25
First-Passage-Time in Discrete Time
Marcin Jaskowski and Dick van Dijk
Econometric Institute,
Erasmus School of Economics,
The Netherlands
January 2015
Abstract
We present a semi-closed form method of computing a rst-passage-time (FPT) density for discrete time Markov
stochastic processes. Our method provides exact solution for one-dimensional processes and approximations for
higher dimensions. In particular, we show how to nd an exact form of FPT for AR(1), and an approximate
FPT for VAR(1). The method is valid for any type of innovation process if multi-period transition probabilities
can be computed. It is intuitively straightforward, avoids the use of complex mathematical tools and therefore
it is suitable for econometric applications. For instance, our method can be applied to form structural models of
duration without the need to invoke simplistic continuous-time Brownian motion models. Finally, the method that
we propose can be eciently implemented by parallelization of computing tasks.
Keywords: First-passage-time distribution, stochastic duration modeling, overshooting, law of total probability,
discrete-time stochastic processes, AR(1), ARG(1), VAR(1).
Introduction
Theoretical models in nancial economics are typically set up with continuous-time Brownian motion processes for
their mathematical convenience and tractability. Unfortunately, mathematical convenience does not necessarily chime
with the convenience of the econometric estimation. It is dicult to estimate continuous-time processes, mainly
because the processes that adequately capture stylized facts also lead to complex ltering problems. This discord
between theory and applications is especially striking in modelling duration times between dierent events.
On the one hand, theoretical models in economics and nance use rst-passage-time (FPT) densities (alternatively
called rst-hitting-models) and they dene duration as the rst time when a stochastic process crosses a specied
threshold. On the other hand, econometricians usually choose a dierent route by modelling duration time, as a new
stochastic process. But, in fact, rst-passage-time densities and duration models dene the same concept - the length
of time separating dierent stochastic events. That is, as Whitmore (1986) pointed out, duration models are just a
reduced-form way of modeling rst-passage time densities.
In econometric literature, with a recent exception of Mixed Hitting Time (MHT) models (Abbring (2010)), there
are hardly any papers using FPT distributions. For example, Engle and Russell (1998) in order to model transaction
times proposed the so-called autoregressive conditional duration models (ACD). This approach has been extended to
log-ACD model by Bauwens and Giot (2000) and to stochastic conditional duration (SCD) model by Bauwens and
Veredas (2004).
1
However, one drawback of the duration approach is that it models transaction prices and transaction times as
distinct from each other phenomena. But Renault, Van der Heijden, and Werker (2014) note that this is too simplistic.
They argue that transaction times are endogenous precisely because they are closely related to the price and volatility
process and empirically they strongly reject the SCD and ACD specications for NYSE-traded stocks.
Thus, Abbring (2010) and Renault et al. (2014) propose to use structural model of duration in order to form a
closer link between theory and empirical applications. Abbring (2010) explains how to use a spectrally negative Levy
process - a continuous-time process where only negative jumps are allowed with a heterogenous threshold. Similarly,
Renault et al. (2014) propose an extension to the MHT model, which is more appropriate for time-series applications.
In contrast to these two papers, we advocate a rst-passage-time approach to durations but with the stochastic
latent process cast in discrete-time. We provide an exact method for nding FPT distributions for stationary Markov
processes. Our method is based on the repeated application of the law of total probability. All one needs in order
to understand and apply the method, are basic probability methods and numerical integration. Theoretically, our
method is based on recursion, however, we show that it need not be a bottleneck. More precisely, we show how to
represent the solution in a matrix form, where all elements can be evaluated either separately or in vectors. This allows
us to circumvent the need to solve for FPT sequentially and, if necessary, signicantly speed up the computations by
parallelization of computing tasks.
First-passage time is a notoriously dicult mathematical problem. Usually formulas for the rst-passage time
density are obtained for Brownian motion under the assumption of continuous-time. Closed form solutions are known
only for several standard models and only those set in continuous-time. In most applications, researchers limit
themselves to a geometric Brownian motion for the simple reason that FPT density is known for this process in closed
form. Few additional closed form solutions are known. For example, in Linetsky (2004) we can nd closed-form
formulas for an Ornstein-Uhlenbeck process and the Feller process, in nance it is better known as the Cox-Ingresoll-
Ross (CIR) process. The"closed-form(ness)" of these solutions can be disputed, as they are based either on inversion
of Laplace transforms or on eigenfunction expansions, both of which can only be implemented numerically.
Density of the rst-passage time has many applications, ranging from biology (Ditlevsen and Lansky (2006)),
physics and chemistry (Larralde (2004)) to economics and statistics (Durbin (1971)). In nance it is used in the credit
risk literature, in pricing of non-European style options, real options (Dixit (1993)) and in the modeling of limit-order
executions (Lo, MacKinlay, and Zhang (2002)). Most, if not all, of these models invoke continuous-time Brownian
motion models.
Abbring (2012) (see also Abbring and Salimans (2012) for a detailed description of the mle estimation) has recently
introduced into the econometric literature a mixed hitting time model, which uses spectrally negative Levy processes
(a spectrally negative Levy process is a Levy process that has no positive jumps) which have convenient Laplace
transforms. This approach, however, also does not provide closed form solutions, especially as inversion of Laplace
transforms also need to be evaluated numerically.
Larralde (2004) shows formally that the universal features of the FPT density for an Ornstein-Uhlenbeck process
under the assumption of continuous-time do not extend to the discrete-time O-U process. There is a simple explanation
for this dierence. Continuous-time processes based on a standard Brownian motion move slowly towards the threshold
and the FPT event happens when the continuous time process "gently" touches the boundary. This is not the case for
discrete-time processes. In discrete time, for processes with continuous support of their innovations, the probability
that a process at time τ nishes exactly at the boundary is equal to zero. In fact, rst-passage time for a discrete time
process will always be equivalent to overshooting the boundary.
As Novikov and Kordzakhia (2008) note, the density and expectation of the rst-passage time for discrete-time
auto-regressive processes are usually approximated via Monte-Carlo simulations or by Markov chain approximations.
We can nd semi-closed-form solutions for AR(p)'s FPT density in Novikov and Kordzakhia (2008), Larralde (2004),
Basak and Ho (2004), Majumdar (2010), Di Nardo (2008) and in Baumgarten (2014). We complement these papers
2
Figure 1: The sample path of process xt+1 = µ + φxt + σεt+1. It starts at x0 = −2.5, and the other parameters areµ = −0.3 and σ = −0.25. There are three dierent constant and horizontal hitting barriers: x.
with an implicit method for nding FPT density for autoregressive models common in applied econometrics.
