# Under which conditions the given random process is martingale and under which submartingale?

Let $$a_t$$ be adapted to the filtration random process $$a_t: P\{\int _0^T|a_t|dt < \infty \} = 1$$ and $$b_t \in M_T^2. \quad$$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\int _0^tb_sdW_s\} \; t \in [0, T]\,$$ is martingale and under which submartingale ?
As I understand, this is a famous example of "exponential martingale" and the answer is:
The process will be martingale for $$a_s = -\frac {b_s^2}{ 2 }$$.
But I don’t understand how to prove it. And what conditions will be for submartingale?
My attempt to prove was:
Let's try to find conditions when $$E(\frac{X_t}{X_s}|\mathcal F_s)= 1$$ .

$$E(\frac{X_t}{X_s}|\mathcal F_s)=exp\{\int _s^ta_sds\} E(exp\{\int _s^tb_sdW_s\})$$
Also, I understand that $$\int _s^tb_sdW_s$$ has Gaussian distribution.
But I do not know what to do next. I would be grateful for any help.

• Ito integral is not Gaussian, in general. It is if, say, $b_s$ is deterministic. – ir7 Jun 14 '20 at 22:18

One can approach this using the Ito lemma. Let $$I_t=\int_0^t a_u du+\int_0^tb_udW_u, (\forall) t\in [0;T]$$. Then, by definition we have that: $$dI_t=a_t+b_tdW_t.$$ Using Ito lemma applied to $$f(I_t)$$, where $$f(x)=e^x$$, we get: $$dX_t=d\left(e^{I_t}\right)=\underbrace{e^{I_t}}_{X_t}dI_t + \frac{1}{2}e^{I_t}d\langle I \rangle_t,$$ where $$\langle I \rangle_t$$ is the quadratic variation of $$(I_t)_{t\geq 0}$$. This quadratic variation can be obtained using the rules of stochastic calculus: $$d\langle I \rangle_t =(b_t)^2 dt.$$ Therefore, $$dX_t=X_tdI_t+\frac{1}{2}X_t(b_t)^2dt=\left(a_t+\frac{b_t^2}{2}\right)dt+X_tb_tdW_t.$$ This is really just a shorthand notation for: $$X_t=X_0+\int_0^t \left(a_u+\frac{b_u^2}{2}\right)du+\int_0^t X_ub_udW_u.$$ But since the last term of the above formula is a stochastic integral (which is a martingale), we have that: $$\mathbb{E}\left[X_t\right]=\mathbb{E}\left[X_0\right]+\mathbb{E}\left[\int_0^t\left(a_u+\frac{b_u^2}{2}\right)du\right].$$ To ensure the martingality of $$(X_t)_{t\geq 0}$$, a necessary condition is: $$\mathbb{E}\left[\int_0^t\left(a_u+\frac{b_u^2}{2}\right)du\right] = 0.$$ This is somewhat different than what you have written above, as the integral $$\int_0^t\left(a_u+\frac{b_u^2}{2}\right)du$$ is a random variable. Your condition is sufficient, but not necessary.
Since the submartingale condition is $$\mathbb{E}\left[X_t|\mathcal{F}_s\right]\geq X_s, \text{for }s\leq t$$ (assuming the filtration is indeed $$\left(\mathcal{F}_t\right)_{t\geq 0}$$), then the sufficient condition for $$(X_t)_{t\geq 0}$$ to be a submartingale should be straightforward to see now.
• $\int_0^t X_ub_udW_u$ is a local martingale but not necessarily a martingale. To ensure that it's a martingale you need to show that $E\int_0^T |X_ub_u|^2du<\infty,$ which you haven't proved. – user39119 Jun 15 '20 at 11:18
• @UBM Do you have any ideas how to prove that $\int_0^T |X_ub_u|^2du<\infty$? – Helen Jun 15 '20 at 12:01
• @Helen I think we can't prove that condition with the information we are given. In my opinion this approach is wrong, it takes you to a dead end. A different approach would be to say that If $a_s=− \frac{b_s^2}{2}$, then the process 𝑋 is the stochastic exponential of the process $\{\int_0^t b_sdW_s; 0 \leq t \leq T\}$, so a sufficient condition for 𝑋 to be a martingale is the Novikov condition. – user39119 Jun 15 '20 at 12:42
• @UBM Something must surely escape me, for I do not see immediately why $\int_0^t X_ub_udW_u$. Is assured to be a local martingale. It would be great if you could point to a reference with a page and theorem/definition number (e.g. Protter). I'm far from being an expert in stochastic calculus and I'd like to gain a better understanding of this. How does $\mathbb{E}\left[\int_0^t|X_ub_u|^2du \right]<+\infty$ ensure martingality? Does the argument use Ito isometry? – fwd_T Jun 15 '20 at 17:41