station." (D5) They may share details of the trip with family members or with other drivers, and they may get riders to pay prior to reaching the destination so that at the drop-off, the driver can just drop them very quickly and leave immediately.

Lastly in terms of safety, drivers adopt various personal protection measures. They avoid wearing seatbelts at night so that they can quickly turn around or get out of the vehicle if necessary. Platform policy is that "we are not allowed to carry any weapon in the car" (D4) but one did admit to having a can of pepper spray, and another stated that, while he himself did not have a weapon, others did: "maybe they have weapons in their cars: that's the best way they can protect themselves. If you attack them [the driver] then they attack you if they have a chance to." (D5) There are certainly reports of drivers using weapons they were keeping their cars [22].

Deactivation Mitigation

The main strategy that drivers adopted here was 'making nice' with clients. As D10 noted above, when a rider says "do this, do that" then drivers feel obliged to comply in order to avoid getting a bad rating and risk being deactivated: "we have no choice".

Driver Group-Initiated Risk Mitigation

To help address some of the income challenges they faced, some drivers had joined what is locally known as Stokvels -rotating savings and credit schemes [START_REF] Menze | Online Stokvels: the use of social media by the marginalized[END_REF]. These have a general intention to encourage drivers to save their earnings, enabling them to make more effective use of their income and therefore reduce the need to accept highrisk trips or riders. They have also been used more specifically to help drivers buy their own car; something which increases income relative to having to rent a car, and thus addresses both income and safety risks, again by reducing the need for high-risk trips:

"Yah especially us Zimbabweans, that's how we bought our own cars. We always do the groups like Stokvel; put the money together until its fine for one person to buy and then the next [time] we buy for someone else." (D7) On a more day-to-day basis, drivers share information either via WhatsApp groups or by meeting up in some set locations in-between trips or when waiting at the airport. Information shared includes locations where the police are impounding license-less vehicles, or where crimes have been committed, or other experiences which can alert drivers to implement risk-mitigation strategies. As well as being a mechanism for 'horizontal' sharing between peers, this was also acting in a more 'vertical' manner to help new drivers learn quickly about key risks and ways to avoid them.

Discussion and Conclusions

Clients face risks in using ride-hailing services: there are occasional reports of robbery or attack or they may be caught up in accidents. Likewise, platforms themselves also face risks such as lawsuits or bankruptcy. However, overall, and as proposed in the literature [START_REF] Heeks | Decent Work and the Digital Gig Economy[END_REF], this research exposed a strong asymmetry of risk between the stakeholders of gig economy platforms in the global South. It is the workers who bear the lion's share of the risks; many of which arise from the specifics of the South Africa context: risks of not earning enough to cover necessary outgoings; risks to personal safety up to and including death; risks of losing their job through deactivation. While these are risks identified in the literature, it is valuable to receive bottom-up corroboration from gig economy workers. Just because it is workers who must face the greatest risks, there is no necessity for workers to have to undertake most of the risk mitigation; a topic about which the literature has so far told us little. Yet this was the situation here. Digital platforms could take on more risk and more responsibility for risk mitigation, but they do not do so because their business model requires them to treat workers as 'independent contractors' and not as employees:

"Uber does not want to come as an employer, it comes out as a service provider by providing us with the application. For example, when it comes to an accident, Uber does not involve itself, they do not compensate hospital or funerals as there are a lot of drivers who have passed away from this." (D18) Similarly, it could easily design away the current information asymmetries between clients and workers; for example, requiring more details from clients when they register, and providing drivers with full information about clients. Both of these would reduce risks. It could also 'red-line' certain areas of the city where drivers have been repeatedly attacked and stop drivers having to travel there. Again, this would reduce risks. It does none of these things because they would potentially reduce business, and platforms themselves are under the pressure of high levels of debt to maximise income and profit.