One advantage of our method is its mathematical simplicity. The main intuition behind our method is based on
the law of total probability. We do not use Laplace transforms, convolutions, collocations or eigenfunction expansions.
We make use of simple calculus and basic matrix algebra. This simplicity should, hopefully, facilitate adaptation and
extensions of our method to any particular case needed in applied statistics.
We show that for discrete time autoregressive processes, typically used in econometrics, it is possible to obtain
exact formulas if transition probabilities are available in closed-form. The intuition for this result comes from the
continuous-time literature (see Buonocore, Nobile, and Ricciardi (1987), Longsta and Schwartz (1995) and Collin-
Dufresne and Goldstein (2001)) that uses Fortet's Lemma. However, what in continuous-time is just an approximation,
in discrete time can be an exact formula. Fortet's lemma translated into the language of discrete time is just the law
of total probability.
1 Description of the Method
We will concentrate on the case when a boundary x is hit by a process xt from below. The opposite case, hitting from
above, is analogous. Let t and s ∈ N, denote time with t < s. Let us assume that at time t the value of process xt is
below the boundary: xt < x and the rst passage time can be dened as
τ ≡ inf s : xs ≥ x (1)
3
Figure 2: Three dierent possible First-Passage-Time distributions for the process xt dened in Figure 1, with hittingbarriers dened by x.
4
Figure 3: The process xt starts from an indicated value x0 and the rst-passage time happens already at time t+ 1.Blue arrows represent dierent possible realizations of process starting at xt = x0 at time t + 1, above the hittingboundary x. Given this realization, say xt+1 = x, red arrows represent possible realizations of process x at time xt+kstarting from xt+1 = x. Finally, green arrows represent dierent possible realizations of the process above the hittingboundary at time t+ k given that we started at xt = x0.
and its probability density function
qτ=s ≡ Pr (τ = s) .
The result follows from the law of total probability. The main intuition is based on the following uncontroversial
observation:
Observation 1.1. If a process at time t is below a certain threshold, xt < x, and at time t+k it is above the threshold
xt+k ≥ x, then at some intermediate time s (t < s ≤ t+ k ) it must have passed through the threshold x.
1.1 Intuition
The intuition behind the method can be explained on an example with just two periods. We are interested in nding
rst passage time probabilities: P (τ = t+ 1) = qt+1 and P (τ = t+ 2) = qt+2.
First, we may notice that Pr (τ = t+ 1) is trivial to obtain and is equal to the transition probability from xt =
x0 < x to the level above the hitting boundary, xt+1 = x ≥ x, which is
qt+1 = Pr (τ = t+ 1) =
∞
x
Pr (xt+1 = x | xt = x0) dx
= Pr (xt+1 ≥ x | xt = x0)
The task of nding qt+2 is only a little more involved. First, sum up the information that we already have. We know
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the transition probability for process x at time t+ 2 to any value x above x :
Pr (xt+2 ≥ x | xt = x0) =
∞
x
Pr (xt+2 = x | xt = x0) dx.
We know also that, if at time t+ 2, the process x is above the boundary of x, then it must have passed this boundary
for the rst time either at time t + 1 or at time t + 2. Therefore, the probability that xt+2 ≥ x | xt = x0 can be
decomposed into a sum of two disjoint events:
1. the process x at time t + 2 is above the boundary of x given that it passed through this boundary already at
time t+ 1, which has probability equal to Pr (xt+2 ≥ x | τ = t+ 1)Pr (τ = t+ 1)
2. the process x at time t+ 2 is above the boundary of x given that it passed through this boundary only at time
t+ 2, which has probability equal to Pr (xt+2 ≥ x | τ = t+ 2)Pr (τ = t+ 2)
This is summarized in the following total probability equation:
Pr (xt+2 ≥ x | xt = x0) = Pr (xt+2 ≥ x | τ = t+ 1)Pr (τ = t+ 1) + Pr (xt+2 ≥ x | τ = t+ 2)Pr (τ = t+ 2) . (2)
We can solve equation (2) for Pr (τ = t+ 2) as all other elements of this equation are known. In particular, we
know Pr (xt+2 ≥ x | τ = t+ 1), Pr (τ = t+ 1) and that Pr (xt+2 ≥ x | τ = t+ 2) = 1, because the probability that
the process x at time t+ 2 is above the hitting boundary, given that the rst-passage time occurred at time t+ 2, can
only be equal to 1.
If we were to nd Pr (τ = t+ 3) , or any other Pr (τ = t+ k), then we could proceed in the same manner by adding
one additional equation for each next Pr (τ = t+ k).
1.2 Proof by construction
Just as in the example above, we exploit observation 1.1 by partitioning the sample space into a sum of k disjoint
events:
Pr (A) =
t+k∑s=t+1
Pr (A ∩Bs) (3)
where A denotes an event that at time t+ k, process xt+k ≥ x, Bs denotes an event that τ = s and A ∩Bs is a setproduct of A and Bs. A ∩Bs corresponds to a situation in which at time t+ k process xt, is above the threshold x
and at time s there was already a rst passage.
Obviously, we have
Pr (A ∩Bs) = Pr (A | Bs)Pr (Bs)
= Pr (A | Bs) qτ=s
where the probabilities Pr (A ∩Bs) and Pr (A | Bs) are dened by transition probabilities for a particular stochastic
process.
Since s runs through all time points between t + 1 and t + k, we know that the sample space can be partitioned
according to the rst passage time:
Ω = Bt+1 ∪Bt+2 ∪ . . . ∪Bt+k ∪Bt+k+u (4)
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where Bt+k+u for u ≥ 1 denotes an event in which the rst-passage time occurs only after t+ k time period. Thus,
an eventA ∩Bt+k+u is an empty set implying that Pr (A ∩Bt+k+u) = 0.
Finally, using sample space partition, (4), and the law of total probability, we may form a sequence of k equations:
Pr (xt+1 ≥ x | xt = x) = Pr (τ = t+ 1)︸ ︷︷ ︸=qt+1
Pr (xt+1 ≥ x | τ = t+ 1)︸ ︷︷ ︸=1
(5)
Pr (xt+2 ≥ x | xt = x) = Pr (τ = t+ 1)︸ ︷︷ ︸=qt+1
Pr (xt+2 ≥ x | τ = t+ 1) + Pr (τ = t+ 2)︸ ︷︷ ︸=qt+2
Pr (xt+2 ≥ x | τ = t+ 2)︸ ︷︷ ︸=1
(6)
and so on until time t+ k:
Pr (xt+k ≥ x | xt = x) = Pr (τ = t+ 1)︸ ︷︷ ︸=qt+1
Pr (xt+k ≥ x | τ = t+ 1) + . . .+ Pr (τ = t+ k)︸ ︷︷ ︸=qt+k
Pr (xt+k ≥ x | τ = t+ k)︸ ︷︷ ︸=1
(7)
Theorem 1.2. Equations (5) to (7) dene an implicit solution for the rst passage time probabilities from qt+1 until
qt+k. We have a formula for t+ 1 :
qt+1 = Pr (xt+1 ≥ x | xt = x) (8)
and for any j in the range between t+ 1 and t+ k, we obtain qt+j recursively:
qt+j = Pr (xt+j ≥ x | xt = x)−j−1∑i=1
qt+iPr (xt+j ≥ x | τ = t+ i) (9)
Proof. The system of equations (5) to (7) we obtain by construction, and equations (8) and (9) follow directly.
