Table Of ContentA Semi-Markovian Modeling of Limit Order Markets
Anatoliy Swishchuk and Nelson Vadori
Department of Mathematics and Statistics, University of Calgary,
2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
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[email protected], [email protected]
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2 March 11 2015
n
a
J
Abstract. R.ContandA.deLarrard[5]introducedatractablestochasticmodelforthedynamics
7 of a limit order book, computing various quantities of interest such as the probability of a price
increase or the diffusion limit of the price process. As suggested by empirical observations, we
]
F extend their framework to 1) arbitrary distributions for book events inter-arrival times (possibly
M non-exponential) and 2) both the nature of a new book event and its corresponding inter-arrival
time depend on the nature of the previous book event. We do so by resorting to Markov renewal
.
n processes to model the dynamics of the bid and ask queues. We keep analytical tractability via
i
f explicit expressions for the Laplace transforms of various quantities of interest. We justify and
-
illustrate our approach by calibrating our model to the five stocks Amazon, Apple, Google, Intel
q
[ and Microsoft on June 21st 2012. As in [5], the bid-ask spread remains constant equal to one tick,
onlythebidandaskqueuesaremodeled(theyareindependentfromeachotherandgetreinitialized
1
after a price change), and all orders have the same size.
v
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1
Key words. limit order book, Markov renewal process, diffusion limit, duration analysis, Weibull,
7
1 Gamma.
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AMS subject classifications. 60K15, 60K20, 90B22, 91B24, 91B70
.
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1 1 Introduction
:
v
i Recently, interest in the modeling of limit order markets has increased. Some research has focused
X
on optimal trading strategies in high-frequency environments: for example [8] studies such optimal
r trading strategies in a context where the stock price follows a semi-Markov process, while market
a
ordersarriveinthelimitorderbookviaapointprocesscorrelatedwiththestockpriceitself. [2]de-
velopsanoptimalexecutionstrategyforaninvestorseekingtoexecutealargeorderusingbothlimit
and market orders, under constraints on the volume of such orders. [13] studies optimal execution
strategiesforthepurchaseofalargenumberofsharesofafinancialassetoverafixedintervaloftime.
On the other hand, another class of articles has aimed at modeling either the high-frequency dy-
namics of the stock price itself, or the various queues of outstanding limit orders appearing on the
1
bid and the ask side of the limit order book, resulting in specific dynamics for the stock price. In
[7], a semi-Markov model for the stock price is introduced: the price increments are correlated and
equal to arbitrary multiples of the tick size. The correlation between these price increments occurs
via their sign only, and not their (absolute) size. In [4], the whole limit order book is modeled (not
only the ask and bid queues) via an integer-valued continuous-time Markov chain. Using a Laplace
analysis, they compute various quantities of interest such as the probability that the mid-price in-
creases, or the probability that an order is executed before the mid-price moves. A detailed section
on parameter estimation is also presented. For a more thorough literature on limit order markets,
we refer to the above cited articles and the references thereof.
The starting point of the present manuscript is the article [5], in which a stochastic model for the
dynamics of the limit order book is presented. Only the bid and ask queues are modeled (they are
independent from each other and get reinitialized after a price change), the bid-ask spread remains
constant equal to one tick and all orders have the same size. Their model is analytically tractable,
and allows them to compute various quantities of interest such as the distribution of the duration
between price changes, the distribution and autocorrelation of price changes, the probability of
an upward move in the price and the diffusion limit of the price process. Among the various
assumptions made in this article, we seek to challenge two of them while preserving analytical
tractability:
i) the inter-arrival times between book events (limit orders, market orders, order cancellations)
are assumed to be independent and exponentially distributed.
ii) the arrival of a new book event at the bid or the ask is independent from the previous events.
Assumption i) is relatively common among the existing literature ([9], [12], [3], [6], [11], [15], [4]).
Nevertheless, as it will be shown later, when calibrating the empirical distributions of the inter-
arrivaltimestotheWeibullandGammadistributions(Amazon,Apple,Google,IntelandMicrosoft
on June 21st 2012), we find that the shape parameter is in all cases significantly different than 1
(∼ 0.1 to 0.3), which suggests that the exponential distribution is typically not rich enough to
capture the behavior of these inter-arrival times.
