# Tagged Questions

39 views

### Deriving the PDE for basket option

The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE, where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$ $dS_2 = rS_2dt + \sigma_2 S_2dW_2$ I need some ...
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### Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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### Feynman Kac solution discontinuity at 0

In most exposition of the Feynman Kac formula $$\frac{\partial u}{\partial t}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) = 0$$ the condition of the ...
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### Why must a stochastic process be at least second order in terms of differential equations?

A first order differential equation in $q(t)$ has a unique path through each possible value of $q(0)$. This is opposed to a stochastic process (e.g. random walk), where any place might be "hopped ...
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### Establishing recurrence and positive recurrence of Markov processes via “barriers”?

I've been reading the book by Wentzell and Freidlin on dynamical systems with small random perturbations. On page 42 it's stated: It is possible to give stronger conditions for recurrence and ...
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### Diffusion process. Distribution vs transition probability.

I need confirmation on the following problem: Take a SDE of the form: $$dX_t=a(X_t,t)dt+b(X_t,t)dW_t$$ where all the conditions, such that the solution $X_t$ is defined ...
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### Can someone explain to me Feyman Kac and walk through an example?

I kind of understand what needs to be done to convert an SDE to a PDE but I don't understand why we're allowed to do it. What is the generator? ie: given $dS(t) = rS(t)dt + \sigma S(t)dB(t)$ we get ...
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### Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
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### Initial Conditions for Finite Difference of PDE

I am having trouble with figuring out what my initial conditions should be for a simple finite difference algorithm I wrote in Matlab. Specifically, I'm trying to show that the regular 1-Dimensional ...
Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}$$ ...