In the following subsections, we derive explicit formulas for the conditional transition probabilities from Theorem
(1.2) and we will also show a simple way to speed up computations which circumvents the need for a recursive loop.
1.3 Probability of overshooting the boundary
Models base on continuous-time diusion processes or more generally spectrally negative Levy processes have the
convenient property that they do not jump over the boundary, but they gently touch it. This is not the case for
stochastic processes in discrete-time. The probability that a particular realization of a stochastic discrete-time process
will end exactly on the boundary is zero. This kind of complication does not trouble researchers using spectrally
negative Levy processes in continuous time. But, below we show that the problem can be addressed and the trouble
with overshooting the boundary should not discourage applied statisticians from the use of discrete-time processes.
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1.3.1 Deriving formulas for conditional transition probabilities
In order to solve equations (5) to (7) for q, rst we need to derive formulas for all other transition probabilities. We
start this derivation by redening rst-passage time probabilities in the following way:
Pr (τ = t+ 1)︸ ︷︷ ︸=qt+1
= Pr (xt+1 ≥ x | xt = x)
Pr (τ = t+ 2)︸ ︷︷ ︸=qt+2
= Pr (xt+1 < x ∩ xt+2 ≥ x | xt = x)
Pr (τ = t+ 3)︸ ︷︷ ︸=qt+3
= Pr (xt+1 < x ∩ xt+2 < x ∩ xt+3 ≥ x | xt = x)
...
Pr (τ = t+ k)︸ ︷︷ ︸=qt+k
= Pr (xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+k−1 < x ∩ xt+k ≥ x | xt = x)
Event Bs is equivalent to
Bs := xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x
and so
Pr (Bs) := Pr (xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x)
it follows that for t < s < t+ k,
A ∩Bs := xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x ∩ xt+k ≥ x
and
Pr (A ∩Bs) := Pr (xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x ∩ xt+k ≥ x) (10)
= Pr (xt+k ≥ x | xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x) (11)
· Pr (xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x) (12)
= Pr (A | Bs)Pr (Bs) (13)
= Pr (xt+k ≥ x | τ = t+ s)Pr (τ = t+ s) (14)
In order to move from equation (11) to equation (13), we make use of the Markovian property of the process xt, which
allows us to write:
Pr (xt+k ≥ x | τ = t+ s) = Pr (xt+k ≥ x | xt+1 < x ∩ xt+2 < x ∩ . . . ∩ xt+s−1 < x ∩ xt+s ≥ x)
= Pr (xt+k ≥ x | xt+s ≥ x)
8
and it is equal to
Pr (xt+k ≥ x | xt+s ≥ x)︸ ︷︷ ︸=Pr(xt+k≥x|τ=t+s)
=
∞
x
Pr (xt+s = x)
Pr (xt+s ≥ x)Pr (xt+k ≥ x | xt+s = x) dx (15)
=1
Pr (xt+s ≥ x)
∞
x
(Pr (xt+s = x)Pr (xt+k ≥ x | xt+s = x)) dx (16)
The1 transition probabilities on the left-hand side of equations (5) to (7) can be dened by a one-dimensional integral
in the following way:
Pr (xt+1 ≥ x | xt = x) = Pr (a1) =
∞
x
Pr (xt+1 = x | xt = x) dx (17)
Pr (xt+2 ≥ x | xt = x) = Pr (a2) =
∞
x
Pr (xt+2 = x | xt = x) dx (18)
... =... (19)
Pr (xt+k ≥ x | xt = x) = Pr (ak) =
∞
x
Pr (xt+k = x | xt = x) dx. (20)
The right hand side is more complicated. Let us dene the following set of equations:
Pr (b1 (x)) = Pr (xt+j ≥ x | τ = t+ j − 1; xt+j−1 = x) (21)
Pr (b2 (x)) = Pr (xt+j ≥ x | τ = t+ j − 2; xt+j−2 = x) (22)
... =... (23)
Pr (bk−1 (x)) = Pr (xt+k ≥ x | τ = t+ 1; xt+1 = x) (24)
Now, using equations (17) to (24), we can rewrite (15) as
Pr (xt+k ≥ x | xt+s ≥ x) =
∞
x
Pr (as (x))
Pr (as)Pr (bk−s (x)) dx
=1
Pr (as)
∞
x
(Pr (as (x))Pr (bk−s (x))) dx
The following example presents one of the equations that needs to be solved.
1In Appendix 3, we explain the intuition behind formula (15).
9
Example 1.3. For instance, we obtain qt+3 implicitly from the following equation:
Pr (xt+3 ≥ x | xt = x)︸ ︷︷ ︸=Pr(a3)
= Pr (xt+1 ≥ x | xt = x)︸ ︷︷ ︸=qt+1
Pr (xt+3 ≥ x | τ = t+ 1)︸ ︷︷ ︸=´∞x
Pr(xt+1=x)Pr(xt+1≥x)
Pr(xt+3≥x|xt+1=x)dx
+ Pr (xt+1 < x ∩ xt+2 ≥ x | xt = x)︸ ︷︷ ︸=qt+2
Pr (xt+3 ≥ x | τ = t+ 2)︸ ︷︷ ︸=´∞x
Pr(xt+2=x)Pr(xt+2≥x)
Pr(xt+3≥x|xt+2=x)dx
+ Pr (xt+1 < x ∩ xt+2 < x ∩ xt+3 ≥ x | xt = x)︸ ︷︷ ︸=qt+3
Pr (xt+3 ≥ x | τ = t+ 3)︸ ︷︷ ︸=´∞x
Pr(xt+3=x)Pr(xt+3≥x)
Pr(xt+3≥x|xt+3=x)dx=1
1.3.2 q in matrix form
Altogether, a one-dimensional process can be represented in matrix form, as described in the following theorem.
