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2
votes
2answers
118 views

What information about the stochastic process is available from path-dependent options?

Assume the stock follows a process, which is defined by the following stochastic differential equation $$\frac{dS}{S}=r(t)dt+\sigma(S,t)dW,$$ so that the stock price process has local volatility. ...
4
votes
2answers
158 views

Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
3
votes
2answers
139 views

Shortcomings of generalized Brownian motion for asset price modelling

I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$ as a model for asset prices. I know ...
0
votes
0answers
78 views

Is there a strong solution to $\frac{dS}{S}=\sigma(S)dw$?

Does someone know if there is a strong solution for this SDE : $$\frac{dS_t}{S_t}=\sigma(S_t)dW_t$$ where $$\sigma(S)=\begin{cases} 1\;\;\;S>1\\2\;\;\;S\leq 1 \end{cases} $$ $S_0=1$ and $W_t$ is ...
1
vote
1answer
101 views

Bracket-Notation in SDEs

I often come across the following notation in my script, and I have not found it anywhere else. While our lecturer insists it is of utmost importance to write this way in his exams, he yet failed to ...
3
votes
2answers
157 views

Geometric Brownian Motion with non-negative random increments

I am attempting to model a cumulative time-series of a positive integer variable across independent entities. The cumulative series appears to follow a process of Geometric Brownian Motion (GBM) based ...