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 Indian Journal of Finance and Banking; Vol. 3, No. 1; 2019 

  ISSN 2574-6081   E-ISSN 2574-609X 

Research Article               Published by Centre for Research on Islamic Banking & Finance and Business, USA 

 

     13 
 

Complex Probability and Markov Stochastic Process 
 

 

Bijan Bidabad 

B.A., M.Sc., Ph.D., Post Doc. 

Professor of Economics and Chief Islamic Banking Advisor 

Bank Melli Iran, Tehran, Iran         

Email: bijan@bidabad.com 

 

Behrouz Bidabad 

Faculty of Mathematics, Polytechnics University  
Hafez Ave., Tehran, 15914, Iran 

Email: bidabad@aut.ac.ir 

 

Abstract  

This note discusses the existence of "complex probability" in the real world sensible problems. By defining a 

measure more general than the conventional definition of probability, the transition probability matrix of discrete 

Markov chain is broken to the periods shorter than a complete step of the transition. In this regard, the complex 

probability is implied. 

Keywords: Prbability, Markov Process, Stochastic Process, Population Census 

1. Introduction 

Sometimes analytic numbers coincide with the mathematical modeling of real world and make the real analysis of 

problems complex. All the measures in our everyday problems belong to R, and mostly to R+. Probability of 
occurrence of an event always belongs to the range [0,1]. In this paper, it is discussed that to solve a special class of 

Markov chain which should have a solution in the real world; we are confronted with "analytic probabilities"!. 

Though the name probability applies to the values between zero and one, we define a special analog measure of 

probability as a complex probability where the conventional probability is a subclass of this newly defined measure.  

 Now define the well-known discrete-time Markov chain  nY  a Markov stochastic process whose state space is 

 Ns ,...,2,1  for which  0,1,2,...T  . Refer to the value of Yn as the outcome of the nth trial. We say Yn 

being in state i if Yn = i. The probability of Yn+1 being in state j, given that Yn is in state i (called a one-step 

transition probability) is denoted by 
1, nn

ijP  , i.e., 

 , 1

1Prn n

ij ij n nP P Y j Y i

                                                                 (1) 

Therefore, the Markov or transition probability matrix of the process is defined by  

 

1

(2)

0 ,

1

ij

ij

N

ij
j

P

P i j s

P i s




  

  

P

 



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The n-step transition probability matrix 
( ) ,n n

ijPP which
n

ijP denotes the probability that the process goes 

from state i to state j in n transitions. Formally, 

 Pr ,n

ij n m mP Y j Y i i j S                       

(3) 

 

According to Chapman – Kolmogorov relation for discrete Markov matrices (Karlin and Taylor (1975)), it can be 

proved that  

 

( ) ( ) (4)n n n N Natural numbers P P  

P
n that is P to the power n is a Markov matrix if P is Markov.  

 Now, suppose that we intend to derive the t-step transition probability matrix P(t) where t≥0 from the above 

(3) and (4) definition of n-step transition probability matrix P. That is, to find the transition probability matrix for 

incomplete steps. On the other hand, we are interested in finding the transition matrix P(t) when t is between two 

sequential integers. This case is not just a tatonnement example. To clarify the application of this phenomenon, 

consider the following example.  

Example 1. Usually, in the population census of societies with N distinct regions, migration information is collected 

in an NxN migration matrix for a period of ten years. Denote this matrix by M. Any element of M, mij is the 

population who left region i and went to region j through the last ten years. By dividing each mij to sum of the ith row 

of M, a value of Pij is computed as an estimate of the probability of transition from ith to jth regions. Thus, the 

stochastic matrix P gives the probabilities of going from region i to region j in ten years (which is one–step 

transition probability matrix). The question is: how we can compute the transition probability matrix for one year or 

one-tenth step and so on.  

 If we knew the generic function of probabilities in a very small period of time, we would be able to solve 

problems similar to example 1. But the generic function (5) is not obtainable. If it were, we would apply the 

continuous time Markov procedure using the generic NxN matrix A as 

( )
lim (5)

h o

h

h





P I
A  

Where P(h) denotes transition probability matrix at time h. Then the transition probability matrix at any time 0t  

might be computed as follows. (Karlin and Taylor (1975)). 

  

P(t) = e At                                 

      (6) 

 

 Therefore a special procedure should be adopted to find the transition probability matrix P(t) at any time t 

from discrete Markov chain information. As will be shown later, the adopted procedure coincides with a transition 

probability matrix with complex elements.  

 



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2.Breaking the time in discrete Markov chain  

 Consider again matrix P defined in (2). Also, assume P is of full rank.  

Assumption 1: P is of full rank.  

This assumption assures that all eigenvalues of P are nonzero, and P is diagonalizable, Searle (1982), Dhrymes 

(1978). This assumption is not very restrictive, since; actually, most of Markov matrices have dominant diagonals. 

That is the probability of transition from state i to itself is more than the sum of probabilities from state i to all other 
states. The matrices having dominant diagonals are non-singular, Takayama (1974). Therefore, P can be 

decomposed as follows (Searle (1982), Klein (1973)). 