Corollary 1.4. A system of equations, dened in (5) to (7), for the rst-passage time probabilities can be represented
in a matrix form in the following way:
Pr (a1)
Pr (a2)
Pr (a3)...
Pr (ak)
︸ ︷︷ ︸
=A
= Θ×
q1
q2
q3...
qk
︸ ︷︷ ︸
=q
(25)
where A is dened by (17) to (20), and k × k matrix Θis equal to
Θ =
1 0 0 · · · 0´∞x
Pr(x1=x)Pr(x1≥x)Pr (b1 (x)) dx 1 0 · · · 0´∞
xPr(x1=x)Pr(x1≥x)Pr (b2 (x)) dx
´∞x
Pr(x2=x)Pr(x2≥x)Pr (b1 (x)) dx 1 · · · 0
......
.... . .
...´∞x
Pr(x1=x)Pr(x1≥x)Pr (bk−1 (x)) dx
´∞x
Pr(x2=x)Pr(x2≥x)Pr (bk−2 (x)) dx
´∞x
Pr(x3=x)Pr(x3≥x)Pr (bk−3 (x)) dx · · · 1
(26)
and we obtain
q =Θ−1 ×A. (27)
Summarizing, the (i, j)′th element of Θ matrix is the following. For i + j = k, we might write element (i, j) of
matrix Θ as:
Θi,j =
∞
x
Pr (xj = x)
Pr (xj ≥ x)Pr (bi=k−j (x)) dx
=1
Pr (xj ≥ x)
∞
x
Pr (xj = x)Pr (bi=k−j (x)) dx.
10
For instance, the (3, 1)′th element2 of matrix (26) is 1
Pr(x1≥x)´∞xPr (x1 = x)Pr (b2 (x)) dx and we have x1=i and
b2=j=k−i and so i+ j = 3. Elements of matrix Θ can be easily computed if the underlying autoregressive process has
closed-form transition-probabilities.
1.3.3 Existence of finite τ and E (τ)
A rst-passage time exists, for a given process xt, if:
Pr (τ =∞) = 0 (28)
which, in terms of our notation, is equivalent to:
limi→∞
qτ=i = 0.
Baumgarten (2014), Basak and Ho (2004) and Novikov and Kordzakhia (2008) describe the necessary and sucient
conditions for the existence of nite rst passage time for AR(1) processes with independent and identically distributed
innovations. In particular, Novikov and Kordzakhia (2008) show that innovations to the process xt need to fulll
only fairly mild assumptions on the left tail of its distribution, for the process xt to possess nite rst passage
time distribution. However, we might be also interested in nding rst passage time distributions for some other
autoregressive processes with innovations which are not necessarily i.i.d., like for instance ARG(1) or an AR(1) process
with a stochastic volatility. Therefore we suggest to take a more pragmatic numerical approach, which can be easily
applied to any stochastic autoregressive process.
There are multiple ways in which we can determine the limit of qτ=i sequence, but one particularly convenient for
numerical applications, is the Cauchy sequence test. A sequence is Cauchy if for every positive real number, ε > 0,
there is a positive integer N , such that for all natural numbers n, m > N , the following condition holds:
|qn − qm| < ε. (29)
If criterion (29) is met for a suciently small ε then we can infer that (28) also holds true.
In addition to nite τ , we might also be interested to see whether the mean rst passage time, equal to
E (τ) =
∞∑i=1
i · qτ=i
is nite as well. The, so-called, Cauchy convergence test provides a simple to implement method. We test for
convergence by forming a sequence out of partial sums, that is
sn =
n∑i=1
i · qτ=i. (30)
Then, according to the test, an innite series is convergent if sn can be shown to be a Cauchy sequence. In practice,
we can choose some very small ε ≈ 0 and then verify that for large enough integer N, we can nd n, m > N such that
|sn − sm| < ε. (31)
We note here also that condition (31) is stronger than condition (28).
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Figure 4: Plot of q density from the model and Monte-Carlo simulation
1.4 AR(1) example
We have a process:
yt+1 = κ+ ρyt + εt+1 (32)
where εt ∼ N(0, σ2
ε
). Time t expected value and variance of yt+k are equal to
Et (yt+k) =(1 + ρ+ . . .+ ρk−1
)κ+ ρkyt
V art (yt+k) =(
1 + ρ2 + . . .+ ρ2(k−1))σ2ε
Since εt has a Gaussian distribution, we can use the above two formulas to describe the transition probability of
process yt to the values above the threshold y. Transition probabilities are equal to
Prt(yt+k ≥ y
)= Prt
(yt+k − Et (yt+k)
V art (yt+k)≥y − Et (yt+k)
V art (yt+k)
)= 1− Φ
(y − Et (yt+k)
V art (yt+k)
)in terms of our notation from the section before we have
Pr (ai) = 1− Φ
(y − Et (yt+i)
V art (yt+i)
)= Φ
(−y − Et (yt+i)
V art (yt+i)
)(33)
yt = y0
2See 6 for a graphic representation.
12
and
Pr (bj (y)) = 1− Φ
(y − Et+k−j (yt+k)
V art+k−j (yt+k)
)= Φ
(−y − Et+k−j (yt+k)
V art+k−j (yt+k)
)(34)
yt+k−j = y
Finally, we may plug appropriate values of (33) and (34) into (25):
Φ(−y−Et(yt+1)
V art(yt+1)
)Φ(−y−Et(yt+2)
V art(yt+2)
)Φ(−y−Et(yt+3)
V art(yt+3)
)...
Φ(−y−Et(yt+k)V art(yt+k)
)
︸ ︷︷ ︸
=AAR(1)
=
1 0 0 · · · 0
Θ2,1 1 0 · · · 0
Θ3,1 Θ3,2 1 · · · 0...
......
. . ....
Θk,1 Θk,2 Θk,3 · · · 1
︸ ︷︷ ︸
=ΘAR(1)
×
q1
q2
q3...
qk
︸ ︷︷ ︸=qAR(1)
where the elements of matrix ΘAR(1) are dened by (26) for Gaussian noise. In this case, for i+ j = k we might write
element (i, j) of matrix ΘAR(1) as
Θi,j =
∞
x
Pr (yj = y)
Pr(yj ≥ y
)Pr (bt+i−j (y)) dy
=1
Φ(−y−Et(yt+j)V art(yt+j)
) ∞
x
φ (yj = y, Et (yt+j) , V art (yt+j)) Φ
(−y − Et+i−j (yt+k | yt+k−j = y)
V art+k−j (yt+k | yt+k−j = y)
)dy
where φ (·) and Φ (·) stand respectively for Gaussian pdf and cdf functions. In order to obtain implicit estimates of
(27) for our AR(1) process:
qAR(1) =(ΘAR(1)
)−1×AAR(1). (35)
Figure (4) compares a density distribution for a particular AR(1) process with (35) to 50.000 draws from a Monte
Carlo simulation for the process with the same parameters.
