Stochastic Differential Equations - Adam Oberman Math

Stochastic Differential Equations

Consider a diffusion process

$dx(t) = \mu(x(t),t)\,dt + \sigma(x(t),t)\,dW(t)$

We are interested in two questions about this process:

Given a target G at the final time T, what is the expected propability of reaching the target starting from the point x, under the evolution above.
Given an initial density p, what is the density at the final time T, under the evolution above.

The first question can be generalized to the expected value of a target function $u T (x)$

Contents

1 Kolmogorov backward equation
2 Kolmogorov forward equation
3 Adjoint Equations
4 Notes from Wikipedia

Kolmogorov backward equation

The answer to the first question is provided by the Kolmogorov backward equation.

$-\frac{\partial}{\partial t}u(x,t)=L u \equiv \mu(x,t)\frac{\partial}{\partial x}u(x,t) + \frac{1}{2}\sigma^2(x,t)\frac{\partial^2}{\partial x^2}u(x,t) \qquad \text{ for } \qquad t\le T$ ,

subject to the final condition

u (x, T) = u T (x)

.

This is derived using Ito's lemma.

This equation can be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of $u s (x)$ over all paths that originate from state x at time t:

$P(X_s \in B | X_t = x) = E[u_s(x) | X_t = x]$

Kolmogorov forward equation

The second question is answered by the Kolmogorov forward equation, or Fokker–Planck equation. It describes the time evolution of the probability density function of the position of a particle,

$\frac{\partial}{\partial t}p(x,t)= L^* u \equiv -\frac{\partial}{\partial x}[\mu(x,t)p(x,t)] + \frac{1}{2}\frac{\partial^2}{\partial x^2} \sigma^2(x,t)p(x,t) \qquad\text{ for } t \ge 0$

with initial condition

p (x,0) = p 0 (x)

.

Adjoint Equations

These equations are Adjoint_Processes, in the sense that if we define

$\langle u, p \rangle = \int u dp$

we have

$\langle L u, p \rangle = \langle u,L^* p \rangle$

which follows from integration by parts.

We are ignoring boundary terms, by assuming, for example, that we are in the periodic setting.

In particular, we have for any $u, p$ satisfying the forward and backwards equations

$\frac{d}{dt} \langle u, p \rangle = \langle u_t, p \rangle + \langle u, p_t \rangle = \langle -Lu, p \rangle + \langle u, L^* p \rangle = 0$

since the operators are adjoints.

Notes from Wikipedia

From [1] or [2]

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