The result, then, is that responsibility for risk mitigation is laid largely at the door of the individual worker; consistent with the gig economy view of an atomised, individualised workforce, and with the idea of risk inequalities between platforms and workers. To some extent, workers accept this, trading off the risks against the benefits of an income and the autonomy and flexibility they perceive such work to offer [see also [START_REF] De Stefano | The rise of the just-in-time workforce: On-demand work, crowdwork, and labor protection in the gig-economy[END_REF]. But this should not be seen as a free choice. The developing country context shapes not just risks and, hence, risk-mitigation strategies but also acceptance of risk and adoption of risk mitigation. As noted, many of these workers are immigrants who move to South Africa due to lack of employment opportunities in their home countries but then find a similar lack of options other than gig work, as two of the Zimbabweans commented: "there are no jobs here in South Africa" (D12); "It's not just like all about that Zimbabweans will just accept any rate that comes up, but the situation is the one that can force us" (D15). Once within the sector, and notwithstanding risk-mitigation strategies, the way in which platforms have constructed the market creates pressure from supplydemand imbalance and leads drivers to undertake high-risk strategies; sometimes at the cost of their own lives.

In terms of recommendations, one would say that there should be more risk-bearing and risk-mitigation by the platform. With workers literally dying for Uber and Bolt there is at least a moral argument for this. Yet there are dangers of things moving in the opposite direction: in early 2020, the South African platforms were under fire for increasing their commission rates, continuing to add new cars and drivers to the platform, and removing driver access to advance rider and trip details [START_REF] Malinga | Uber, Bolt drivers plan shutdown of ride-hailing services[END_REF]. Each of these will increase not decrease risks. This creates an urgent need to place a greater emphasis on workers and their perspective, as outlined in this paper. That, for example, is the aim of the Fairwork Foundation, which has been active in South Africa for the past two years. It rates platforms against decent work standards that include their risk-mitigation actions, and seeks to use public, client and government pressure to make platforms improve [10]. Similar initiatives are needed to pressurise platforms to do more to reduce worker risks.

A second recommendation would relate to worker groupings. It is notable that one time-honoured group-initiated risk-mitigation strategy is not much in evidence in South Africa: the formation of trade unions or even of worker associations. The South African e-Hailing Association in Cape Town and The Movement in Johannesburg have both sought to collectivise workers, but with only limited success. One problem -reflected in views of those interviewed here -is that many drivers have bought into the 'independent contractor' identity and regard themselves as entrepreneurs or proto-entrepreneurs; an identity at odds with membership of any formal worker association. Sadly, it may take greater materialisation of the risks to change this.

Finally, we can note some directions for future research. The first will be a more general repetition of this work in other cities, countries and gig economy sectors to get a broader and more systematic understanding of risk and its mitigation; perhaps using the categorisation of risks presented above as a basis for deductive enquiry. The second would arise from some sense of differentiation that emerged in the interview data e.g. that younger drivers may take more risks, or that Bolt drivers faced more risks than those driving for Uber. Future research could therefore investigate various possible dimensions of difference in terms of risk profiles and risk-mitigation strategies: older vs. younger workers; men vs. women; immigrant vs. local workers; owners vs. renters; and between different platforms.

Table 1 .

 1 Overview of interviewees

	No.	Country of Origin		Platform			Ownership
		Zim-	SA	Other	Uber	Bolt	Both	In-	Owner	Renter	Rent
		babwe						driver			to own
	1

This refers to driver interview no.10; other interviewee references in the text follow a similar format.


Anushruti Vagrani

Saroj Bijarnia

Vigneswara Ilavarasan

Silvia Masiero

Appraising WhatsApp in the Indian Context: Understanding the Rural Sentiment

Keywords: Digital Platforms, WhatsApp, Social Capital, Sentiment Analysis, Rural India

Web-based instant messaging applications are predominant as means of Internet usage in rural settings. This becomes particularly interesting in the Indian rural context where applications like WhatsApp are accepted and used extensively, while the Internet is still rather an alien concept for a large part of the population. While adoption of these applications is motivated by ease of access, ease of use, and other facilitating conditions, to better understand the collective acceptance, the role of social influence needs to be contextualized for the rural areas in point. To this end, we use the analytical lens of social capital to understand and elaborate on the rural sentiment around the usage of WhatsApp. We triangulate a thematic analysis of qualitative details from an ethnographic study with the sentiment analysis of selected parts of interviews conducted during the study. This mixed-method study throws light on how social constructs in rural households play a role in defining technological usability and habits, as well as individual-level perceptions and sentiments. Our study, which illustrates the usefulness of mixed methods in ICT4D, has implications for various stakeholders to understand the usage patterns and perception of digital platforms in rural contexts.