1.5 ARG(1) process
ARG(1) process has been introduced in Gouriéroux and Jasiak (2006) and Darolles, Gourieroux, and Jasiak (2006).
For constants c > 0, v > 0, we dene the conditional density of zt+1 given zt as the Poisson mixture of standard
gamma distributions:
zt+1
c| (P, zt) ∼ gamma (v + P) , where P | zt ∼ Poisson (ρzt/c)
13
Figure 5: Plot of q density from the model and Monte-Carlo simulation for ARG(1) process
and the conditional density is3
f (zt+1 | zt) =1
c
∞∑k=0
[(ρztc
)k!
e−ρztc ×
( zt+1
c
)v+k−1e−
ρzt+1c
Γ (v + k)
](36)
We need also transition probabilities for any j time periods ahead. In order to obtain f (zt+j | zt), we make use of4
the ane property of ARG(1) process. We have
zt+jct+j|t
| (P, zt) ∼ gamma (v + P) , where P | zt ∼ Poisson(ρt+j|tzt/ct+j|t
)and the conditional density for multiple periods is
f (zt+j | zt) =1
c
∞∑k=0
(ρt+j|tct+j|t
zt
)k!
e−ρt+j|tct+j|t
zt ×
(zt+jct+j|t
)v+k−1e−ρt+j|tct+j|t
zt+j
Γ (v + k)
(37)
3As was mentioned before, ARG(1) is a discrete time version of a the continuous time CIR process. We can map ARG(1) parametersinto CIR parameters by choosing the following:
c =1
2σ2dt
v =2κθ
σ2
ρ = 1− κdt
and if we would let dt approach 0 then zt would approach the CIR process:
dzt = κ (θ − zt) dt+ σ√ztdWt
4see Proposition 2 in Gouriéroux and Jasiak (2006)
14
where
ρt+j|t = ρj (38)
ct+j|t = c(1 + ρ+ . . .+ ρj−1
)= c
1− ρj
1− ρ(39)
2 First-passage-time for a two-dimensional process.
2.1 The general method
Suppose that xt and yt dene two stationary stochastic process. In addition, assume that we have a joint distribution
of two stochastic processes:(xt, yt) and its joint transition probability from time t to any time t+k in the future. This
time, just like before, we are interested in nding FPT for the process xt with boundary dened by x. Obtaining an
exact solution for the two-dimensional process appears to be computationally impossible. That is, in order to obtain
an exact solution we would need to integrate over all possible paths of yt and separately for each time point between
the starting time t and the end point t + k. Following Buonocore et al. (1987) and Collin-Dufresne and Goldstein
(2001) we circumvent this problem by discretizing process yt. Let yt+v for every v belong to one of ny states. This
way, for the discretized process yt we are able to consider all dierent n2y paths.
For brevity, we will abuse slightly notation and write y(s)t+v instead of yt+v = σ(s) to denote events in which process
y at time t+ v is at state s.
Using sample space partition, (4), and the law of total probability we may adjust for the second random variable
and form a sequence of k × ny equations. Starting from t+ 1 we obtain a set of ny equations which dene qt+1,
Pr(xt+1 ≥ x, y(1)t+1 | xt = x
)= qt+1
(y(1)t+1
)(40)
... =...
Pr(xt+1 ≥ x, y(n)t+1 | xt = x
)= qt+1
(y(n)t+1
)(41)
and for t+ 2, we also have ny equations:
Pr(xt+2 ≥ x, y(1)t+2 | xt = x
)=
n∑s=1
qt+1
(y(s)t+1
)Pr(xt+2 ≥ x, y(1)t+2 | τ = t+ 1, y
(s)t+1
)+ qt+2
(y(1)t+2
)... =
...
Pr(xt+2 ≥ x, y(n)t+2 | xt = x
)=
n∑s=1
qt+1
(y(s)t+1
)Pr(xt+2 ≥ x, y(n)t+2 | τ = t+ 1, y
(s)t+1
)+ qt+2
(y(n)t+2
)and so on until time t+ k:
Pr(xt+k ≥ x, y(1)t+k | xt = x
)=
n∑u=1
qt+1
(y(u)t+1
)Pr(xt+k ≥ x, y(1)t+k | τ = t+ 1, y
(u)t+1
)+ . . .+ qt+k
(y(1)t+k
)(42)
... =...
Pr(xt+k ≥ x, y(n)t+k | xt = x
)=
n∑u=1
qt+1
(y(u)t+1
)Pr(xt+k ≥ x, y(n)t+k | τ = t+ 1, y
(u)t+1
)+ . . .+ qt+k
(y(n)t+k
)(43)
This is summarized by the following theorem.
15
Theorem 2.1. Equations (40) to (43) dene implicit solution for the rst passage time probabilities from qt+1 until
qt+k. We have a formula for t+ 1 :
qt+1
(y(i)t+1
)= Pr
(xt+1 ≥ x, y(i)t+1 | xt = x, yt
)∀i ∈ 1, . . . , n (44)
and for any j in the range between t+ 1 and t+ k, we obtain qt+j recursively:
qt+j
(y(i)t+j
)= Pr
(xt+j ≥ x, y(i)t+j | xt = x, yt
)−j−1∑v=1
n∑u=1
qt+v
(y(u)t+v
)Pr(xt+j ≥ x, y(i)t+j | τ = t+ v, y
(u)t+v
)(45)
∀j ∈ 1, . . . , k ,∀i ∈ 1, . . . , n
and consequently an unconditional value of qt+j is given by
qt+j =
ny∑i=1
qt+j
(y(i)t+j
)Proof. The system of (40) to (43) equations we obtain by construction and equations (44) and (45) follow directly
from them.
Similarly to equation (25) which denes implicitly FPT for a one dimensional process, we can represent equations
(44) and (45) in a matrix form. However, this time, the number of equations will be n times bigger. That is, vectors
A and q will have (n · k)× 1 dimension and matrix Θ will have a dimension equal to (n · k)× (n · k), where n denotes
a number of distinct states for random process y, while k is a number of time-periods.
Corollary 2.2. A system of equations dened in theorem (2.1) can be represented in a matrix form in the following
way:
Pr(a1, y
(1)t+1
)...
Pr(a1, y
(1)t+1
)Pr(a2, y
(1)t+2
)...
Pr(a2, y
(n)t+2
)...