Regarding Assumption ii), we split the book events into 2 different types: limit orders that in-
crease the size of the corresponding bid or ask queue, and market orders/order cancellations that
decrease the the size of the corresponding queue. Assimilating the former to the type ”+1” and
the latter to the type ”−1”, we find empirically that the probability to get an event of type ”±1”
is not independent of the nature of the previous event. Indeed, we present below the estimated
transition probabilities between book events at the ask andthe bid for the stock Microsoft on June
21st 2012. It is seen that the unconditional probabilities P(1) and P(−1) to obtain respectively an
event of type ”+1” and ”−1” are relatively close to 1/2, as in [5]. Nevertheless, denoting P(i,j)
the conditional probability to obtain an event of type j given that the last event was of type i, we
observe that P(i,j) can significantly depend on the previous event i. For example, on the bid side,
P(1,1)=0.63 whereas P(−1,1)=0.36.
2
Microsoft Bid Ask
P(1,1) 0.63 0.60
P(−1,1) 0.36 0.41
P(−1,−1) 0.64 0.59
P(1,−1) 0.37 0.40
P(1) 0.49 0.51
P(−1) 0.51 0.49
Estimatedprobabilitiesforbookeventarrivals. June21st 2012.
On another front, we will show that we can obtain diffusion limit results for the stock price with-
out resorting to the strong symmetry assumptions of [5]. In particular, the assumption that price
increments are i.i.d., which is contrary to empirical observations, as shown in [7] for example.
The paper is organized as follows: section 2 introduces our semi-Markovian modeling of the limit
order book, section 3 presents the main probabilistic results obtained in the context of this semi-
Markovianmodel(durationuntilthenextpricechange,probabilityofpriceincreaseandcharacteri-
zationoftheMarkovrenewalprocessdrivingthestockpriceprocess),section4dealswithdiffusion
limit results for the stock price process, and section 5 presents some calibration results on real
market data.
2 A Semi-Markovian modeling of limit order markets
Throughout this paper and to make the reading more convenient, we will use - when appropriate
- the same notations as [5], as it is the starting point of the present article. In this section we
introduce our model, highlighting when necessary the mains differences with the model in [5].
Let s , sa, sb be respectively the mid, the ask and the bid price processes. Denoting δ the ”tick
t t t
size”, these quantities are assumed to be linked by the following relations:
1
s = (sa+sb), sa =sb+δ.
t 2 t t t t
Wewillalsoassumethatsb isdeterministicandpositive. Inthiscontext,sa =sb+δands =sb+δ
0 0 0 0 0 2
arealsodeterministicandpositive. Asshownin[5], theassumptionthatthebid-askspreadsa−sb
t t
is constant and equal to one tick does not exactly match the empirical observations, but it is a
reasonable assumption as [5] shows that - based on an analysis of the stocks Citigroup, General
Electric, General Motors on June 26st 2008 - more than 98% of the observations have a bid-ask
spread equal to 1 tick. This corresponds to a situation where the order book contains no empty
levels (also called ”gaps”).
Thepriceprocesss isassumedtobepiecewiseconstant: atrandomtimes{T } (wesetT :=0),
t n n≥0 0
it jumps from its previous value s to a new value s = s ±δ. By construction, the same
Tn− Tn Tn−
holds for the ask and bid price processes sa and sb. These random times {T } correspond to the
t t n
times at which either the bid or the ask queue get depleted, and therefore, the distribution of these
3
times {T } will be obtained as a consequence of the dynamics that we will choose to model the bid
n
andaskqueues. Letusdenoteqa andqb thenonnegativeinteger-valuedprocessesrepresentingthe
t t
respective sizes of the ask and bid queues at time t, namely the number of outstanding limit orders
at each one of these queues. If the ask queue gets depleted before the bid queue at time T - i.e.
n
qa =0 and qb >0 - then the price goes up: s =s +δ and both queue values (qb ,qa ) are
Tn Tn Tn Tn− Tn Tn
immediatelyreinitializedtoanewvaluedrawnaccordingtothedistributionf, independentlyfrom
all other random variables. In this context, if n , n are positive integers, f(n ,n ) represents the
b a b a
probability that, after a price increase, the new values of the bid and ask queues are respectively
equal to n and n . On the other hand, if the bid queue gets depleted before the ask queue at time
b a
T - i.e. qa > 0 and qb = 0 - then the price goes down: s = s −δ and both queue values
n Tn Tn Tn Tn−
(qb ,qa ) are immediately reinitialized to a new value drawn according to the distribution f(cid:101), inde-
Tn Tn
pendently from all other random variables. Following the previous discussion, one can remark that
theprocessesqb,qa willnevereffectivelytakethevalue0,becausewheneverqb =0orqa =0,we
t t Tn Tn
”replace” the pair (qb ,qa ) by a random variable drawn from the distribution f or f(cid:101). The precise
Tn Tn
construction of the processes (qb,qa) will be explained below.
