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### how to calculate pdf and cdf for an Ornstein-Uhlenbeck process

I have the Task. For Ornstein-Uhlenbeck process generate a path and plot a) cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile. My solution. From the literature we ...
• 257
1 vote
125 views

• 552
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### Question on Ito's lemma involving $\mathrm{d}W(t)$

I am new to Ito-calculus, so please forgive me if the question is stupid. Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
• 471
238 views

### In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
• 43
1 vote
56 views

• 121
1 vote
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### Milstein Scheme for Jump-Diffusion models

Hey in this report (Approximation of Jump Diffusions in Finance and Economics by Bruti-Liberati and Platen) is described the Milstein formula (3.5) for simulation SDE with jump component. How it is ...
• 423
1 vote
145 views

### Simulation of Gamma process (distribution of increments)

The gamma process is a Levy process $X$, where $X_t$ has gamma distribution with parameters $at,b>0$ and density $$f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx}$$ I want to ...
• 423
2k views

### Heston stochastic volatility, Girsanov theorem

How can we apply Girsanov's theorem to a stochastic volatility model? In Heston's model the dynamics are given by \begin{align*} dS_t &= \mu S_t dt + \sqrt{v_t}S_t d\widehat{W}^\mathbb{P}_{1,t}, ...
• 141
75 views

### Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion

Now, I'm given an SDE $$dS_t = 2S_t\,dt + 4 S_t\,dW_t$$ which I need to find the solution of. I have the solution on paper, but I want to know if there's any way I can solve this directly in python. ...
• 101
1 vote