Pr(ak, y
(1)t+k
)...
Pr(ak, y
(n)t+k
)
︸ ︷︷ ︸
=A
=
1n×n 0n×n 0n×n · · · 0n×n
Θn×n2,1 1n×n 0n×n · · · 0n×n
Θn×n3,1 Θn×n
3,2 1n×n · · · 0n×n
......
.... . .
...
Θn×nk,1 Θn×n
k,2 Θn×nk,3 · · · 1n×n
︸ ︷︷ ︸
=Θ(n·k)×(n·k)
×
qt+1
(y(1)t+1
)...
qt+1
(y(n)t+1
)qt+2
(y(1)t+2
)...
qt+2
(y(n)t+2
)...
qt+k
(y(1)t+k
)...
qt+k
(y(n)t+k
)
︸ ︷︷ ︸
=q
(46)
where the elements of vector A are dened by
Pr(aj , y
(u)t+j
)= Pr
(xt+j ≥ x, y(u)t+j | xt = x, yt
)∀j ∈ 1, . . . , k ,∀u ∈ 1, . . . , n
and matrix Θ(n·k)×(n·k) consists of k × k submatrices each of n × n dimension. Any (v, j)'th submatrix, Θn×nv,j ,
is equivalent to any (v, j)'th element of matrix from (26). The dierence is that it conditions on the transition
16
probabilities between dierent states of process σt,
Θn×nv,j =
Θv,j
(yt+j = y(1) | yt+v = y(1)
)· · · Θv,j
(yt+j = y(1) | yt+v = y(n)
)...
. . ....
Θv,j
(yt+j = y(n) | yt+v = y(1)
)· · · Θv,j
(yt+j = y(n) | yt+v = y(n)
) . (47)
For brevity we will write
Θn×nv,j =
Θ
1|1v,j Θ
1|2v,j · · · Θ
1|nv,j
Θ2|1v,j Θ
2|2v,j · · · Θ
2|nv,j
......
. . ....
Θn|1v,j Θ
n|2v,j · · · Θ
n|nv,j
, (48)
where the upper script denotes transition between the states of the second stochastic process. For instance, 1 | 2 in
Θ1|2v,j , means that here we condition on yt+v moving from state y(2) at time t+v, to state y(1) at time t+ j. Altogether,
we can obtain a vector of FPT probabilities in a vector form:
q =Θ−1 ×A (49)
2.2 Example of finding FPT density for a Gaussian VAR(1) proces
We are interested in
xt+1 = ψxxxt + ψxyyt + εx,t+1 (50)
yt+1 = ψyxxt + ψyyyt + εy,t+1 (51)
where we allow the innovations to be correlated: cov (εx,t, εy,t) = σxy. In order to obtain transition probabilities for
processes xt and yt we may rewrite them as one joint VAR(1) process zt:
zt =
[xt
yt
], (52)
Ψ =
[ψxx ψxy
ψyx ψyy
](53)
εt =
[εx,t
εy,t
](54)
Σ =
[σ2x σxy
σxy σ2y
](55)
and altogether we have
zt+1 = Ψzt + εt+1 (56)
17
and so
Et (zt+k) = Ψkzt (57)
V art (zt+k) = Ψk−1ΣΨ′k−1 + . . .+ ΨΣΨ′ + Σ
=
k−1∑j=0
ΨjΣΨ′j (58)
We assumed that the pair (xt, yt) follows a joint Gaussian process. Therefore, from the properties bivariate Gaussian
processes we know that
Et [xt+k | yt+k] = Et [xt+k] +Covt [xt+k, yt+k]
V art [yt+k](yt+k − Et [yt+k]) (59)
V art [xt+k | yt+k] = V art [xt+k] +Covt [xt+k, yt+k]
2
V art [yt+k](60)
where the formulas for Et [xt+k] ,V art [xt+k] , Et [yt+k] , V art [yt+k] and Covt [xt+k, yt+k] are obtained from (57) and
(58).
Next we proceed following the steps from 1.4. Since εt has a Gaussian distribution, we can use the above two
formulas to describe the transition probability of process xt to the values above the threshold x. Transition probabilities
are equal to
Prt
(xt+k ≥ x, yt+k = y(1)
)= Prt
(xt+k ≥ y | xt+k = x(1)
)Prt
(yt+k = y(1))
)= Prt
(xt+k − Et (xt+k)
V art (xt+k)≥ x− Et (xt+k)
V art (xt+k)| yt+k = y(1)
)Prt
(yt+k = y(1))
)=
[1− Φ
(x− Et (xt+k)
V art (xt+k)| yt+k = y(1)
)]Prt
(yt+k = y(1)
)= Φ
(−x− Et (xt+k)
V art (xt+k)| yt+k = y(1)
)Prt
(yt+k = y(1)
)in terms of our notation from the section before we have
Pr(ai, yt+i = y(1)
)= Pr
(ai | yt+i = y(1)
)Prt
(yt+i = y(1)
)(61)
and
Pr (bj (x)) = Φ
(−x− Et+k−j (xt+k)
V art+k−j (yt+k)
)(62)
xt+k−j = x
18
Finally, we may plug appropriate values of (61) and (62) into (25):
Φ(−y−Et(yt+1)
V art(yt+1)
)Φ(−y−Et(yt+2)
V art(yt+2)
)Φ(−y−Et(yt+3)
V art(yt+3)
)...
Φ(−y−Et(yt+k)V art(yt+k)
)
︸ ︷︷ ︸
=AVAR(1)
=
1n×n 0n×n 0n×n · · · 0n×n
Θn×n2,1 1n×n 0n×n · · · 0n×n
Θn×n3,1 Θn×n
3,2 1n×n · · · 0n×n
......
.... . .
...
