Deterministic interpretation of stochastic differential equation

Question

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$\delta S=\mu\ S\ \delta t\ +\sigma \ S\ \phi \ (\delta t)^{1/2}$$ where $\phi$ is drawn from a standardized normal distribution.

This reasoning which seems to follow naturally from the definition of a Wiener process triggered some thoughts and questions I cannot solve. Think of the following general diffusion process: $$dS=a(t,S(t))\ dt\ +b(t,S(t))\ dX$$ Now transform the second term in a similar fashion as above and drop the stochastic component:$$dS=a(t,S(t))\ dt\ +b(t,S(t))\ (dt)^{1/2}$$

NB: The last term is no Riemann-Stieltjes integral (that would e.g. be $d(t)^{1/2}$)

My questions
(1) How would you interpret the last formula and how can the (now deterministic) differential equation be solved analytically? Will you get an additional term with a second derivative like in Ito's lemma?
(2) Is the last term a fractional integral of order 1/2 (which is a Semi-Integral)? Or is this a completely different concept?
(3) Will there be a different result in the construction of the limit like with Ito integrals (from left endpoint) and Stratonovich integrals (average of left and right endpoint)?

Note: This is a cross-posting from mathoverflow where I got no answer to these questions.

Answer 1

(1) You analytically solve a stochastic differential equation (SDE) using Ito's lemma. Your second equation (the discretized one) is how you could model one path over one step. To find the solution, you would model many of these paths over many steps and then take the expectation (i.e., Monte Carlo methods). The solution to the SDE models all of these paths simultaneously in expectation. You can't switch directly to discretized version and solve without some numerical technique like Monte Carlo. The differential notation is really just short hand for the more formal way to write the Ito process. For example: $$S_t = x + \int_0^t \mu_s S_s ds + \int_0^t \sigma_s S_s dW_s \Leftrightarrow dS_t = \mu_t S_t dt + \sigma_t S_t dW_t$$

Then use the expectation operator to find the expected stock price $S$ at time $t$ given the parameters and original stock price $x$. Do you have a specific problem? Maybe someone here could help you along. The stuff's pretty tricky, but notes that Steve Shreve later built into a textbook series are still available for free. His textbooks are pretty approachable if you'd like to learn more.

(2) Different concept. Ito calculus is a bit different from the calculus we learn in high school and undergrad. The power of $1/2$ is in your second equation because of the variance-standard deviation conversion. I think the wikipedia and Shreve links above are the best place to start.

(3) I am not familiar with Stratonovich integrals.

Answer 2

(1) You can easily solve it in the case of constant coefficients. The answer will be $\infty$.

In fact, this equation has no solution on any interval. The intuition is the following. For the SDE like $$ dS_t = \mu(t,S_t)dt+ \sigma(t,S_t)dw_t $$ you can mention that $dw_t = \xi_t\sqrt{t}$, where $\xi_t\sim\mathcal{N}(0,1)$ are standard gaussian i.i.d. random variables.

When you do this "infinite summation" like for the integral, it will be like $$ \int\limits_0^T \sigma(t,S_t)\xi_t\sqrt{dt}. $$ Due to the fact that expectation of $\xi$ is zero and thanks to the Law of Large Numbers, this integral makes sense. Because the integrand is so often positive and so often negative on any small interval.

On the other hand in your equation you reject the term $\xi_t$ - and now on some small intervals your integrand will be positive or negative. Then this integral will diverge.

You can also consider the following motivation. When you discretize the time and simulate $$ \Delta S_t = \mu(t,S_t)\Delta t +\sigma(t,S_t)\xi_t\sqrt{\Delta t} $$ then for any $\Delta t$ you will have nice behaviour of the trajectories. On the other hand try to simulate $$ \Delta S_t = \mu(t,S_t)\Delta t +\sigma(t,S_t)\sqrt{\Delta t} $$ and decrease the time step. You will see immediately that even on the segment $[0,1]$ your trajectory will diverge for sufficiently small time steps.

(2) Surely, these are different notions.

(3) - if you talk about your deterministic equation - then there will be no difference, integrals will blow up.

I hope that it was useful - otherwise please comment and ask.

Answer 3

1) This last DE is implicit equation. This can't be solved analytically. I guess you can solve it by finite difference method.

2) The last term is indeed a differential term of order 1/2. However, it is the term for time difference and it can remain in the equation as it is. In the final formula as well it will come out to be as difference term, implying that we use difference of time when substituting in the formula to compute stock price, or option price for that matter.

3) I am not sure about this part.