 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Mathematics course of the University of Oxford

Year 4 of the mathematics course of the University of Oxford

Stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford

2007

{c}

The deterministic <stochastic integral> in the solution is centered Gaussian with variance
$$
v(t)=\int_0^ta(s)^{-2}\sigma(s)^2ds.
$$
Therefore $X(t,x)$ has mean $a(t)x$ and variance
$$
V(t)=a(t)^2v(t)
=\int_0^t e^{2\int_s^t\mu(r)dr}\sigma(s)^2ds.
$$
For $t>0$ this is strictly positive, since $\sigma>0$. Its density at $y$ is
$$
\boxed{\frac1{\sqrt{2\pi V(t)}}
\exp\left[-\frac{(y-a(t)x)^2}{2V(t)}\right].}
$$
The boundedness assumptions on $\sigma$ and its reciprocal in particular ensure nondegeneracy; compact-interval <continuity> is already sufficient for the displayed finite-time construction.

Solution

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 4 / b / Solution

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 4 / b / Solution

 Ancestors (13)

 Discussion (0) New discussion

 Articles by others on the same topic (0)

Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 4 / b / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion