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I am quite new with Stochastic Calculus, and I was wondering if it is possible to (formally, i.e. mathematically rigorously) if it's possible to get the differential of the stochastic process that I will define below. So I have the initial assumptions of a probability space $(\Omega, \mathbb{P}, \mathcal{F} )$ together with a filtration $\left\{ \mathcal{F}_{t} \right\}$ and a Brownian motion $W$:

1 - We have a time interval $[0,T]$ and discrete set of time points that belong to this interval: $0=t_{0} < t_{1} < \cdots <t_{n}$.

2 - For these discrete time points we have random variables $X_{t_{1}}, \ldots X_{t_{n}}$. At this point, I assume that the r.v. $X$ have a common distribution (for example, normal with same mean and variance).

3 - We have a stochastic proces $r(t)$ whose s.d.e. is given by $dr(t)=\mu(t)dt + \sigma(t) dW(t)$.

Having these, I define the process:
$$ C(t,\omega) = \sum_{t:t_{i}\leq t} X_{t_{i}}(\omega) e^{\int_{t_{i}}^{ \min \left\{t_{i+1},t \right\} } r(s)ds} $$

Is it possible to (formally) get the differential of this jump-process? Basically, what is $dC(t)$?
Thanks

CA-Math
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1 Answers1

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That should not be a big problem, as at each time it seems only one of the terms in the sum changes with time. So when we are at $t_i < t < t_{i+1}$ you will get $$ \mathrm dC(t) = X_{t_i}\cdot\mathrm d\mathrm{exp}\left(\int_{t_i}^t r(s)\mathrm ds\right) $$ which you can easily get from Ito's formula. You only need to work around $\mathrm dC(t_i)$, but your process does actually seem to be continuous, so I think nothing special will happen at those points.

Formally speaking, I think you need to assume that $X$ is a sequence of variables somehow adapted to the filtration you work with, and likely to assume its independence from $W$.

SBF
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  • Hi Ilya. Thanks for your comments. Yes, in between the times t_{i} the process is continous, and your formula applies there... but the challenging case (for me) is on these points where there is a discontinuity. How do you express the differential there? And how do you express it for all times? – CA-Math Sep 26 '19 at 18:14
  • @CA-Math do you think the process is discountinuous at some points of time? How would you write a differential of $C(t) = |t|$ in one go? – SBF Sep 27 '19 at 09:42
  • Hi Ilya. Thanks again for your reply. To answer your second comment first, I see that in your example $| t |$ is not differentiable at $0$, but given that I am new to SDE, and that in this context a SDE represents an integral, perhaps in this case there was somehow a way to write it? For your first question, yes, I think this process is discontinuous a the $t_{i}$. For example, take the random variable $X_{t_{0}}(\omega)=5$ then assume that the path of $r(s,\omega)$ is such that $C(t_{1}-)=7$. Then, at time $t_{1}$ the random variable $X_{t_{1}}(\omega)=10$. Then $C(t_{1})=17$. – CA-Math Sep 27 '19 at 10:05
  • @CA-Math doing it wrong. Please write explicitly how did you find that $C(t_1) = 17$ – SBF Sep 27 '19 at 14:39
  • Hi Ilya,once again, thanks for your reply. And yes, I'll explain my reasoning, given that you so kindly ask. My process $C$ above tries to mimic a bank account that accrues interest continuously. The r.v. $X$ captures the amount of money I deposit/withdraw at times $t_{i}$. At time $t_{0}=0$ I randomly determine I'll deposit 5 in the account. I let the money sit in the account for one year (time $t_{1}$). Assume that $r$ is constant in this year, and such that: $5 exp(\int_{0}^{1}r(s)ds = 7$. So the account grew to 7. I now check my rv $X(t_{1})$ and it is 10. (continues below) – CA-Math Sep 29 '19 at 08:03
  • So the account now at time $t_{1}=1yr$ reads:

    $C(1) = X(t_{0}) e^{\int_{0}^{1}r(s) ds} + X(1) e^{\int_{1}^{1}r(s) ds} = 5 e^{\int_{0}^{1}r(s) ds} + 10 = 7 + 10 =17$

    This is my reasoning. Not sure if you spotted something I misinterpreted. If so, please kindly let me know. Cheers,

     – CA-Math Sep 29 '19 at 08:04
  • Ah, you're right - I've missed the exponent effect there, my bad. Well, the you only need to add those jumps, me or you need to look how this is formally written, e.g. in Shreve's stochastical finance 2. – SBF Oct 01 '19 at 09:23
  • Thanks. I'll check that book and see if I can find an answer. Otherwise I close the question. Cheers – CA-Math Oct 01 '19 at 19:09