1P XΛX                                                                                                     (7) 

Where X is an NxN matrix of eigenvectors  1,..., ,,i Ni x  

1[ ,...., ]NX x x                                                                                                     (8) 

and Λ the NxN diagonal matrix of corresponding eigenvalues, 

         1{ ,..., }Ndiag  Λ                                                                                              (9) 

Using (7), (8), and (9) to break n-step transition probability matrix P to any smaller period of time t 0, we do as 

follows. If 0it  for all iЄ{1,…,K}are fractions of n–step period and 
1

k

i
i

t n


  for any n belonging to natural 

numbers then,  

1

1

k

ik
n i

j

t
it 



 


P P P                                                                                    (10) 

 On the other hand, the transition probability matrix of n-step can be broken to fractions of n, if the sum of 

them is equal to n. Therefore, any 0t  fraction of the one-step transition probability matrix can be written as,  

 

1t t P XΛ X                           

              (11) 

where,  

 1 ,....,t t t

Ndiag  Λ                                                                                   (12) 

Before discussing the nature of eigenvalues of P, let us define the generalized Markov matrix. 

Definition 1. Matrix Q is a generalized Markov matrix if the following conditions are fulfilled: 



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 

 

1

1

1) ,

2) Re ( ) 0,1 ,

3) Im( ) 1,1 ,

4) Re ( ) 1

5) Im( ) 0

ij

ij

ij

N

ij
j

N

ij
j

q C i j S

q i j S

q i j S

q i S

q i S





  

  

   

  

  

 

 

Remark 1. According to definition 1, matrix Q can be written as:  

 

i Q U V                     

                      (13) 

Where U and V are NxN matrices of real and imaginary parts of Q with .1i   

 

Remark 2. Matrix U has all Properties of P defined by (2); thus, P Q. 

 

Theorem 1. If P is a Markov matrix, then Pt also satisfies Markovian properties.  

Proof: According to Chapman–Kolmogorov relation for continuous Markov chain (Karlin and Taylor (1975)), we 

have 

 

( ) ( ) ( ) , 0t s t s t s  P P P                                                                (14) 

 

That is, if P(t) and P(s), transition probability matrices at times t and s are Markovs, then the product of them P(t+s) 

is also Markov. Let t=1, then P(1) is a one-step transition probability matrix which is equivalent to (2). Hence, our 
discrete Markov matrix P is equivalent to its continuous analog P(1). So  

 

(1)P P              

          (15) 

 

If we show that  

 

( )t tP P                                                                                                              (16) 



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Then according to (14) 

 

t s t s P P P            

         (17) 

We can conclude that if P is Markov then Pt, Ps and Pt+s are also Markovs for 0, st  and the theorem is 

proved.  

Rewrite P(t) in (6) as (18).  

 

1( ) ( )t t P XΛ X            

        (18) 

Where ,i i S  are the eigenvalues of A defined by (5), and  

 

 1( ) exp( ),...,exp(t t

Nt diag  Λ               

(19) 

 

And X  is the corresponding eigenmatrix of A. Take the natural logarithm of (18), 

 

1( ) ( )ln t t P XΦ X                                                      

(20) 

Where, 

 1( ) ,..., Nt t diag                                                                                    (21) 

So, 

1( )ln t t P XψX                                                                                             (22) 

where 

 1 ,..., Ndiag  ψ                                                                                             (23) 

 

Write (22) for t=1 and multiply both side by t,  

 



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1(1)t ln t P XψX                     

        (24) 

 

By comparison of (22) and (24) conclude that  

 

( ) (1)ln t t lnP P                                    

(25) 

or, 

( ) (1)tt P P                  

        (26) 

 

Given (15), equation (26) is the same as (16) Q.E.D. 

 

Result 1. Matrix Pt fulfills definition 1. Thus, 
t P Q . This comes from the following remarks. 

  

Remark 3. Sum of each row of Pt is equal to one. Since Pt satisfies Markovian properties (theorem 1).  

 

Remark 4. Sum of imaginary parts of each row is equal to zero. This immediately comes from remark 3. 

 

Remark 5. If qij denotes the ijth element of Pt for 0,t   then 1ijq  for all i and j belonging to S. This remark can 

be concluded from Theorem 1.  

 

Remark 6. If , 0t t Q P  equals to the complex matrix defined by (13), then 1 , .jkV j k S   Since, 

 

2 2

2 2

1 1

1 1 .

jk jk jk jk jk

jk jk jk

q u iv u iv

u v v

     

   
 

 

Remark 7. Given Q as in remark 6, then ujk[0,1]. This also comes immediately from Theorem 1.  

 

 



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3. Discussion on broken times 

 The broken time discrete Markov chain is not always a complex probability matrix defined by definition 1. 

Matrix P
t has different properties with respect to t and eigenvalues. i  may be real (positive or negative) or 

complex depending on the characteristic polynomial of P. 

