Show that the solution \(X_t\) of
\[ dX_t=X_t^2 dt + X_t dB_t\]
where \(X_0=1\) and \(B_t\) is a standard Brownian motion has the representation
\[ X_t = \exp\Big( \int_0^t X_s ds -\frac12 t + B_t\Big)\]
Show that the solution \(X_t\) of
\[ dX_t=X_t^2 dt + X_t dB_t\]
where \(X_0=1\) and \(B_t\) is a standard Brownian motion has the representation
\[ X_t = \exp\Big( \int_0^t X_s ds -\frac12 t + B_t\Big)\]
Comments Off on SDE Example: quadratic geometric BM
Posted in Ito Formula, SDE examples
Tagged JCM_math545_HW4_S23
Define
\[ X_t =X_0 + \int_0^t B_s dB_s\]
where \(B_t\) is a standard Brownian Motion. Show that \(X_t\) can also be written
\[ X_t=X_0 + \frac12 (B^2_t -t)\]
Comments Off on Practice with Ito and Integration by parts
Posted in Ito Formula, SDE examples
Tagged JCM_math545_HW3_S23
Consider \(I(t)\) defined by \[I(t)=\int_0^t \sigma(s,\omega)dB(s,\omega)\] where \(\sigma\) is adapted and \(|\sigma(t,\omega)| \leq K\) for all \(t\) with probability one. Inspired by problem “Homogeneous Martingales and Hermite Polynomials” Let us set
\begin{align*}Y(t,\omega)=I(t)^4 – 6 I(t)^2\langle I \rangle(t) + 3 \langle I \rangle(t)^2 \ .\end{align*}
Comments Off on BDG Inequality
Posted in Ito Formula, Ito Integrals, Martingales
For functions \(f(x)\) and\( g(x) \) and constant \(\beta>0\), define \(X_t\) as the solution to the following SDE
\[dX_t = – \beta X_t dt + h(X_t)dt + g(X_t) dW_t\]
where \(W_t\) is a standard Brownian Motion.
Comments Off on Ito Variation of Constants
Posted in Ito Formula, SDE examples, Stochastic Calculus
For \(\alpha \in \mathbf R\) and \(\beta >0\), Define \(X_t\) as the solution to the following SDE
\[dX_t = – \beta X_t dt + \alpha dW_t\]
where \(W_t\) is a standard Brownian Motion.
Comments Off on Ornstein–Uhlenbeck process
Posted in Ito Formula, SDE examples
Recall that if \(u(t)\) and \(v(t)\) are deterministic functions which are once differentiable then the classic integration by parts formula states that
\[ \int_0^t u(s) (\frac{dv}{ds})(s)\,ds = u(t)v(t) – u(0)v(0) – \int_0^t v(s) (\frac{du}{ds})(s)\,ds\]
As is suggested by the formal relations
\[ (\frac{dv}{ds})(s)\,ds=dv(s) \qquad\text{and}\qquad (\frac{du}{ds})(s)\, ds=du(s)\]
this can be rearranged to state
\[ u(t)v(t)- u(0)v(0)= \int_0^t u(s) dv(s) + \int_0^t v(s) du(s)\]
which holds for more general Riemann–Stieltjes integrals. Now consider two Ito processes \(X_t\) and \(Y_t\) given by
\[dX_t=b_s ds + \sigma_s dW_t \qquad\text{and}\qquad dY_t=f_s ds + g_s dW_t \]
where \(W_t\) is a standard Brownian Motion. Derive the “Integration by Parts formula” for Ito calculus by applying Ito’s formula to \(X_tY_t\). Compare this the the classical formula given above.
Comments Off on Ito Integration by parts
Posted in Ito Formula