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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...

The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations

Fabienne Castell, Jessica Gaines (1996)

Annales de l'I.H.P. Probabilités et statistiques

We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $d X_{t} = 1_{...}$

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Temps d'explosion d'équations différentielles stochastiques du type Doléans-Dade

The abstract Riemannian path space.

The approximate Euler method for Lévy driven stochastic differential equations

The Banach fixed point method for Ito stochastic differential equations

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

The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary

The noise made by a Poisson snake.

The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations

The Picard boundary value problem for a third order stochastic difference equation

The smallest g-supermartingale and reflected BSDE with single and double $L^{2}$ obstacles

The steepest descent method for forward-backward SDEs.

The stochastic logistic equation : stationary solutions and their stability

The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities.

Théorème du support pour les diffusions réfléchies de type Ventcell

Three examples of brownian flows on $ℝ$

Tightness results for laws of diffusion processes application to stochastic mechanics

Time reversal for drifted fractional Brownian motion with Hurst index $H > 1 / 2$ .

Time-dependent backward stochastic evolution equations.

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