Proving the discounted stock price is martingale

Let $$\mathcal{K}_s$$ be $$\mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $$\tilde{V}_t(\theta)$$ is the discounted value process of the self financing strategy $$\theta$$, $$\tilde{V}_t(\theta)=x+\sum_{j=1}^k\sum_{i=0}^{m-1} a^j_{t_i}(\tilde{S}^j_{t_{i+1}\wedge t}-\tilde{S}^j_{t_i\wedge t}).$$

Now let $$\mathcal{U}$$ be $$\mathcal{U}=\{f-h:f\in\mathcal{K}_s,\;h\in L_+^\infty\},$$ where $$L_+^\infty(P) = \{ Z\in L^\infty(P):P(Z\geq0)=1\}.$$

Suppose there exists $$g \in \mathcal L^q$$ with $$P(g>0)=1$$ such that $$\int fgdP\leq0\hspace{1cm}\forall f\in\mathcal{U}.$$

I want to show that this implies that the discounted stock prices $$(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$$ are $$Q$$-martingales where $$dQ=gdP$$.

My attempt: I know that if $$\int fg dP=0$$ for all $$f$$ in $$\mathcal K_s$$ then $$(\tilde S_t^1,\tilde S_t^2,....,\tilde S_t^k)$$ are $$Q$$-martingales because $$1_A (\tilde S_t^i-\tilde S_s^i) \in \mathcal {K}_s$$ for $$A\in \mathcal G_s$$ and $$0 \leq s \leq t$$. Can anyone please help me with the above case.

• Is this a question for a Ph.D. type of problem in stochastic analysis, or is this a question to understand why the discounted tradable stock price should be a martingale? Basically I am saying that if you are just starting out in finance the book you're reading may not be the best introduction. On the other hand if you're already an expert in stochastic analysis I'm sure a forum member can help you with the technicalities. Aug 19 '21 at 12:43
• @FridoRolloos I have had courses in stochastic calculus and some introductory finance before and this question is from the proof of a theorem
– abc
Aug 19 '21 at 12:50
• Ok, I hope somebody here can help you further with the fundamental theorem proof, perhaps the same person that gave you an answer already here: math.stackexchange.com/questions/4223935/… Aug 19 '21 at 12:58

Simple steps:

1. Take a stochastic process for $$S(t)$$ and the money-savings account $$M(t)$$. Both under the risk-free measure Q.
2. Apply Ito's lemma to find the dynamics of $$\frac{S(t)}{M(t)}$$.
3. You will find that the dynamics $$d\left(\frac{S(t)}{M(t)}\right)$$ has no drift thus discounted stock process $$\frac{S(t)}{M(t)}$$ is a martingale.

I skipped here all the regularity conditions but that should be enough for you to follow that path. Good luck.

• Welcome @Lech, I edited your post using LaTex, hope you don't mind. Aug 20 '21 at 12:53
• @Lech DO u mean risk neutral measure in step 1?
– abc
Aug 20 '21 at 14:48
• @abc, Yes. This is what I mean. Please take a look at Lecture no 3 from Computational Finance course where I discuss these properties (1h03m). Good luck.
– Lech
Aug 22 '21 at 14:55
• @Lech Ok I will go through it.But as far as I understand in this proof we are proving the existence of a risk neutral measure so how can we assume it in step 1?
– abc
Aug 22 '21 at 17:32
• hi @abs, for discussion on the existence of such measures you need to look at work of J.M. Harrison and D.M. Kreps.
– Lech
Aug 23 '21 at 8:38