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The problem is the following (Problem 3.10 from Karatzas/Shreve):

Let $Z_t = \exp \left(\int_0^t X_u \, dW_u-\frac{1}{2}\int_0^t X^2_u \, du \right)$. Define $Y_t=1/Z_t$. Show the following stochastic differential equation holds \begin{align*} dY_t=Y_tX_t^2\,dt-Y_tX_t\,dW_t, \quad Y_0=1 \end{align*} where we have assumed $\int_0^t X_u^2 \, du<\infty$ almost surely for all $0<t<\infty$, and $W_t$ is the standard Brownian motion.

My questions are:

  1. How can we prove $Y_t$ is a semi-martingale?
  2. For the differential equation, this is my try (which I assume $Y_t$ is a semi-martingale

\begin{align*} Y_tZ_t=1 \ \Longrightarrow Z_t\,dY_t + Y_t\,dZ_t + d\langle Y_t, Z_t \rangle = 0 \end{align*} Since $Y_tZ_t=1$, we have $\langle Y_t, Z_t\rangle = 0$. Also, from the textbook, it is shown $dZ_t= Z_tX_t \,dW_t$ So from the above we have \begin{align*} Y_tZ_t \,dY_t = -Y_t^2 \, dZ_t \ \Longleftrightarrow\ dY_t = -Y_t^2\,dZ_t = -Y_tX_t\,dW_t \end{align*} since $dZ_t = Z_tX_t\,dW_t$ and $Z_tY_t=1$. However, it seems the $dt$ term is missing.... I can't really see what is wrong in the calculation though. Does anyone have any comments?

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    $\begingroup$ Hint: Apply Ito's formula to $f(Z_t)$ where $f(x) = 1/x$. $\endgroup$ Aug 2, 2021 at 15:40
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    $\begingroup$ The problem with your solution is that $Y_t Z_t = 1$ does not imply that $\langle Y_t, Z_t \rangle = 0$ $\endgroup$ Aug 2, 2021 at 17:47

2 Answers 2

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The issue with your solution is that $Y_t Z_t = 1$ does not imply that $\langle Y_t, Z_t \rangle = 0$ for the covariation. See the answer by @ChristopherK for a computation of $\langle Y_t, Z_t \rangle$. Here's a way to solve your original problem.

Let $$R_t = \underbrace{\int_0^t X_u dW_u}_{(\star )} - \underbrace{\frac{1}{2}\int_0^t X_u^2 du}_{(\star \star)}$$ Note that with the assumption that $\int_0^t X_u^2 du < \infty$, the process $R_t$ is well-defined. Moreover, $R_t$ is a semi-martingale, because $(\star)$ is a local martingale and $(\star \star)$ is a process of bounded variation.

Let $g(x) = e^{-x}$ and observe that $Y_t = g(R_t)$. Itô's lemma implies that the class of semi-martingales is stable under the application of $C^2$ maps, which $g$ clearly is. That solves (1).

To solve (2), observe that $g_x(x) = - e^{-x}, g_{xx}(x) = e^x$. Further observe that the dynamics of $R_t$ can be written as $$dR_t = X_t dW_t - \frac{1}{2} X_t^2 dt$$ By Itô's lemma: $$\begin{align*} dY_t &= g_x(R_t)(dR_t) + \frac{1}{2} g_{xx}(R_t)(dR_t)^2 \\ &= - \exp (-R_t) \left( X_t dW_t - \frac{1}{2}X_t^2 dt\right) + \frac{1}{2} \exp(-R_t) X_t^2 dt \\ &=\exp (-R_t) X_t^2 dt - \exp (-R_t) X_t dW_t \\ &=Y_t X_t^2 dt - Y_tX_tdW_t \end{align*}$$

As required.

As an additional technical point, setting $Y_t = \frac{1}{Z_t}$ is a perfectly valid thing to do you because $P \left(\inf_{0 \leq s \leq t} Z_s > 0\right) = 1 $.

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A note on the calculation, based on the comment: $d[Y,Z]_{t} = -X_{t}^{2}\, dt$.

Letting $$U_{t} = \int_{0}^{t} X_{u}\, dW_{u} - \frac{1}{2}\int_{0}^{t} X_{u}^{2}\, du$$ and applying Itô's lemma for $f(u) = e^{u}$,

\begin{align*} dZ_{t} = d(f(U_{t})) &= f'(U_{t})\, dU_{t} + \frac{1}{2}f''(U_{t})\, d[U,U]_{t} \\ &= e^{U_{t}}\, (X_{t}\, dW_{t} - \frac{1}{2}X_{t}^{2}\, dt) + \frac{1}{2}e^{U_{t}}X_{t}^{2}d[W,W]_{t} \\ &= Z_{t}X_{t}\, dW_{t}, \end{align*} as claimed.

Doing same for $Y_{t} = Z_{t}^{-1} = e^{-U_{t}}$ and $g(u) = e^{-u}$,

\begin{align*} dY_{t} = d(g(U_{t})) &= g'(U_{t})\, dU_{t} + \frac{1}{2}g''(U_{t})\, d[U,U]_{t} \\ &= -e^{-U_{t}}\, (X_{t}\, dW_{t} - \frac{1}{2}X_{t}^{2}\, dt) + \frac{1}{2}e^{-U_{t}}X_{t}^{2}d[W,W]_{t} \\ &= -Y_{t}X_{t}\, dW_{t} + Y_{t}X_{t}^{2}\, dt. \end{align*}

Then you can calculate the quadratic variation

\begin{align*} d[Y,Z]_{t} = -Y_{t}Z_{t}X_{t}^{2} d[W,W]_{t} = -X_{t}^{2}\, dt. \end{align*}

It follows by the product rule for Itô calculus that

\begin{align*} d(Y_{t}Z_{t}) &= Y_{t}\, dZ_{t} + Z_{t}\, dY_{t} + d[Y,Z]_{t} \\ &= Y_{t}Z_{t}X_{t}\, dW_{t} + Z_{t}\, (-Y_{t}X_{t}\, dW_{t} + Y_{t}X_{t}^{2}\, dt) - X_{t}^{2}\, dt \\ &= 0, \end{align*} as indeed the derivative of a constant should be.

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