Θn×nk,1 Θn×n
k,2 Θn×nk,3 · · · 1n×n
︸ ︷︷ ︸
=ΘVAR(1)
×
q1
q2
q3...
qk
︸ ︷︷ ︸=qVAR(1)
where the elements of matrix ΘV AR(1) are dened by (26) for Gaussian noise. In this case, for i + j = k we might
write element (i, j) of matrix ΘV AR(1) as
Θi,j
(y(v)t+i | y
(s)τ=t+j
)=
∞
x
Prt(xt+j = x, y
(s)t+j
)Prt
(xt+j ≥ x, y
(s)t+j
)Pr (bt+i−j (x) , y(v)t+i | y
(s)t+j
)dx
=
∞
x
Prt(xt+j = x | y(s)τ=t+j
)Prt
(xt+j ≥ x | y
(s)τ=t+j
)Pr (bt+i−j (x) | y(v)t+i, y(s)t+j
)Prt
(y(v)t+i | y
(s)t+j
)dy
=
∞
x
φ(xj = x, Et (xt+j) , V art (xt+j) | y
(s)τ=t+j
)Φ
(−x−Et(xt+j)V art(xt+j)
) Φ
(−x− Et+i−j
(xt+k | xt+k−j = x
)V art+k−j
(xt+k | xt+k−j = x
) | y(v)t+i, y(s)t+j
)Prt
(y(v)t+i | y
(s)t+j
)dx
3 Conclusions
The main contribution of this paper is to provide a method for computing rst-passage-time (FPT) distribution for
stationary Markov processes in discrete-time. We show that it is possible to nd an exact FPT distribution in case
of one-dimensional stochastic processes and an approximate solution in the case of two-dimensional processes. We
present a few examples with detailed description. That is, in one dimensional case, we show how to nd FPT for an
AR(1) process with Gaussian innovations and the so-called ARG(1), which is an autoregressive gamma process - a
discrete-time equivalent of the square root (CIR) process in continuous-time. In the two dimensional case, we show
how to compute FPT for a VAR(1) process with Gaussian innovations. FPT probabilities are obtained implicitly from
equations following from the law of total probability. The main ingredient of our method is the transition probability
for examined processes. Our method is applicable to all types of innovation distributions, as long we can compute
multiperiod transition probabilities either in closed form or by means of Fourier transforms. The proposed method
avoids the use of complex mathematical concepts and it is easy to understand and apply in practice.
Finally, we show how to solve the problem of overshooting the boundary. Most of the models in continuous-time
literature assume processes that cannot jump over the boundary. That is exactly the reason why spectrally negative
Levy processes, which have jumps only in one direction, are the basis for the Mixed Hitting Time models. Diusion
models in continuous time always assume Gaussian distribution and they cannot generate skewed or leptokurtic asset
returns. This restriction can be circumvented only by adding a separate process with jumps. However, in discrete
time all realizations of the process are jumps and there is no need to conne oneself to Gaussian distributions. In this
sense discrete-time models are often more general then their continuous-time counterparts, or at least they can save
on the dimensionality of the problem. Our method can be applied both for models where the process behind FPT
events is observed as well as for cases when it is latent.
19
Appendix
Intuition behind the overshooting formula (15)
Let us suppose that the process xt+s ≥ x at time t+ s can take only a nite and discrete number of n states:
Bt+s :=xt+s = x1
∪xt+s = x2
∪ . . . ∪ xt+s = xn
=
n⋃i=1
xt+s = xi
where
x ≤ x1 < x2 < . . . < xn
and then
Pr (xt+k ≥ x | xt+s ≥ x) = Pr (At+k | Bt+s)
= Pr
(xt+k ≥ x |
n⋃i=1
xt+s = xi
)
=Pr(xt+k ≥ x ∩
⋃ni=1
xt+s = xi
)Pr(⋃n
j=1 xt+s = xj)
=Pr(xt+k ≥ x ∩
xt+s = x1
∪ . . . ∪ xt+k ≥ x ∩ xt+s = xn
)Pr(⋃n
j=1 xt+s = xj)
=Pr(⋃n
i=1
xt+k ≥ x ∩
xt+s = xi
)Pr(⋃n
j=1 xt+s = xj)
=
n∑i=1
Pr(xt+k ≥ x ∩
xt+s = xi
)Pr(⋃n
j=1 xt+s = xj)
=
n∑i=1
Pr(xt+s = xi
)Pr(xt+k ≥ x |
xt+s = xi
)Pr(⋃n
j=1 xt+s = xj) (63)
and here we can see thatPr(xt+s=xi)Pr(xt+k≥x|xt+s=xi)
Pr(⋃ni=1xt+s=xi)
from (63) is a discrete time analogue of the integrand
in (15) equation.
Intuition for the two dimensional process
Let us suppose that the process xt+s ≥ x at time t+ s can take only a nite and discrete number of n states:
Bt+s :=xt+s = x1
∪xt+s = x2
∪ . . . ∪ xt+s = xn
=
n⋃i=1
xt+s = xi
where
x ≤ x1 < x2 < . . . < xn
20
and then
Pr(xt+k ≥ x ∩
yt+k = y(u)
| xt+s ≥ x ∩
yt+s = y(v)
)=
= Pr
(xt+k ≥ x ∩
yt+k = y(u)
|
n⋃i=1
xt+s = xi
∩yt+s = y(v)
)
=Pr(xt+k ≥ x ∩
yt+k = y(u)
∩⋃n
i=1
xt+s = xi
∩yt+s = y(v)
)Pr(⋃n
j=1 xt+s = xj∩yt+s = y(v)
)=
∑ni=1 Pr
(xt+k ≥ x ∩
yt+k = y(u)
∩xt+s = xi
∩yt+s = y(v)
)Pr(⋃n
j=1 xt+s = xj∩yt+s = y(v)
)=
∑ni=1 bd
(xi)ξd(xi)ad(xi)Pr(yt+s = y(v)
)Pr(⋃n
j=1 xt+s = xj|yt+s = y(v)
)Pr(yt+s = y(v)
)=