t t
Let τ := T −T the ”sojourn times” between two consecutive price changes, N := sup{n :
n n n−1 t
T ≤t}=sup{n:τ +...+τ ≤t}thenumberofpricechangesuptotimet,X :=s −s the
n 1 n n Tn Tn−1
consecutive price increments (which can only take the values ±δ). With these notations we have:
(cid:88)Nt
s = X .
t k
k=1
Let us now present the chosen model for the dynamics of the bid and ask queues. As mentioned in
introduction, we seek to extend the model [5] in the two following directions, as suggested by our
calibration results:
i) inter-arrival times between book events (limit orders, market orders, order cancellations) are
allowed to have an arbitrary distribution.
ii) the arrival of a new book event at the bid or the ask and its corresponding inter-arrival time
are allowed to depend on the nature of the previous event.
In order to do so, we will use a Markov renewal structure for the joint process of book events and
corresponding inter-arrival times occurring at the ask and bid sides. Formally, for the ask side,
consider a family {Rn,a} of Markov renewal processes given by:
n≥0
Rn,a :={(Vn,a,Tn,a)} .
k k k≥0
For each n, Rn,a will ”drive” the dynamics of the ask queue on the interval [T ,T ) where the
n n+1
stock price remains constant. {Vn,a} and {Tn,a} represent respectively the consecutive
k k≥0 k k≥0
book events and the consecutive inter-arrival times between these book events at the ask side on
the interval [T ,T ). At time T where one of the bid or ask queues gets depleted, the stock
n n+1 n+1
price changes and the model will be reinitialized with an independent copy Rn+1,a of Rn,a: it will
therefore be assumed that the processes {Rn,a} are independent copies of the same Markov
n≥0
4
renewal process of kernel Qa, namely for each n:
P[Vn,a =j,Tn,a ≤t|Tn,a,Vn,a :p≤k]=Qa(Vn,a,j,t), j ∈{−1,1}
k+1 k+1 p p k
P[Vn,a =j]=va(j), j ∈{−1,1}
0 0
P[Tn,a =0]=1.
0
We recall that as mentioned earlier, we consider two types of book events Vn,a: events of type +1
k
which increase the ask queue by 1 (limit orders), and events of type −1 which decrease the ask
queueby1(marketordersandordercancellations). Inparticular,thelatterassumptionsconstitute
a generalization of [5] in the sense that for each n:
• Vn,a depends on the previous queue change Vn,a: {Vn,a} is a Markov chain.
k+1 k k k≥0
• the inter-arrival times {Tn,a} between book events can have arbitrary distributions. Fur-
k k≥0
ther, they are not strictly independent anymore but they are independent conditionally on
the Markov chain {Vn,a} .
k k≥0
We use the same notations to model the bid queue - but with indexes a replaced by b - and we
assume that the processes involved at the bid and at the ask are independent.
In [5], the kernel Qa is given by (the kernel Qb has a similar expression with indexes a replaced by
b):
λa
Qa(i,1,t)= (1−e−(λa+θa+µa)t), i∈{−1,1}
λa+θa+µa
θa+µa
Qa(i,−1,t)= (1−e−(λa+θa+µa)t), i∈{−1,1}.
λa+θa+µa
Giventhesechosendynamicstomodeltoaskandbidqueuesbetweentwoconsecutivepricechanges,
we now specify formally the ”state process”:
L(cid:101)t :=(sbt,qtb,qta)
which will keep track of the state of the limit order book at time t (stock price and sizes of the bid
and ask queues). In the context of [5], this process L(cid:101)t was proved to be Markovian. Here, we will
needto”add”tothisprocesstheprocess(Vb,Va)keepingtrackofthenatureofthelastbookevent
t t
atthebidandtheasktomakeitMarkovian: inthissensewecanviewitasbeingsemi-Markovian.
The process:
L :=(sb,qb,qa,Vb,Va)
t t t t t t
constructed below will be proved to be Markovian.
