# For which p>0 does $S_t=W_t+t^p$ admit an equivalent martingale measure?

Let W be a brownian motion and p>0.

For which p does $S_t=W_t+t^p$ admit an equivalent martingale measure?

I recently saw at my lectures that

NFLVR cond: There does not exist a sequence $\{H_n\}_{n \geq 1}$ of predictable processes, integrable wrt S, such that there exists $t_0 , b , \epsilon >0$ $$\int_0^{t_0} H_n(s) S_s > -1/n$$ and $$P(\int_0^{t_0} H_n(s) S_s >b)> \epsilon$$

Is equivalent to having such a measure and I think that is pretty much my only tool, so I guess it has to be that. On the other hand I can't get any ideas on how to use the condition.

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cameron-martin deals with $W_t + f(t)$ and their condition is $\int^t_0 | f^{\prime} |^2 < \infty$. – mike May 14 '13 at 15:48
Hi mike, thanks for the comment. I'm sure Cameron-Martin is not in our tool kit, but it's nice to know the "solution". – Henrik May 14 '13 at 16:29
o.k. let's put it this way: I think NFLVR is too much and a brutally direct effort to construct the likeihood ratios show they want to be $e^{\int f^{\prime}(t) dW_t - \frac 12 \int (f^{\prime})^2 (t) dt}$. Also, you can rule out small p by law of iterated logarithm – mike May 15 '13 at 11:44
You are right, I have actually solved it and will post my answer soon. It is much simpler than NFLVR. – Henrik May 15 '13 at 15:35

It can bee seen as a corollary to (at least the proof i have seen) of Girsanov that if we are looking for a equivalent martingale measure for a continuous semimartingale $S_t = M_t + A_t$ where $M_t$ is the local martingale part and $A_t$ is the bounded variation part, then $A_t$ must satisfy that there exists predictable $h_t$ such that $A_t=\int _0^t h_s d<M_s>$ and $\int_0^t h_s^2 d<M_s> < \infty$. ($<M>$ is quadratic variation). In our case this rules out $p\leq 1/2$ since $A_t=\int_0^t p\cdot s^{p-1} ds$ which is only square integrable if $p > 1/2$.
Assuming from now on $p>1/2$. We will see an Martingale measure does indeed exist. This is simply and application of Girsanov - e.g. by applying Novikovs condition which is here trivial - on $h_s = - p s^{p-1}$. This directly yields that $W_t - \int h_s d<W>_s=W_t + t^p$ is a brownian motion under $\mathbb{Q}$ (in particular a local martingale) and $\mathbb{P}$ and $\mathbb{Q}$ are equivalent.