The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes.

Under this model, these assets have continuous prices evolving continuously in time and are driven by Brownian motion processes.[1] This model requires an assumption of perfectly divisible assets and a frictionless market (i.e. that no transaction costs occur either for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models.

Contents

1 Financial market processes

1.1 Definition

1.2 The augmented filtration

1.3 Bond

1.4 Stocks

1.5 Dividend rate

2 Portfolio and gain processes

2.1 Definition

2.2 Motivation

3 Income and wealth processes

3.1 Definition

4 Viable markets

4.1 Definition

4.2 Implications

5 Standard financial market

5.1 Definition

5.2 Comments

6 Complete financial markets

6.1 Definition

6.2 Motivation

6.3 Corollary

7 See also

8 Notes

9 References

Financial market processes[edit]

Consider a financial market consisting of

N

+

1

{\displaystyle N+1}

financial assets, where one of these assets, called a bond or money market, is risk free while the remaining

N

{\displaystyle N}

assets, called stocks, are risky.

Definition[edit]

A financial market is defined as

M

=

(

r

,

b

,

δ

,

σ

,

A

,

S

(

0

)

)

{\displaystyle {\mathcal {M}}=(r,\mathbf {b} ,\mathbf {\delta } ,\mathbf {\sigma } ,A,\mathbf {S} (0))}

:

A probability space

(

Ω

,

F

,

P

)

{\displaystyle (\Omega ,{\mathcal {F}},P)}

A time interval

[

0

,

T

]

{\displaystyle [0,T]}

A