Introduction

With more than half a billion people using the Internet in India in 2019 (IMRB, 2019), the growth of Internet adoption in the country sees rural adoption acquiring increasing importance. This significant increase in Internet penetration has led governments to deliver various services in online mode and also to invest in building digital platforms such that inclusion is achieved (Mir, Kar, Dwivedi, Gupta, & Sharma, 2019; Mir, Kar, Gupta, & Sharma, 2019). However, it is well understood that the types and frequency of the usage, impact, and also the pattern of consumption of online services are significantly different in urban and rural areas owing to the different socioeconomic profiles of urban and rural internet users [START_REF] Venkatesh | Digital divide initiative success in developing countries: A longitudinal field study in a village in India[END_REF][START_REF] Kenny | Superfast broadband: is it really worth a subsidy?[END_REF][START_REF] Galloway | Can broadband access rescue the rural economy?[END_REF]. ICT4D research has attempted to understand these differences through theoretical means including the Sustainable Livelihoods Approach [START_REF] Scott | The economic impact of telecommunications on rural livelihoods and poverty reduction: a study of rural communities in India (Gujarat)[END_REF], the choice framework [START_REF] Kleine | ICT4WHAT?-Using the choice framework to operationalise the capability approach to development[END_REF], and social network theory [START_REF] Venkatesh | Digital divide initiative success in developing countries: A longitudinal field study in a village in India[END_REF]) among others. However, studies with a focus on the social aspect around ICT adoption and impact in rural areas are still relatively scarce, with few examples constituting exceptions [START_REF] Selouani | Social impact of broadband internet: A case study in the Shippagan Area, a rural zone in Atlantic Canada[END_REF][START_REF] Thapa | Exploring the link between ICT intervention and human development through a social capital lens: The case study of a wireless project in the mountain region of Nepal[END_REF].

With rural people spending more time online, it becomes more interesting to understand how their offline social structure plays a role in online interactions and how these interactions affect their social structure. In any social structure, these interactions depend on personal, material, and social factors. Understanding these factors demand more micro-level, in-depth studies. In rural areas of India, with an increasing number of smartphones and other mobile devices, and with increasing levels of Internet penetration, the study of how social structures are reflected in online interactions and vice versa, presents an interesting case; on which limited ICT4D research has been conducted so far.

WhatsApp is a Web-based instant messaging application that makes it for a large part of global Internet usage, with more than 200 million active users in India before 2018 (Statista Report, 2017). A large part of online interactions takes place through WhatsApp across many communities including those in rural India (Sánchez-Moya, & Cruz-Moya, 2015; Digital Empowerment Foundation, 2018). Research shows that WhatsApp has gained popularity due to reasons including the design features that allow 'emergent users' [START_REF] Devanuj | Technology adoption by "emergent" users: the user-usage model[END_REF] like those in developing countries to overcome the barriers towards technology usage [START_REF] Balkrishan | Making and breaking the user-usage model: WhatsApp adoption amongst emergent users in India[END_REF]. In this study, we focus on understanding types of usages that WhatsApp is serving in rural areas and explore the sentiments related to these usages. We ask the following research questions:

1. What are the rural sentiments related to various usages of WhatsApp? 2. How does social capital play a role in the generation and development of these sentiments?

To answer our questions, we triangulate a thematic analysis of data collected from an ethnographic study of rural Rajasthan (North India) with sentiment analysis of selected parts of interviews conducted during the study. Our data reveals that offline social relations in the rural social environment are reflected in the patterns of usage of WhatsApp and the sentiments associated with them. One-to-one or group interactions on WhatsApp leverage upon bridging, bonding, and linking social capital [START_REF] Putnam | Bowling alone: The collapse and revival of American community[END_REF][START_REF] Woolcock | The place of social capital in understanding social and economic outcomes[END_REF] in ways that we discuss below. Social capital theory helps to understand the interconnected links between personal, material, and social factors in a rural setting. Based on the results of our analysis, the links we establish in the discussion help in understanding the appraisal of instant messaging platforms in the context of the rural sector in developing countries.

This paper is structured as follows. Section 2 contextualizes sentiment in the information systems' space, justifying the choice of social capital as a theoretical lens for the study. Section 3 explains the research flow, followed by our research method. In Section 4 we analyze our data, and in section 5 we discuss the implications and conclusions from the study.

