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Two dimensional probabilities with a given conditional structure

Josef Štěpán;  Daniel Hlubinka — 1999

Kybernetika

A properly measurable set 𝒫 ⊂ X × M 1 ( Y ) (where X , Y are Polish spaces and M 1 ( Y ) is the space of Borel probability measures on Y ) is considered. Given a probability distribution λ ∈ M 1 ( X ) the paper treats the problem of the existence of X × Y -valued random vector ( ξ , η ) for which ℒ ( ξ ) = λ and ℒ ( η | ξ = x ) ∈ 𝒫 x λ -almost surely that possesses moreover some other properties such as “ ℒ ( ξ , η ) has the maximal possible support” or “ ℒ ( η | ξ = x ) ’s are extremal measures in 𝒫 x ’s”. The paper continues the research started in [7].

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 ) = 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 ≥ 0 , where Y is the exponential...

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

Josef Štěpán;  Petr Dostál — 2003

Kybernetika

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.

Kermack-McKendrick epidemic model revisited

Josef Štěpán;  Daniel Hlubinka — 2007

Kybernetika

This paper proposes a stochastic diffusion model for the spread of a susceptible-infective-removed Kermack–McKendric epidemic (M1) in a population which size is a martingale N t that solves the Engelbert–Schmidt stochastic differential equation (). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coefficients depend on the size N t . Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer...

Absorption in stochastic epidemics

Josef Štěpán;  Jakub Staněk — 2009

Kybernetika

A two dimensional stochastic differential equation is suggested as a stochastic model for the Kermack–McKendrick epidemics. Its strong (weak) existence and uniqueness and absorption properties are investigated. The examples presented in Section 5 are meant to illustrate possible different asymptotics of a solution to the equation.

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