The process L is piecewise constant and changes value whenever a book event occurs at the bid or
at the ask. We will construct both the process L and the sequence of times {T } recursively on
n n≥0
n ≥ 0. The recursive construction starts from n = 0 where we have T = 0, sb > 0 deterministic,
0 0
and (qb,qa,Vb,Va) is a random variable with distribution f ×vb×va, where f is a distribution
0 0 0 0 0 0 0 0
on N∗×N∗, and both vb and va are distributions on the two-point space {−1,1}, that is vb(1) =
0 0 0
5
P[Vb = 1] is given and vb(−1) = 1−vb(1) (and similarly for the ask). We will need to introduce
0 0 0
the following processes for the bid side (for the ask side, they are defined similarly):
k
T¯n,b :=(cid:88)Tn,b, Nn,b :=sup{k :T +T¯n,b ≤t}.
k p t n k
p=0
With these notations, the book events corresponding to the interval [T ,T ) occur at times
n n+1
T +T¯n,b (k ≥0)untiloneofthequeuesgetsdepleted,andNn,b countsthenumberofbookevents
n k t
on the interval [T ,t], for t∈[T ,T ).
n n n+1
ThejointconstructionofLandofthesequenceoftimes{T } isdonerecursivelyonn≥0. The
n n≥0
following describes the step n of the recursive construction:
• For each T ∈{T +T¯n,b} , the book event vb :=Vn,b occurs at time T at the bid side.
n k k≥1 n,T Nn,b
T
If qb +vb >0, there is no price change at time T and we have:
T− n,T
(sb,qb,qa,Vb,Va)=(sb ,qb +vb ,qa ,vb ,Va ).
T T T T T T− T− n,T T− n,T T−
If on the other hand qb +vb = 0, there is a price change at time T and the model gets
T− n,T
reinitialized:
(sb,qb,qa,Vb,Va)=(sb −δ,x˜b,x˜a,vb ,va ),
T T T T T T− n n 0,n 0,n
where {(x˜b,x˜a)} are i.i.d. random variables, independent from all other random vari-
k k k≥0
ables, with joint distribution f(cid:101)on N∗ ×N∗, and {v0b,k,v0a,k}k≥0 are i.i.d. random variables,
independent from all other random variables, with joint distribution vb ×va on the space
0 0
{−1,1}×{−1,1}. We then set T = T and move from the step n of the recursion to the
n+1
step n+1.
• For each T ∈{T +T¯n,a} , the book event va :=Vn,a occurs at time T at the ask side.
n k k≥1 n,T Nn,a
If qa +va >0, there is no price change at time T andTwe have:
T− n,T
(sb,qb,qa,Vb,Va)=(sb ,qb ,qa +va ,Vb ,va ).
T T T T T T− T− T− n,T T− n,T
If on the other hand qa +va = 0, there is a price change at time T and the model gets
T− n,T
reinitialized:
(sb,qb,qa,Vb,Va)=(sb +δ,xb,xa,vb ,va ),
T T T T T T− n n 0,n 0,n
where{(xb,xa)} arei.i.d. randomvariables,independentfromallotherrandomvariables,
k k k≥0
withjointdistributionf onN∗×N∗,and{vb ,va } arethei.i.d. randomvariablesdefined
0,k 0,k k≥0
above. We then set T =T and move from the step n of the recursion to the step n+1.
n+1
ItresultsfromtheaboveconstructionandtheMarkovrenewalstructureoftheprocesses{Rn,a} ,
n≥0
{Rn,b} that the process L is Markovian.
n≥0 t
6
Sincetheprocesses{Rn,a} areindependentcopiesofthesameMarkovrenewalprocessofkernel
n≥0
Qa, we will drop the index n when appropriate in order to make the notations lighter. Following
this remark, we will introduce the following notations for the ask, for i,j ∈ {−1,1} (for the bid,
they are defined similarly):
Pa(i,j):=P[Va =j|Va =i],
k+1 k
Fa(i,t):=P[Ta ≤t|Va =i],
k+1 k
Ha(i,j,t):=P[Ta ≤t|Va =i,Va =j],
k+1 k k+1
(cid:90) ∞
ha(i,j):= tHa(i,j,dt),
0
ha :=ha(1,1)+ha(−1,−1), ha :=ha(−1,1)+ha(1,−1),
1 2
(cid:90) ∞
ma(s,i,j):= e−stQa(i,j,dt), s∈C,
0
(cid:90) ∞
Ma(s,i):=ma(s,i,−1)+ma(s,i,1)= e−stFa(i,dt), s∈C.
0
Throughout this paper, we will use the following mild technical assumptions:
(A1) 0<Pa(i,j)<1, 0<Pb(i,j)<1, i,j ∈{−1,1}.