Literature Review and Theoretical Framework

Moving from the classical to the neo-classical theory of capital introduces value to the masses in the form of Human Capital, Cultural Capital, and also, Social Capital [START_REF] Lin | Building a network theory of social capital[END_REF]. While human capital is a form of investment in human factors (e.g. skill, knowledge, experience), that provides profit or benefit to an individual; social capital derives its value based on the collective interactions of a group, in turn providing benefits to the society and its members.

Social capital is a term used in many different ways in various fields of research. The central idea means that all the various associations of a person or a group work as assets to them. Bourdieu was one of the antecedents to discuss social capital while defining it as "the aggregate of the actual or potential resources that are linked to possession of a durable network of more or less institutionalized relationships of mutual acquaintance and recognition" [START_REF] Bourdieu | The forms of capital[END_REF]). Later Coleman focuses on its functions [START_REF] Coleman | Social capital in the creation of human capital[END_REF] and Putnam discusses two different forms of social capital called bridging and bonding social capital. While bridging social capital is inclusive and is often related to weak ties between individuals, bonding social capital is exclusive and represents strong ties in close relationships with family, friends or neighbors [START_REF] Putnam | Bowling alone: The collapse and revival of American community[END_REF][START_REF] Putnam | Bowling together[END_REF] The initial approach in terms of social capital for online interactions was the same applied for media in general; that of time displacement. It suggested that the time spent online comes most likely from an earlier offline activity which was social [START_REF] Nie | Internet use, interpersonal relations, and sociability[END_REF]. Later on, it was argued that unlike traditional media, the internet is an interactive space having sociability within [START_REF] Williams | On and off the'Net: Scales for social capital in an online era[END_REF][START_REF] Dimaggio | Social implications of the Internet[END_REF]. Several of the studies focus on understanding how various factors from the real-life affect the online social behavior while using online social network sites and applications including WhatsApp. These studies establish the role played by experience (Leong, Ibrahim, Dalvi-Esfahani, Shahbazi, & Nilashi, 2018) and psychological factors including social support [START_REF] Aharony | The importance of the WhatsApp family group: an exploratory analysis[END_REF][START_REF] Gazit | Factors explaining participation in WhatsApp groups: an exploratory study[END_REF]. Aharony also studies the kind of social capital that WhatsApp interactions generate [START_REF] Aharony | What's App: a social capital perspective[END_REF]. In this study, we focus on the sentiments around WhatsApp usage and explore the role that social capital plays.

Sentiment Analysis

Opinions expressed by people are subjective expressions of their sentiments [START_REF] Liu | Sentiment analysis and subjectivity[END_REF]. With the growing trend of people expressing their opinion on various web platforms, numerous techniques have been developed that enable analysts to process these in text form and dig out meaningful information using linguistics and natural language processing [START_REF] Liu | Sentiment analysis and opinion mining[END_REF]. Content analysis and sentiment analysis represent types of such techniques that have been used extensively in information systems research (cf. Rathore, Kar, & Ilavarasan, 2017; Bijarnia, Ilavarasan, & Kar, 2020 ). A number of these studies are based on the analysis of a large amount of data from social media such as Twitter and Facebook. The data analysis is used to get insights on various business and prediction aspects including market , 2014). In this work, we use content analysis including sentiment analysis on interview transcripts and triangulate it with thematic analysis.

Method

The research flow of this mixed-method study is shown in Fig. 1 below. Subsequently, the research methodology is explained in detail. An ethnographic study was conducted by the lead researcher in a rural Gram Panchayat (village area) in Rajasthan, a state of north India, to explore the interplay between broadband and various socio-economic and cultural factors in an Indian rural setup. The researcher spent around eight months during the year 2018 in the core village and the adjacent town while collecting data through in-depth interviews, observations during interviews, and informal interactions. The interviews were audio-recorded, transcribed, and coded with multiple rounds of thematic analysis with the help of NVIVO12. During the process of coding, the usage of the messaging application WhatsApp emerged as one of the prominent elements. The recurrent presence of this topic in the discussions led our focus in the direction of examining rural sentiment around WhatsApp specifically. Thematic analysis and content analysis are two common approaches used for an in-depth understanding of a phenomenon. For an overall detailed understanding of the same phenomenon, these approaches can complement each other quite well. Thus, we parallelly conducted a content analysis of the text from the transcripts. While a word cloud (Fig. 2) indicated major concepts emerging in interviews, a sentiment analysis (Fig. 3) in the form of polarity analysis helped in understanding whether emotions were positive or negative. A detailed Emotion analysis then provides a detailed score of eight different kinds of emotions. We see the results from these parallel analysis processes together and then discuss the implications. The process is described in Fig. 1.

