Download POPULATION STUDIES-Concept, Scope and Coverage

Markov Process
Vidya Sagar
M. Sc (student of IVth semester)
Roll no. :- 5
Department of Statistics
CONTENTS




Introduction
Stochastic process
Martingale process
classification Martingale process
 Submartingale Process
 supermartingale Process
 Application of Martingale process
INTRODUCTION
• Martingale process is play vital role in field of
management and many others fields. The concept
of martingale in probability theory was
introduced by Paul Lévy in 1934, though he
did not name them: the term "martingale"
was introduced later by Ville (1939), who
also extended the definition to continuous
martingales. Much of the original
development of the theory was done
by Joseph Leo Doob among others. Part of
the motivation for that work was to show
the impossibility of successful betting
strategies.
stochastic process
A stochastic process is the
collection of random variable
{x(t),tϵT} defined on the given
probability space indexed by
parameter t where t varies from an
index of T.
Martingale process
In probability theory, a martingale is a
model of a fair game where knowledge
of past events never helps predict the
mean of the future winnings and only
the current event matters.
The expected position of a martingale
some time in the future is exactly where
it is now.
Martingale process
A stochastic process {X1, X2, X3…. } is a
martingale if
Xt = E[Xt+1 |Ft ]
where t≥0 and Ft = {X1, X2, X3…. Xt }
=>a martingale is a sequence of random
variables (i.e., a stochastic process) for
which, at a particular time in
the realized sequence, the expectation of
the next value in the sequence is equal to
the present observed value even given
knowledge of all prior observed values.
Example of Markov process
•
•
•
An unbiased random walk is an example of a
martingale.
A gambler's fortune (capital) is a martingale if
all the betting games which the gambler plays
are fair.
Suppose Xn is a gambler's fortune
after n tosses of a fair coin, where the gambler
wins $1 if the coin comes up heads and loses
$1 if it's tails. The gambler's conditional
expected fortune after the next trial, given the
history, is equal to his present fortune. This
sequence is thus a martingale.
Classification of Martingale process

Martingale process has been classified into two
type

submartingale

supermartingale

Every martingale is also a submartingale and a
supermartingale. Conversely, any stochastic
process that is both a submartingale and a
supermartingale is a martingale.
Submartingale process
A stochastic process {X1, X2, X3…. } is a
martingale if
E[Xt+1 |Ft ] ≥ Xt
where t≥0 and Ft = {X1, X2, X3…. Xt }

In this case of subrmartingale, the
gain is favourable for player.
Supermartingale process
A stochastic process {X1, X2, X3…. } is a
martingale if
E[Xt+1 |Ft ] ≤Xt
where t≥0 and Ft = {X1, X2, X3…. Xt }

In this case of supermartingale, the
gain is unfavourable for player.
Example of supermartingale and sub
martingale Process
•Consider again the gambler who wins $1 when a coin
comes up heads and loses $1 when the coin comes up tails.
Suppose now that the coin may be biased, so that it comes
up heads with probability p.
–If p is equal to 1/2, the gambler on average neither wins
nor loses money, and the gambler's fortune over time is a
martingale.
–If p is less than 1/2, the gambler loses money on
average, and the gambler's fortune over time is a
supermartingale.
–If p is greater than 1/2, the gambler wins money on
average, and the gambler's fortune over time is a
submartingale.
Application of Martingale process







Security price modelling
Portfolio insurance Claims
forecasting Asset-liability matching
Premium rating
Asset and Option pricing
Investment strategy
Control of large funds
Application of Martingale process








Equitable profit allocation
Mortality/Morbidity modelling
Macroeconomic growth modelling
Market product pricing
Risk theory
Solvency and Ruin calculations
Early warning systems
Consumption/Investment problems
Dynamic hedging strategies.
Thank you