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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 $σ (X, t) = \tilde{σ} (X (t)) .$ A martingale option pricing method is presented.

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

Josef Štěpán, Petr Dostál (2003)

Kybernetika

This paper continues the research started in [J. Štěpán and P. Dostál: The $d X (t) = X b (X) d t + X σ (X) d W$ equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price $X (t)$ born by the above semilinear SDE with $σ (x, t) = \tilde{σ} (x (t)),$ we suggest two methods how to compute the price of a general option $g (X (T))$ . The first, a more universal one, is based on a Monte Carlo procedure while the second one provides explicit formulas. We in this case need an information on the two dimensional distributions of $ℒ (Y (s), τ (s))$ for $s \geq 0,$ where $Y$ is the exponential...

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

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

The Donsker delta function of a Lévy process with application to chaos expansion of local time

The existence and exponential stability for random impulsive integrodifferential equations of neutral type.

The fBm Itô's formula through analytic continuation.

The fermion number processes as a functional central limit of quantum hamiltonian models

The Feynman-Kac formula for a system of parabolic equations

The gauge and conditional gauge theorem

Currently displaying 1 – 20 of 77