Demographic Description

Baral is located in the Ajmer district of Rajasthan State in India, 70 KM from the district headquarter. With around 500 households at the core of this rural area, the population is a good mixture of two religions and various castes. As a common practice, men in households work in the mills on temporary or permanent jobs, while females take care of agriculture and livestock. Literacy is said to be close to a hundred percent in Ajmer district, however, the level of education varies by a large magnitude. The majority of the elderly population has two to ten years of school education. Younger adults, 20-35 years old ones, have mostly completed school education at the least. Occupation choices involve the job of a painter, carpenter, confectioner, tailor, construction worker, property dealer, general store owner, mobile and electronics store owner, and teacher among others. In the other part of the Gram Panchayat, which is attached to the town, teachers, banking professionals, charted accountants, doctors, electronic, automobile, cloth, and other merchandise shop owners are included in the list of occupations. On income standards, the majority of the population falls under the lower middle class and below that. On average, every two out of ten households were such that one or more members contained a keypad phone, but no smartphone. In households where the household head is relatively younger, within the range of 20-40 years, the household heads mostly own a smartphone. In other cases, the young adult male/s in the household most frequently own a smartphone. Female adults in the household in the majority of the cases own a keypad phone for their use. Younger and newlywed females and those who are about to get married are often an exception to this case. Some girls who are earning or are continuing higher studies have their smartphones. WhatsApp has become one of the most common applications used in recent years. Before 2016 there were relatively very few WhatsApp users there in the village. For this study extracts from the interview transcripts of 34 respondents who talked about WhatsApp usage in different forms are used. Respondents belong to age 15 to 60 years.

Content Analysis

Content analysis [START_REF] Harwood | An Overview of Content Analysis[END_REF]) is used to find out the hidden meaning from large text data. This type of analysis uses principles from text mining and natural language processing [START_REF] Kayser | Extending the knowledge base of foresight: The contribution of text mining[END_REF]. We have used two content analysis techniques i.e. Word cloud (Heimerl, Lohmann, Lange, & Ertl, 2014) and sentiment analysis [START_REF] Pang | Opinion mining and sentiment analysis[END_REF]. Word cloud is a set of words that are most frequent in the text data. Using NVIVO12, we created a word cloud with the selected text consisting of the responses of the qualitative interviews regarding WhatsApp. In Sentiment analysis, we try to identify how sentiments are expressed in a text. For the sentiment analysis, we perform polarity analysis and Emotion analysis. Polarity analysis suggests whether the sentiments expressed in the text are overall negative or positive, while in Emotion analysis the text is evaluated in terms of emotions including anger, anticipation, disgust, fear, joy, sadness, surprise, and trust. Sentiment analysis is performed with R programming using syuzhet and dplry packages. For each Sentiment type, the range of scores is 1 to 8. We first selected all the interviews where a respondent has talked about WhatsApp usage in any form. From these interview transcripts, those portions were manually picked where details around WhatsApp usage are being discussed, this gave us the dataset for content analysis. These segments of conversations were classified majorly into five different categories that were frequently being discussed: video calling, chat with close connections, communication with social connections, job search, and education-related communication, and work-related communication. Content analysis was then performed on the filtered data.

Thematic Analysis

The thematic analysis provides a controlled way to understand and structure the ideas emerging from ethnographically collected data [START_REF] Aronson | A pragmatic view of thematic analysis[END_REF]. Through a thematic analysis, we identify, organize, and find meaningful patterns in our data and understand the shared experiences concerning the particular research questions for the study (Braun, & Clarke, 2012). We followed the following steps:

Step1. Familiarizing with the data: we evaluated the transcripts and memos and wherever needed, the audio files repetitively while trying to read the assumptions that the respondent had at the time of interviews.

Step 2. Initial codes: Initial codes were generated by highlighting the relevant text sections. This part led us to statements and emotions that were shared commonly, and those that stand out as exceptions. "Take WhatsApp for example. I used to scold my kids for using it. Some scandal took place in our village because of WhatsApp, there were some boys... A girl went to a hotel with her friend. The guy asked her to send the video on WhatsApp which got sent to two groups. It became an issue as there were so many members of the groups. Then the girl who was married had to face divorce. So I used to be scared of all these things. I used to tell my kid not to use WhatsApp, it contains dirty stuff. Later I got to know that things don't work this way. Whatever we put there is there only. My son, both my kids, keep on telling me that I should try and understand things first before I scold them. The whole world is using these things."

