Using Girsanov’s Theorem to Change a Stochastic Process’s Drift
Summary
The document explores whether Girsanov’s theorem can change the drift of a process defined using Brownian motion, and asks how to identify the corresponding change of measure. The author substitutes a drift shift into the process, attempts to choose the shift that removes the drift, and raises questions about the resulting process and the martingale condition for the Radon–Nikodym density. It also asks whether the Novikov expectation is taken under the original or changed measure.
The text is an attempted derivation and a set of questions, not a verified solution. In particular, the stated shift appears to confuse a time-dependent Brownian value with its terminal value, and expressions involving a stochastic differential should be checked carefully. Any application requires the integrand and stopping assumptions to be well-defined and the density process to satisfy suitable conditions. The document gives no proof that its proposed measure exists.
Key ideas
- Girsanov’s theorem can be used to study changes in drift under a change of probability measure.
- A candidate Brownian drift shift must be substituted consistently into the stochastic differential.
- The Radon–Nikodym density must satisfy conditions that make it a martingale.
- The document questions the measure under which the Novikov condition is evaluated but gives no resolution.
- The proposed derivation is uncertain and should not be treated as an established result.
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Full text
# Change of measure to get a determined drift
# Change of measure to get a determined drift
let's say I have a real stochastic process $dX_t=dt+\frac{1}{B_t}dB_t$ on $[0,T]$, with $B_t$ Brownian in $\mathbb{P}$ (not centered in 0) in $[0,\tau]$ with $\tau$ some adequate stopping time that makes everything good and smooth.
My exercise asks to find another prob. measure $\mathbb{Q}$ such that under this measure $X_t$ has a certain drift $\mu$, let's say $\mu=0$. Now the thing here is trying to apply Girsanov theorem. If that probability exists than we would have a shift of the Brownian of the form $dBt=dW_t+Y_tdt$.
Replacing $dB_t$ in the previous expression gives $dX_t=(1+\frac{Y_t}{B_t})dt+\frac{1}{B_t}dW_t \Rightarrow$ Imposing $drift=0$ we get $Y_t=-B_T$.
So with the substitution $dW_t=dB_t+B_tdt$ our process becomes driftless.
So we know that if exists, this probability is has the form of the R_N derivative $\dfrac{d\mathbb{Q}}{d\mathbb{P}}=Z_t$, where $Z_t$ is the Dooleans-Dade exponential and that condition for that to happen according to Girsanov Theorem is that $Z_t$ is a martingale. For example, if Novikov conditions holds, i.e. if $E[e^{1/2(\int_0^tB_t^2ds)}]< \infty$, then it is a martingale and everything works.
Now my question is: what i just did makes some sense? I mean, when I impose the Brownian sustitution, than my $X_t$ becomes $dX_t=\frac{dW_t}{dB_t}$. This expression does make sense? $B_t$ was a brownian with respect to $\mathbb{P}$, now under the new probability measure i'm not sure it is even well defined as a stochastic process. I thought that being $\mathbb{Q}<<\mathbb{P}$ maybe this is not an issue, but i'm not sure. The alternative would be to replace $W_t$ also in $B_t$ but that would make the computations for $Y_t$ impossibles.
My second question is regarding the Novikov condition: $E[e^{1/2(\int_0^tB_t^2ds)}]< \infty$, this expectation should be take under $P$ or under $Q$?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.