Results 311 to 320 of about 732,831 (345)

Stochastic Differential Equations

2016
This chapter is devoted to stochastic differential equations, which motivated Ito’s construction of stochastic integrals. After giving the general definitions, we provide a detailed treatment of the Lipschitz case, where strong existence and uniqueness statements hold.
Setsuo Taniguchi, Hiroyuki Matsumoto
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Stochastic Differential Equations

1985
We now return to the possible solutions X t (ω) of the stochastic differential equation (5.1) where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X t satisfies the stochastic integral equation or in differential form (5.2) .
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Stochastic Integrals and Differential Equations

2004
This chapter provides the tools needed for option pricing. The field of stochastic processes in continuous time, which are defined as solutions of stochastic differential equations, has an important role to play. To illustrate these notions we use repeated approximations by stochastic processes in discrete time and refer to the results from Chapter 4.
Wolfgang Karl Härdle, Wolfgang Karl Härdle, Christian M. Hafner, Jürgen Franke   +3 more
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Stochastic Differential Equations

1997
Starting with coefficients a(t, x) = ((a ij (t, x)))1≤i, j≤d and b(t, x) = (b i (t,x))1≤i≤d, we saw in Chapter 3 how the associated parabolic equation $$ \frac{{\partial u}} {{\partial t}} + L_t u = 0 $$ (0.1) can be a source of a transition probability function on which to base a continuous Markov process.
Srinivasa R. S. Varadhan, Daniel W. Stroock   +1 more
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Stochastic differential equations

2008
Elementary concepts of stochastic differential equations (SDE) and algorithms for their numerical solution are reviewed and illustrated by the physical problems of Brownian motion (ordinary SDE) and surface growth (partial SDE). Discretization schemes, systematic errors and instabilities are discussed.
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Stochastic Differential Equations

1984
This is a brief introduction to Langevin equations (stochastic differential equations (SDE) with white noise terms)[1–3], with particular emphasis on its use as a calculational tool. We also discuss recently developed (matrix) continued fraction methods for solving certain types of stochastic differential equations and their associated Fokker-Planck ...
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Stochastic Differential Equations

2006
Stochastic differential equations provide a powerful mathematical framework for the continuous time modeling of asset prices and general financial markets. We consider both scalar and vector stochastic differential equations which allow us to model feedback effects in the market. Explicit solutions will be given in certain cases. Furthermore, questions
Eckhard Platen, David Heath
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Stochastic Differential Equations

2017
In this chapter we establish the well-posedness and a priori estimates for SDEs. Weak solutions of SDEs will also be studied briefly.
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Stochastic Differential Equations

1987
In this paragraph we shall consider (real) random processes ξ(t), t ≥ t0, characterized by the stochastic differential $$ \begin{array}{*{20}{c}} {d\xi \left( t \right) = \infty \left( t \right)dt + \beta \left( t \right)d\eta \left( t \right),}\\ {\alpha \left( t \right) = a\left( {t\xi \left( t \right)} \right),\beta \left( t \right) = b\left( {t\
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Stochastic Differential Equations

2001
Virtually all continuous stochastic processes of importance in applications satisfy an equation of the form $$d{X_t} = \mu (t,{X_t})dt + \sigma (t,{X_t})d{B_t}with\;{X_0} = {x_0}$$ (9.1) .
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