Tagged Questions

29 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
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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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243 views

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 ...
399 views

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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How to solve this differential equation? (Steady State Solution of Forward Kolmogorov Equation)

Here's the full question and my attempt at answering it by solving the differential equation. Consider the following SDE $$d\sigma = a(\sigma,t)dt + b(\sigma,t)dW$$ The Forward Equation (FKE) is ...
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Transition density and distribution: (Ornsteinâ€“Uhlenbeck process)

Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}$$ ...
99 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$\partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0?$$ This appeared as a condition for $$X_t=u(t,B_t)e^{\int_0^tv(B_s)ds}$$ to be a martingale. With $B$ a ...
The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
Given the heat equation: $$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$ with the boundary conditions: $$\Phi(x,0)=\Phi_0$$ and a Neumann boundary condition of the kind: ...