0
votes
1answer
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 ...
1
vote
0answers
14 views

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 ...
1
vote
0answers
33 views

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 ...
0
votes
0answers
24 views

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 ...
0
votes
0answers
20 views

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 ...
1
vote
2answers
181 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 ...
1
vote
1answer
69 views

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 ...
11
votes
1answer
372 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 ...
2
votes
0answers
62 views

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 ...
2
votes
1answer
790 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
1answer
299 views

Help understanding the Feynman-Kac formula

From wikipedia: Suppose we wish to find the expected value of the function $e^{-\int_0^t V(x(\tau)) d\tau}$ in the case where $x(\tau)$ is some realization of a diffusion process starting at $x(0) = ...
4
votes
0answers
219 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 $\textit{infinitesimal generator}$ ...
3
votes
1answer
220 views

getting the fundamental solution of Laplace's equation from the heat kernel

Since Laplace's equation is related to the steady state of heat flow problems, I'm guessing that there is a way to get from the heat kernel to the fundamental solution of Laplace's equation by letting ...
1
vote
1answer
219 views

non divergence form vs divergence form operator

Can the non divergence form operator $\mathcal{L}u= u_{xx}+u_{yy} + u_x=\Delta u + u_x$ be put in divergence form? In general, can any constant coefficient non divergence form operator be put into ...
1
vote
1answer
77 views

Martingale Problem and PDE's

Let $X$ be a RCLL Markov Process with generator $A$. Then I know that $$ M^f = f(X)-f(X_0)-\int Af(X_s)ds $$ is a martingal for every $f\in \mathcal{D}_A$. If we suppose that $Af=0$, we see that ...
2
votes
0answers
262 views

Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?

It seems not to be a easy problem. I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
1
vote
0answers
193 views

independent vs. uncorrelated

If I have two stochastic processes $X_t$ and $Y_t$ that are dependent however $dX_tdY_t=0$ where $dX_t=a(z,x,t)dt$ and $dY_t=b(z,x,t)dt$ and $dZ_t=dW_t$(Brownian motion) then from Ito formula ...
2
votes
2answers
133 views

How to solve a PDE with quasi-periodic Poisson process?

For classic GBM stock price model, $$\frac{dS}{S} = \mu \cdot dt + \sigma \cdot dW$$ we have the solution: $$S(t)=S(0)\, \exp\left(\frac{\mu-\sigma^2}{2} t+\sigma W(t)\right).$$ During the ...
2
votes
1answer
206 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
1
vote
2answers
284 views

Expectation of Stochastic Process Given First Hitting Time Information

Let $V_t$ satisfy the SDE $dV_t = -\gamma V_t dt + \alpha dW_t$. Let $\tau$ be the first hitting time for 0, i.e., $\tau $ = min$(t | V_t = 0)$. Let $s =$ min$(\tau, 5)$. Let $\mathcal{F}_s$ be the ...
1
vote
1answer
206 views

Kolmogorov Backward Equation Boundary Value Problem

I need to solve the backward equation $u_t - \gamma x u_x + \frac{1}{2} b^2 u_{xx} $ subject to the final condition $ u(x,T) = (x-a)^2 $. Here a and b and $\gamma$ are constants. I am given a strong ...
1
vote
2answers
142 views

How to model multi-step cell differentiation

Can I better explain cell lineages using PDEs or stochastic?
3
votes
2answers
127 views

Reaction-diffusion equations and stochastic processes

The solution to the Fokker-Planck equation can be thought of as a macroscopic description of the dynamics of a diffusion process. Various results make this heuristic more precise - Ito integration, ...
7
votes
1answer
269 views

PDE - Feynman-Kac vs. finite difference methods

I've heard that in greater than three dimensions, it's more efficient to solve a second-order parabolic PDE using a Monte-Carlo method based on the Feynman-Kac formula that it is to use finite ...
10
votes
4answers
332 views

Roadmap to SPDEs

I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory. I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and ...
8
votes
2answers
547 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...