Girsanow's theorem

In probability theory , Girsanow's theorem is used to change stochastic processes . This is done using a Maßwechsels of the canonical measure P to the equivalent martingale measure Q . This phrase has a special meaning in financial mathematics , as under the equivalent martingale the discounted price of an underlying instrument, such as a share , Martingale are. In the area of stochastic processes, the change of measure is important because the following statement can then be made: If Q is an absolutely continuous probability measure with regard to P , then every P -semimartingale is a Q -semimartingale.

history

The theorem was first proven in 1945 by Cameron and Martin and then in 1960 by Igor Vladimirovich Girsanov . The theorem was generalized by Lenglart in 1977.

sentence

Let be a probability space provided with the natural filtration of the standardized Wiener process . Be an adapted process , so that P- is almost certain and the process is defined by ${\ displaystyle \ {\ Omega, {\ mathcal {F}}, P, \ {{\ mathcal {F}} _ {t} \} _ {0 \ leq t \ leq T} \}}$ ${\ displaystyle ({B} _ {t}) _ {0 \ leq t \ leq T}}$ ${\ displaystyle (\ theta _ {t}) _ {0 \ leq t \ leq T}}$ ${\ displaystyle \ int _ {0} ^ {T} \ theta _ {s} ^ {2} \, \ mathrm {d} s <\ infty}$ ${\ displaystyle (L_ {t}) _ {0 \ leq t \ leq T}}$

{\ displaystyle L_ {t} = \ exp \ left (\ int _ {0} ^ {t} \ theta _ {s} \, \ mathrm {d} B_ {s} - {\ frac {1} {2} } \ int _ {0} ^ {t} \ theta _ {s} ^ {2} \, \ mathrm {d} s \ right)}

be a martingale .

Then it holds under the probability measure with the density with respect to that the process is defined by a standardized Wiener process . ${\ displaystyle P ^ {(L)}}$ ${\ displaystyle L_ {T}}$ ${\ displaystyle P}$ ${\ displaystyle (W_ {t}) _ {0 \ leq t \ leq T}}$ ${\ displaystyle W_ {t} = B_ {t} - \ int _ {0} ^ {t} \ theta _ {s} \, \ mathrm {d} s}$

Remarks

The process is the stochastic exponential of the process with , that is, it solves the stochastic differential equation , . He is always a non-negative local martingale , so also a super martingale . Generally, the hardest part in applying the above sentence is the premise that there is actually a martingale. A sufficient condition for a martingale to be is: ${\ displaystyle L_ {t}}$ ${\ displaystyle (X_ {t})}$ ${\ displaystyle \ mathrm {d} X_ {t} = \ theta _ {t} \, \ mathrm {d} B_ {t}}$ ${\ displaystyle \ mathrm {d} L_ {t} = \ theta _ {t} L_ {t} \, \ mathrm {d} B_ {t}}$ ${\ displaystyle L_ {0} = 1}$ ${\ displaystyle L_ {t}}$ ${\ displaystyle (L_ {t}) _ {0 \ leq t \ leq T}}$

{\ displaystyle \ operatorname {E} \ left (\ exp \ left ({\ frac {1} {2}} \ int _ {0} ^ {T} \ theta _ {t} ^ {2} \, \ mathrm {d} t \ right) \ right) <\ infty \ ,.}

This condition is also called the Novikov condition .

swell

C. Dellacherie, P.-A. Meyer: Probabilités et potentiel - Théorie des Martingales. Chapter VII, Hermann, 1980.
Damien Lamberton, Bernard Lapeyre: Introduction to Stochastic Calculus Applied to Finance. Chapter IV, p. 66, Chapman & Hall, 2000, ISBN 0-412-71800-6 .

credentials

^ AI Yashin: An Extension of the Cameron-Martin Result , Journal of Applied Probability (1993), Volume 30, Number 1, Pages 247-251
^ Rose-Anna Dana, Monique Jeanblanc: Financial Market in Continuous Time. Springer, Berlin 2003, ISBN 978-3-540-43403-0 ( limited preview in the Google book search).

Web links

Notes on Stochastic Calculus with a condensed proof. (PDF file; 488 kB)

[1] AI Yashin: An Extension of the Cameron-Martin Result , Journal of Applied Probability (1993), Volume 30, Number 1, Pages 247-251

[2] Rose-Anna Dana, Monique Jeanblanc: Financial Market in Continuous Time. Springer, Berlin 2003, ISBN 978-3-540-43403-0 ( limited preview in the Google book search).