(A2) Fa(i,0)<1, Fb(i,0)<1, i∈{−1,1}.
(A3) (cid:82)∞t2Ha(i,j,dt)<∞, (cid:82)∞t2Hb(i,j,dt)<∞, i,j ∈{−1,1}.
0 0
Some brief comments on these assumptions: (A1) implies that each state ±1 is accessible from
eachstate. (A2)meansthateachinter-arrivaltimebetweenbookeventshasapositiveprobability
to be non zero, and (A3) constitutes a second moment integrability assumption on the cumulative
distribution functions Ha and Hb.
3 Main Probabilistic Results
Throughout this section and as mentioned earlier, since the processes {Rn,a} are independent
n≥0
copiesofthesameMarkovrenewalprocessofkernelQa,wewilldroptheindexnwhenappropriate
in order to make the notations lighter on the random variables Tn,a, T¯n,a, Vn,a (and similarly for
k k k
the bid side).
3.1 Duration until the next price change
Given an initial configuration of the bid and ask queues (qb,qa) = (n ,n ) (n ,n integers), we
0 0 b a b a
denote σ the first time at which the bid queue is depleted:
b
k
σ =T¯b , k∗ :=inf{k :n + (cid:88) Vb =0}.
b k∗ b m
m=1
7
Similarly we define σ the first time at which the ask queue is depleted. The duration until the
a
next price move is thus:
τ :=σ ∧σ .
a b
In order to have a realistic model in which the queues always get depleted at some point, i.e.
P[σ <∞]=P[σ <∞]=1, we impose the conditions:
a b
Pa(1,1)≤Pa(−1,−1), Pb(1,1)≤Pb(−1,−1).
These conditions correspond to the condition λ ≤ θ+µ in [5], and the proof of the proposition
below shows that they are respectively equivalent to P[σ < ∞] = 1 and P[σ < ∞] = 1. Indeed,
a b
as s → 0 (s > 0), the Laplace transform La(s) := E[e−sσa] of σa tends to P[σa < ∞]. The
proposition below shows that if Pa(1,1) > Pa(−1,−1), this quantity is strictly less than 1, and if
Pa(1,1)≤Pa(−1,−1), this quantity is equal to 1. We have the following result which generalizes
the Proposition 1 in [5] (see also Remark .2 below):
Proposition .1. The conditional law of σ given qa =n≥1 has a regularly varying tail with:
a 0
• tail exponent 1 if Pa(1,1)<Pa(−1,−1).
• tail exponent 1/2 if Pa(1,1)=Pa(−1,−1).
More precisely, we get: if Pa(1,1)=Pa(−1,−1)=p :
a
αa(n)
P[σ >t|qa =n]t→∼∞ √
a 0 t
with:
αa(n):= 1√ (n+ 2pa−1va(1))(cid:112)p (1−p )(cid:113)p ha+(1−p )ha.
p π p −1 0 a a a 1 a 2
a a
If Pa(1,1)<Pa(−1,−1), we get:
βa(n)
P[σ >t|qa =n]t→∼∞
a 0 t
with:
βa(n):=va(1)ua+va(−1)ua+(n−1)ua,
0 1 0 2 3
Pa(1,1)
ua :=ha(1,−1)+ (ua+ha(1,1))
1 1−Pa(1,1) 3
1−Pa(−1,−1)
ua :=−ha(1,1)+ (ua+ha(1,1))+Pa(−1,−1)ha+(1−Pa(−1,−1))ha,
2 1−Pa(1,1) 3 1 2
1−Pa(1,1)
ua :=ha(1,1)+ (Pa(−1,−1)ha+(1−Pa(−1,−1))ha).
3 Pa(−1,−1)−Pa(1,1) 1 2
8
Similar expressions are obtained for P[σ >t|qb =n], with indexes a replaced by b.
b 0
Remark .2. We retrieve the results of [5]: if Pa(1,1) = Pa(−1,−1), then within the con-
text/notations of [5] we get p =1/2 and:
a
(cid:90) ∞ 1
ha(i,j)= 2tλe−2λtdt= ,
2λ
0
and so αa(n)= √n . For the case Pa(1,1)<Pa(−1,−1) (λ<θ+µ with their notations), we find:
πλ
n
βa(n)= ,
θ+µ−λ
which is different from the result of [5] that is βa(n) = n(θ+µ+λ) .. We believe that they made a
2λ(θ+µ−λ)
small mistake in their Taylor expansion on page 10: in the case λ<θ+µ, they should find:
sx
s→0
L(s,x) ∼ 1− .