The statement talks about the initial perception that was generated possibly during various conversations that the respondent had had with her peers and surrounding people about an unfortunate incident in the village. It also talks about how interactions with her kids later restructured her biases towards the usage of these web platforms.

Step 3. Structuring the data in sub-themes: from the codes, we identified different patterns, clubbed them, and created a hierarchy. We looked for patterns that were indicated by various "conversation topics, vocabulary, recurring activities, meanings, feelings, or folk sayings and proverbs" (Taylor & Bogdan, 1989).

Step 4. Structuring themes: various components of ideas and experiences that were meaningless in exclusion are bridged together to make sense in a collective view in this stage [START_REF] Leininger | Ethnography and ethnonursing: Models and modes of qualitative data analysis[END_REF]. The sub-themes were linked to themes.

Step 5. Relevant themes and their names: We identified those themes that were relevant to our research questions and named them to suitably depict the emotions that they carry. These represented respondents' views that indicated the social structure and its interactions with their thought process. During this process, a background of literature guided us while understanding the pattern [START_REF] Aronson | A pragmatic view of thematic analysis[END_REF]. Another aspect can be seen in the upper left part of the diagram where words like 'friends', 'family', 'kids', 'status', 'group', 'conversation' are connected closely. These indicate the conversations between close friends and family. In the upper-right part of the figure words like 'work', 'document', 'job', 'publicity' 'classes' 'share' can be seen in close connection, indicating another prominent usage of WhatsApp in connection with a job, work and education. We identify five broader concepts that we use as a guideline in the emotion analysis below.

Analysis and Findings

Polarity Analysis

The second technique of content analysis was sentiment analysis which is performed in two different ways: polarity analysis and Emotion Analysis. Fig. 3 shows the results of the polarity analysis conducted using NVIVO12. The diagram shows that a majority of the views given by the respondents are 'moderately positive' in nature. The number of 'moderately negative' sentiments deducted by the software are relatively few. Strong reactions, both positive and negative are fewer in number yet significant. These results show the overall acceptance of WhatsApp usage in rural areas. Rural people have more positive sentiments regarding WhatsApp usage than negative sentiments. We further investigate these sentiments through Emotion analysis.

Emotion Analysis

Fig. 4 shows the graphical representation of results from Emotion analysis performed on different categories of WhatsApp usage, which were earlier defined manually. Using R programming the text in these categories was rated between 0-8 on eight different emotions anger, anticipation, disgust, fear, joy, sadness, surprise, and trust. Also, overall positive or negative sentiment is indicated on the same range of value here. The categories were guided by the word cloud, results for each of the five categories are explained below.

WhatsApp Video Calling. Overall sentiments are positive for video calling. Among emotions, trust value is high, respondents have largely shown trust in the WhatsApp video calling feature. One of the potential reasons given by the respondents for this finding is that they can see the person on the other side (See Fig. 4(a)).

Communication with Social Connections. This includes connections within society, within people with same caste or ethnicity who usually stay connected through WhatsApp groups. The messages in these groups are generally forwards that are perceived as informative and relevant. Overall sentiments are positive, however significant negative sentiments are also observed here. This might be since these are considered time killing engagement. Anticipation is observed high in the responses in these categories. Small areas in yellow depict fear in some of the responses (see Job search and Education. The responses on this category show a good mix of sentiments, both positive and negative sentiments can be seen in the graph. Positive sentiments still can be seen in the majority. Students and employment seekers use the application and join a group where they can communicate and get relevant information like study material, details about new openings, process, and date of application, etc. ( Fig. 4(e)).

Thematic Analysis

Our thematic analysis results in eight different themes around WhatsApp namely usage, concerns, emotions, stigmas, resistance, addiction, constraints, and motivators (See Fig. 1). Ideas in these themes are discussed below.