 Since P is a non–negative matrix, Frobenius theorem (Takayama (1974), Nikaido (1970)) assures that P has 

a positive dominant eigenvalue  

 

01      (Frobenius root)                                                                      (27) 

and 

 1 2, ...,i i N                         

(28) 

 

Furthermore, if P is also a Markov matrix then its Frobenius root is equal to one, (Bellman (1970), Takayama 

(1974)). Therefore,  

 

1 1              

       (29) 

1i i S               

      (30) 

 

With the above information, consider the following discussions.  

 

) (0,1]ia i S                                                                  

 

 In this case all 0t

i  for 0t  and no imaginary part occurs in matrix P
t. i  are all positive for i 

belonging to S if we can decompose the matrix P to two positive semi-definite and positive definite matrices B and 

C of the same size (Mardia, Kent, Bibby (1982)) as 

 

1P C B  

 ) 1,1 , 0,i ib i S       

 



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0, tt

i  belongs to sets of real and imaginary numbers based on the value of t. In this case, Pt belongs to the 

class of generalized stochastic matrix Q of definition 1. For i R  , it is sufficient that P be positive definite.  

 

) , (0,1]i ic C i S      

 

P
t in this case for 0t  and t N  belongs to the class of generalized Morkov matrices of definition 1. 

 

Ntd ) (Natural numbers) 

 

In all cases of a, b, and c we never coincide with complex probabilities. Since P
t can be driven by simply 

multiplying P, t times.  

 

)e t Z (integer numbers) 

 

In this case, Pt is a real matrix but does not always satisfy condition 2 of definition 1.  

)f t R   

 

P
t is a complex matrix but does always satisfy conditions 2 and 3 of definition 1.  

 

4. Complex probability justification  

 Interpretation of the "Complex probability" as defined by definition 1 is not very simple and needs more 

elaborations. The interesting problem is that it exists in operational works of statistics, as example 1 discussed. 

Many similar examples like the cited may be gathered.  

 With this definition of probability, the moments of a real random variable are complex. Although the t–step 

distribution t  of initial distribution 0  with respect to Pt may be complex, they have the same total as .0  That 

is, if  

0 01 0( ,..., )N π           

       (32) 

Then,  

t

t o o o oi   π π P π Q π U π V         



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      (33) 

 

And we have the following remark accordingly,  

 

Remark 8. Sum of t-step distribution is equal to sum of initial distribution. That is,  

 

1 1

N N

oj tj
j j

 
 

             

       (34) 

 

This can be derived based on (32) and (33) as  

 

1 1
1 1 1 1

( ,..., ) ( ,..., ) ( ,..., )
N N N N

t tN oj j oj jn oj ji oj jN
j j j j

U U i v v     
   

                   (35) 

 

And, the sum of t–step distribution is  

 

1 1
1 1 1

( ... ) ( ,..., )
N N N

tj oj j jN oj j jn
j j j

u u i v v  
  

                   (36) 

 

The two parentheses in (36) are one and zero, respectively based on conditions 4 and 5 of definition 1. Thus, (36) 

and (34) are the same.  

 The above remark 8 states that though there exists imaginary transition probabilities to move from state j to 

k, the total sum of “imaginary transitions” is equal to zero. On the other hand, after the tth step transition, the total 

distribution has no imaginary part.  

5. Summary  

By summarizing the discrete and continuous times Markov stochastic processes, a class of real-world problems was 

introduced which cannot be solved by each of the procedures. The solutions of these problems coincide with 

“Complex probabilities” of transitions that are inherent in the mathematical formulation of the model. Complex 

probability is defined, and some of its properties with respect to the cited class are examined. Justification of the 

idea of complex probability needs more work that is left for further research.  

6. Acknowledgments 

The authors are indebted to Dr. A. Monajemi, who read the manuscript and gave valuable remarks. 

 



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References 

R. Bellman (1970), Introduction to matrix analysis, McGraw–Hill.  

P.J. Dhrymes (1978), Mathematics for econometrics. Springer-Verlag.  

W. Feller (1970, 1971), An Introduction to probability theory and its applications, Vols. 1,2, Wiley, New York.  

P.G. Hoel, S.C. Port, C. J. Stone (1972), Introduction to stochastic processes. Houghton Mifflin, New York.  

S. Karlin, H.M.Taylor (1975), A first course in stochastic processes. Academic Press.  

E. Klein (1973), Mathematical methods in theoretical economics, topological and vector space foundations of 

equilibrium analysis, Academic Press.  

K.V. Mardia, J.T. Kent, J.M. Bibby (1982), Multivariate analysis, Academic Press. 

H. Nikaido (1970), Introduction to sets and mapping in modern economics. North Holland Publishing Co.  

S.S. Searle (1982), Matrix algebra useful for statistics. Wiley.  

A.Takayama (1974) Mathematical economics. Dryden Press, Illinois.  

E. Wentzel, L. Ovcharov (1986) Applied problems in probability theory. Mir, Moscow.  

 

 

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