0
votes
0answers
31 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 ...
0
votes
1answer
53 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 ...
0
votes
0answers
26 views

Maximal principle for elliptic or linear integro-differential operator

Consider $L$ the operator forming as $$ Lg= -g^{'}(x)+(g(x+1)-g(x)) $$. $h$ on $[0,\infty)$ satisfies the following integro-differential equation $$ Lh \geq 0 $$ with boundary condition: $$ ...
1
vote
0answers
117 views

Feynman-Kac uniqueness

The Feynman-Kac formula relates parabolic differential equations to stochastic processes. For example, if \begin{equation} ...
3
votes
1answer
159 views

Advices on learning SDE/PDE for junior undergrad

Everyone. I am about to take an ODE Course in the summer. I wonder if it will help my understanding in stochastic differential equation and partial differential differential equation. My future plan ...
4
votes
1answer
185 views

Black-76 pde hedging argument wrong

I want to obtain the PDE for the Black-76 model. I believe it has to be the following PDE: $$\left(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}F^{2}\frac{\partial^{2} V}{\partial ...
1
vote
2answers
283 views

Diffusion process. Distribution vs transition probability.

I need confirmation on the following problem: Take a SDE of the form: \begin{equation} dX_t=a(X_t,t)dt+b(X_t,t)dW_t \end{equation} where all the conditions, such that the solution $X_t$ is defined ...
11
votes
1answer
406 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 ...
4
votes
2answers
2k views

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 ...
2
votes
1answer
1k views

Transition density and distribution: (Ornstein–Uhlenbeck process)

Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$ dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0} $$ ...
2
votes
2answers
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 ...
4
votes
0answers
254 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

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 $\textit{infinitesimal generator}$ ...
2
votes
1answer
268 views

Stochastic Heat Equation

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: ...