Usage. usages of WhatsApp for the people in this area are categorized into three major categories. The forms of social capital bonding, bridging, and linking social capital directly reflect in these three usage patterns. The first category includes communication with family and friends (bonding) which majorly include video calling and sending and receiving photographs of the members and responding to those. Video calling also provides indirect access to less educated people who are not comfortable using other text-based features of the application. For example, an elder lady who uses a keypad phone otherwise gets access through her kids who make video calls to her relatives for her:

"Kids do make a call for us to our family, relatives. I talk to my mother and father…A video call, there is our picture on the screen and theirs as well. I: So they make the call for you or you do it yourself? R1: They make the call I only talk. I: What more do you do? R1: We just talk, how are you doing, what did you cook, what did you do all day and things like that. What else to do?"

The second category is that of social connections which usually include distant relatives, friends from the same locality, people from the same caste or ethnicity among others (bridging). The most common information flow is in the form of forwarded messages. This also works as a source of information for current affairs and news, also local information flow. Even after being time-consuming, people continue to be a part of these groups for the sake of connectedness.

"More than half is Jhutha Prachar (false advertising) on WhatsApp. Small things blow up. They write small things in such a big way to make it viral."

The third category includes work, employment search, and education-related usages in this context (linking). Students and job seekers join these groups usually created by e-Mitra and coaching centers who keep on sending details on vacancies for a job, examinations, and study content on these groups. Many of the people instead of going on different websites to look for options rely on these groups for the majority of the information in this regard. Some of the local business owners use WhatsApp groups to market their products in nearby areas.

Emotions. The emotions associated with WhatsApp are a mixture of positive and negative. Though people have accepted usage in their day to day life, there are different notions around it including gender-based biases and stigmas, apprehension and fear of doing something wrong with it, and exposure to misinformation, inappropriate content especially for young teens, frustration because of it being time-consuming and being a popular 'time pass'. Positive notions include a sense of connectedness, empowerment-based on the ability to use, and to learn and based on the information gained. When used for connecting with close friends and family it provides joy, belongingness, and contentment.

Concerns. The concerns around WhatsApp usages are majorly formed by collective notions based on experiences of people around or based on the stories that float across the afternoon and evening chit-chats in the village. These concerns include 'fake news', inappropriate content, misuse of the platform, addiction, and too much time consumption. Also, the inadequate network is another concern in those areas, though mobile internet works fine across the majority of the parts of the village.

Stigmas. The stigmas that exist there in the local rural society are reflected in their usage of WhatsApp as well. One example being gender inequality, in the majority of households a young adult or sometimes teenage male is the first one to get mobile phone access. Females still carry keypad mobile phones so that they can talk to the family. Even when they do have a smartphone, they are expected to use it with restraint.

"Video calling as well the kids they do, I don't pay much attention to that. I: You never used it? R1: No I haven't…I: And your daughter has a smartphone… R1: No my daughter doesn't do that either... never…" (sounded more like she ought not to indulge in all this).

Resistance. Resistance towards learning as well as usage post-learning is more evident in older generations. One of the major reasons is that adoption is still new in these areas and people in their circle are not using similar technology. Various stigmas discussed above also cause resistance.

Addiction. Addiction is not that common among the surveyed individuals; however, people do have the habit of having to check messages on their device, mostly in the evening hours. Interviews display a realization that these habits can lead to addiction.

Constraints. Apart from lack of access, lack of education forms major constraints for text-based applications. Learning to use these applications take many efforts for uneducated people even to use the video calling options. The stigmas and concerns sometimes work as constraints towards access and usage.

Motivators. Though WhatsApp is a text-based technology, it contains features that let people overcome the barriers that they traditionally experience. Pictures, video, and video calling options are usable even for indirect users. A sense of connectedness, information access, a sense of empowerment, joy of connecting to the close family motivates such users often. Another motivating factor is the possibility of saving money through different usages.

Our study illustrates and appraises the core sentiments associated with the usage of WhatsApp in a rural area of north India. The combination of sentiment analysis with thematic analysis of the interviews conducted leads us to identify the key sentiments into play, then deepen our awareness of them through an examination of the eight thematic constructs. Our findings depict a mixture of positive and negative sentiments, each associated with various types of users and with the past experiences lived by interviewees. Putnam's concepts of bridging, bonding, and linking social capital serve to illuminate people's use of the messaging service, providing a basis for further investigation of the multiple topical trends in the analysis.