θ+µ−λ
Proof. Let s>0 and denote L(s,n,i):=E[e−sσa|qa =n,Va =i]. We have:
0 0
k∗ k
(cid:88) (cid:88)
σ = Ta, k∗ :=inf{k :n+ Va =0}.
a m m
m=1 m=1
Therefore:
L(s,n,i)=E[e−sT1aE[e−s(σa−T1a)|q0a =n,V0a =i,V1a,T1a]|q0a =n,V0a =i]
=E[e−sT1aE[e−s(σa−T1a)|qTaa =n+V1a,V0a =i,V1a,T1a]|q0a =n,V0a =i]
1
(cid:124) (cid:123)(cid:122) (cid:125)
L(s,n+Va,Va)
1 1
=E[e−sT1aL(s,n+V1a,V1a)|q0a =n,V0a =i]
(cid:90) ∞ (cid:90) ∞
= e−stL(s,n+1,1)Qa(i,1,dt)+ e−stL(s,n−1,−1)Qa(i,−1,dt)
0 0
=ma(s,i,1)L(s,n+1,1)+ma(s,i,−1)L(s,n−1,−1)
Denote for sake of clarity a := L(s,n,1), b := L(s,n,−1). These sequences therefore solve the
n n
system of coupled recurrence equations:
a =ma(s,1,1)a +ma(s,1,−1)b , n≥0
n+1 n+2 n
b =ma(s,−1,1)a +ma(s,−1,−1)b
n+1 n+2 n
a =b =1.
0 0
Simplealgebra(computinga −ma(s,−1,−1)a ontheonhandandma(s,1,1)b −b onthe
n+1 n n+1 n
other hand) gives us that both a and b solve the same following recurrence equation (but for
n n
different initial conditions):
ma(s,1,1)u −(1+∆a(s))u +ma(s,−1,−1)u , n≥1
n+2 n+1 n
9
with:
∆a(s):=ma(s,1,1)ma(s,−1,−1)−ma(s,−1,1)ma(s,1,−1).
The parameter ∆a(s) can be seen as a coupling coefficient and is equal to 0 when the random
variable (Va,Ta) doesn’t depend on the previous state Va , for example in the context of [5].
k k k−1
If we denote R(X) the characteristic polynomial associated to the previous recurrence equation
R(X):=ma(s,1,1)X2−(1+∆a(s))X+ma(s,−1,−1), then simple algebra gives us:
R(1)=(Ma(s,1)−1)(1−ma(s,−1,−1))+ma(s,1,−1)(Ma(s,−1)−1)<0
(cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125)
<0 >0 >0 <0
Note that Ma(s,i) < 1 for s > 0 because Fa(i,0) < 1. Since ma(s,1,1) > 0, this implies that R
has only one root <1 (and an other root >1):
(cid:112)
1+∆a(s)− (1+∆a(s))2−4ma(s,1,1)ma(s,−1,−1)
λa(s):= .
2ma(s,1,1)
Because we have a ,b ≤1 for s>0, then we must have for n≥1:
n n
a =a λa(s)n−1 b =b λa(s)n−1
n 1 n 1
The recurrence equations on a , b give us:
n n
ma(s,1,−1) ma(s,−1,1)a +∆a(s)
a = b = 1
1 1−λa(s)ma(s,1,1) 1 ma(s,1,1)
Finally, letting L(s,n):=E[e−sσa|qa =n], we obtain:
0
(cid:88)
L(s,n)= L(s,n,i)va(i)=a va(1)+b va(−1).
0 n 0 n 0
i
The behavior of P[σ > t|qa = n] as t → ∞ is obtained by computing the behavior of L(s,n)
a 0
as s → 0, together with Karamata’s Tauberian theorem. By the second moment integrability
assumption on Ha(i,j,dt), we note that:
(cid:90) ∞ (cid:90) ∞
ma(s,i,j)= e−stQa(i,j,dt)=Pa(i,j) e−stHa(i,j,dt)
0 0
(cid:90) ∞
s→∼0Pa(i,j)−sPa(i,j) tHa(i,j,dt)=Pa(i,j)−sPa(i,j)ha(i,j).
0
Now, assume Pa(1,1) = Pa(−1,−1) = p . A straightforward but tedious Taylor expansion of
a
L(s,n) as s→0 gives us:
√ √
L(s,n)s→∼01− παa(n) s.
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