∑ni=1 bd
(xi)ξd(xi)ad(xi)
Pr(⋃n
j=1 xt+s = xj|yt+s = y(v)
)where
bd(xi)
= Pr(xt+k ≥ x |
yt+k = y(u)
∩xt+s = xi
∩yt+s = y(j)
)ξd(xi)
= Pr(yt+k = y(u)
|xt+s = xi
∩yt+s = y(j)
)ad(xi)
= Pr(xt+s = xi
|yt+s = y(j)
)and so analogously for x with a continuous support, we have
Θi,j
(y(u)t+i | y
(v)τ=t+j
)=
= Pr(xt+k ≥ x, yt+k = y(u) | xt+s ≥ x, yt+s = y(v)
)=Pr(xt+k ≥ x, yt+k = y(u), xt+s ≥ x, yt+s = y(v)
)Pr(xt+s ≥ x ∩
yt+s = y(v)
)=
´∞x Pr
(xt+k ≥ x, yt+k = y(u), xt+s = x, yt+s = y(v)
)dx
Pr(xt+s ≥ x ∩
yt+s = y(v)
)=
´∞x Pr
(xt+k ≥ x | yt+k = y(u), xt+s = x, yt+s = y(v)
)Pr(yt+k = y(u), xt+s = x, yt+s = y(v)
)dx
Pr(xt+s ≥ x ∩
yt+s = y(v)
)=
´∞x Pr
(xt+k ≥ x | yt+k = y(u), xt+s = x, yt+s = y(v)
)Pr(yt+k = y(u) | xt+s = x, yt+s = y(v)
)Pr(xt+s = x, yt+s = y(v)
)dx
Pr(xt+s ≥ x ∩
yt+s = y(v)
)=
´∞x Pr
(xt+k ≥ x | yt+k = y(u), xt+s = x, yt+s = y(v)
)Pr(yt+k = y(u) | xt+s = x, yt+s = y(v)
)Pr(xt+s = x | yt+s = y(v)
)dx
Pr(xt+s ≥ x |
yt+s = y(v)
)=
´∞x b (x) c (x) a (x) dx
Pr(xt+s ≥ x |
yt+s = y(v)
)where
21
b (x) = Pr(xt+k ≥ x | yt+k = y(u), xt+s = x, yt+s = y(v)
)= Φ
(−x− Et+k−s
(xt+k | yt+k = y(u), xt+s = x, yt+s = y(v)
)V art+k−j
(xt+k | yt+k = y(u), xt+s = x, yt+s = y(v)
) )c (x) = Pr
(yt+k = y(u) | xt+s = x, yt+s = y(v)
)a (x) = Pr
(xt+s = x | yt+s = y(v)
)= φ
(xt+s = x, Et
(xt+s | yt+s = y(v)
), V art
(xt+s | yt+s = y(v)
))We can observe that there is a pattern in matrix Θ, which can facilitate faster computations. We can show only the
upper left corner elements of matrix Θ:
Θ =
1 0 0 0 0 0
a1b1c1 1 0 0 0 0
a1b2c2 a2b1c1 1 0 0 0
a1b3c3 a2b2c2 a3b1c1 1 0 0
a1b4c4 a2b3c3 a3b2c2 a4b1c1 1 0
a1b5c5 a2b4c4 a3b3c3 a4b2c2 a5b1c1 1
= a b c,
which can be decomposed into a piece-by-piece product of matrices a, b and c:
a =
1 0 0 0 0 0
a1 1 0 0 0 0
a1 a2 1 0 0 0
a1 a2 a3 1 0 0
a1 a2 a3 a4 1 0
a1 a2 a3 a4 a5 1
and
b =
1 0 0 0 0 0
b1 1 0 0 0 0
b2 b1 1 0 0 0
b3 b2 b1 1 0 0
b4 b3 b2 b1 1 0
b5 b4 b3 b2 b1 1
,
c =
1 0 0 0 0 0
c1 1 0 0 0 0
c2 c1 1 0 0 0
c3 c2 c1 1 0 0
c4 c3 c2 c1 1 0
c5 c4 c3 c2 c1 1
,
22
Once again we can show only the upper left corners of multidimensional matrices a and b. We will do it for n = 2:
a =
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
)a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
) 1 0
0 1
0 0
0 0
0 0
0 0
a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
)a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
) a2(y(s)t=0 → y
(1)τ=2
)a2(y(s)t=0 → y
(2)τ=2
)a2(y(s)t=0 → y
(1)τ=2
)a2(y(s)t=0 → y
(2)τ=2
) 1 0
0 1
0 0
0 0
a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
)a1(y(s)t=0 → y
(1)τ=1
)a1(y(s)t=0 → y
(2)τ=1
) a2(y(s)t=0 → y
(1)τ=2
)a2(y(s)t=0 → y
(2)τ=2
)a2(y(s)t=0 → y
(1)τ=2
)a2(y(s)t=0 → y
(2)τ=2
) a3(y(s)t=0 → y
(1)τ=3
)a3(y(s)t=0 → y
(2)τ=3
)a3(y(s)t=0 → y
(1)τ=3
)a3(y(s)t=0 → y
(2)τ=3
) 1 0
0 1
and
b =
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
b1(y(1)τ=1 → y
(1)t=2
)b1(y(2)τ=1 → y
(1)t=2
)b1(y(1)τ=1 → y
(2)t=2
)b1(y(2)τ=1 → y
(2)t=2
) 1 0
0 1
0 0
0 0
0 0
0 0
b2(y(1)τ=1 → y
(1)t=3
)b2(y(2)τ=1 → y
(1)t=3
)b2(y(1)τ=1 → y
(2)t=3
)b2(y(2)τ=1 → y
(2)t=3
) b1(y(1)τ=2 → y
(1)t=3
)b1(y(2)τ=2 → y
(1)t=3
)b1(y(1)τ=2 → y
(2)t=3
)b1(y(2)τ=2 → y
(2)t=3
) 1 0
0 1
0 0
0 0
b3(y(1)τ=1 → y
(1)t=4
)b3(y(2)τ=1 → y
(1)t=4
)b3(y(1)τ=1 → y
(2)t=4
)b3(y(2)τ=1 → y
(2)t=4
) b2(y(1)τ=1 → y
(1)t=3
)b2(y(2)τ=1 → y
(1)t=3
)b2(y(1)τ=1 → y
(2)t=3
)b2(y(2)τ=1 → y
(2)t=3
) b1(y(1)τ=3 → y
(1)t=4
)b1(y(2)τ=3 → y
(1)t=4
)b1(y(1)τ=3 → y
(2)t=4
)b1(y(2)τ=3 → y
(2)t=4
) 1 0
0 1
,
and now a multidimensional version of Θ :
1n×n 0n×n 0n×n · · · 0n×n
Θn×n2,1 1n×n 0n×n · · · 0n×n
Θn×n3,1 Θn×n
3,2 1n×n · · · 0n×n
......
.... . .
...
Θn×nk,1 Θn×n
k,2 Θn×nk,3 · · · 1n×n
=
1 0
0 10n×n 0n×n · · · 0n×n
Θ2,1
(σ(1)t=2 | σ
(1)τ=1
)Θ2,1
(σ(1)t=2 | σ
(2)τ=1
)1n×n 0n×n · · · 0n×n
Θ2,1
(σ(2)t=2 | σ
(1)τ=1
)Θn×n
3,1 Θ2,1
(σ(2)t=2 | σ
(2)τ=1
)Θn×n
3,2 1n×n · · · 0n×n
......
.... . .
...
Θn×nk,1 Θn×n
k,2 Θn×nk,3 · · · 1n×n
23
Figure 6: This graph corresponds to the element Θ3,1 from matrix (26). The process xt starts from an indicated valuex0 and the rst-passage time happens at time t + 1. Blue arrows represent dierent possible realizations of processstarting at xt = x0 at time t+ 1, above the hitting boundary x. Given this realization, let's say xt+1 = x, red arrowsrepresent possible realizations of process x at time xt+3 starting from xt+1 = x..
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