Our analysis reveals that the social constructs around which respondents' opinion cluster play an important role in shaping people's habits and usage of WhatsApp. Though these connections are quite evident in places, the indirect influence on other aspects needs to be explored step by step. For example, while the surveyed individuals show anticipation towards distant connections, bonding social capital in close connections is reflected in terms of trust, anticipation, and joy. Sentiments of fear and anger are also reflected indicating the concerns and stigmas around the usages, these sentiments recur among users who are less acquainted with the new technology (e.g. older people) and those whose awareness of instant messaging is mostly indirect.

There are two orders of implications from this study. Firstly, the stylized facts reported here -of which we have sought to reach primary sensemaking through Putnam's theory -display a high degree of heterogeneity, hence providing a first picture of the range of sentiments associated with new technology in a rural area. These sentimentsand the relations of their variety with different categories of users within the same context -require greater analyses to understand the roots of their formation, and to appraise their implications for usage of WhatsApp and other instant messaging services. The stylized picture produced in this study constitutes a solid basis to do so, in further analyses that will focus on sentiment formation -building causal theory on the roots of sentiments -and the consequences of the associations of specific sentiments with technology usage.

Secondly, this study serves as a means to throw light on the advantages of using mixed methods in ICT4D. Mixed methods, which traditionally see a limited application in our discipline, have recently been taken up in the Information Systems (IS) discipline (e.g. [START_REF] Zachariadis | Methodological implications of critical realism for mixed-methods research[END_REF]) as a means to illuminate the same phenomenon through multiple, complementary angles. Our study has sought to do so by combining the quantitative insights of sentiment analysis with the qualitative features of thematic analysis, hence providing primary explanations for the quantitative results displayed. This has led us to observe a quantitative picture of the data collective and, at a second stage, provide qualitative explanations for it, hence integrating two methods that do not find large complementary applications in ICT4D to date.

It is the benefits of this mixed-method application that we want to throw light on as we conclude the paper. In a domain like ICT4D, where interpretive research predominates, mixed methods offer the possibility to explore the same phenomenon under two concomitant lights, which illuminate each other by providing complementary insights on the same observation. The importance of doing so, already illustrated in IS at large [START_REF] Zachariadis | Methodological implications of critical realism for mixed-methods research[END_REF], is enhanced in ICT4D by the importance of appraising users' view of phenomena, to avoid the design-reality gaps (Heeks 2002) that lead to failure. We hope, on these grounds, that our study can serve as a basis for wider applications of mixed methods to ICT4D, with a view of generating more in-depth appraisals of user perspectives in their context of generation.

  research eg. product co-creation (Rambocas, & Pacheco, 2018; Rathore, Ilavarasan, & Dwivedi, 2016; Chamlertwat, Bhattarakosol, Rungkasiri, & Haruechaiyasak, 2012), customer engagement (Bijarnia, Khetan, Ilavarasan, & Kar, 2019), financial market study eg. stock prediction (Mittal, & Goel, 2012), election prediction (Singh, Dwivedi, Kahlon, Pathania, & Sawhney, 2020) among others. Recent mixed-method studies have used these techniques on text data from other sources like reports and qualitative interview transcripts (Parmar, Maturi, Dutt, & Phate, 2018; Güven, S., Steiner, M., Ge, N., & Paradkar, A.

Fig. 1 :

 1 Fig. 1: Flow of Research

4. 1

 1 Word CloudA word cloud with the responses of the qualitative interviews discussing the WhatsApp application is shown in Fig.2. A word cloud highlights the most repeatedly used words in the responses, and the size of the word shows how frequently the word was used in the conversations. Word cloud provides us with a preliminary filter to refine the broader concepts from the text (DePaolo, & Wilkinson, 2014). The words situated close to each other represent that they are often used in the same or related context. For example 'communication', 'network', 'information', 'connected' are in close connection, depicting one aspect of WhatsApp usage being staying connected in a network supporting information flow.

Fig. 2 .

 2 Fig. 2. Word Cloud showing the ideas reflected upon in the responses

Fig. 3 .

 3 Fig. 3. Polarity Analysis Results

Fig. 4 (

 4 Fig. 4(b)).

Fig. 4 .

 4 Fig. 4. Emotion Analysis for various categories of WhatsApp Usage Chat with close connections. Responses in this category were fewer in number since talking to close connections, call or video calls are the preferred mode. Among emotions trust along with joy and anticipation are prominent (See Fig. 4(c)).