The d X ( t ) = X b ( X ) d t + X σ ( X ) d W equation and financial mathematics. I

Josef Štěpán; Petr Dostál

Kybernetika (2003)

  • Volume: 39, Issue: 6, page [653]-680
  • ISSN: 0023-5954

Abstract

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The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients b and σ being generally C ( ℝ + ) -progressive processes. Any weak solution X is called a ( b , σ ) -stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution μ σ of X in C ( ℝ + ) in the special case of a diffusion volatility σ ( X , t ) = σ ˜ ( X ( t ) ) . A martingale option pricing method is presented.

How to cite

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Štěpán, Josef, and Dostál, Petr. "The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. I." Kybernetika 39.6 (2003): [653]-680. <http://eudml.org/doc/33673>.

@article{Štěpán2003,
abstract = {The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients $b$ and $\sigma $ being generally $C(\{\mathbb \{R\}\}^+)$-progressive processes. Any weak solution $X$ is called a $(b,\sigma )$-stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution $\mu _\sigma $ of $X$ in $C(\{\mathbb \{R\}\}^+)$ in the special case of a diffusion volatility $\sigma (X,t)=\tilde\{\sigma \}(X(t)).$ A martingale option pricing method is presented.},
author = {Štěpán, Josef, Dostál, Petr},
journal = {Kybernetika},
keywords = {weak solution and uniqueness in law in the SDE-theory; $(b,\sigma )$-stock price; its Girsanov and DDS-reduction; investment process; option pricing; weak solution; uniqueness; SDE-theory; -stock price; investment process; option pricing},
language = {eng},
number = {6},
pages = {[653]-680},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. I},
url = {http://eudml.org/doc/33673},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Štěpán, Josef
AU - Dostál, Petr
TI - The $dX(t)=Xb(X)dt+X\sigma (X)dW$ equation and financial mathematics. I
JO - Kybernetika
PY - 2003
PB - Institute of Information Theory and Automation AS CR
VL - 39
IS - 6
SP - [653]
EP - 680
AB - The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients $b$ and $\sigma $ being generally $C({\mathbb {R}}^+)$-progressive processes. Any weak solution $X$ is called a $(b,\sigma )$-stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution $\mu _\sigma $ of $X$ in $C({\mathbb {R}}^+)$ in the special case of a diffusion volatility $\sigma (X,t)=\tilde{\sigma }(X(t)).$ A martingale option pricing method is presented.
LA - eng
KW - weak solution and uniqueness in law in the SDE-theory; $(b,\sigma )$-stock price; its Girsanov and DDS-reduction; investment process; option pricing; weak solution; uniqueness; SDE-theory; -stock price; investment process; option pricing
UR - http://eudml.org/doc/33673
ER -

References

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