DIFFUSION PROCESSES Definition of a Diffusion Process

• A Markov process consists of three parts: a drift (deterministic), a random process and a jump process. • A diffusion process is a Markov process that has continuous sample paths (trajectories). Thus, it is a Markov process with no jumps. • A diffusion process can be defined by specifying its first two moments: Definition of a Diffusion Process

Definition 1. A Markov process Xt with transition probability P (Γ, t|x, s) is called a diffusion process if the following conditions are satisfied. i) (Continuity). For every x and every ε > 0 Z P (dy, t|x, s) = o(t − s) (1) |x−y|>ε uniformly over s < t. ii) (Definition of drift coefficient). There exists a function a(x, s) such that for every x and every ε > 0 Z (y − x)P (dy, t|x, s) = a(x, s)(s − t) + o(s − t). (2) |y−x|6ε uniformly over s < t. Definition of a Diffusion Process

iii) (Definition of diffusion coefficient). There exists a function b(x, s) such that for every x and every ε > 0 Z (y − x)2P (dy, t|x, s) = b(x, s)(s − t) + o(s − t). (3) |y−x|6ε uniformly over s < t. The Backward Kolmogorov Equation

• Under additional regularity assumptions we can show that a diffusion process is completely determined from its first two moments.

Theorem 1. (Kolmogorov) Let f(x) ∈ Cb(R) and assume that Z 2 u(x, s) := f(y)P (dy, t|x, s) ∈ Cb (R).

Assume furthermore that the functions a(x, s), b(x, s) are continuous in both x and s. Then u(x, s) ∈ C2,1(R × R+) and it solves the final value problem ∂u ∂u 1 ∂2u − = a(x, s) + b(x, s) , lim u(s, x) = f(x). (4) ∂s ∂x 2 ∂x2 s→t The Forward Kolmogorov Equation

• Assume that the transition function has a density: Z P (Γ, t|x, s) = p(s, x, t, y) dy. Γ • Under some regularity assumptions we can derive the Fokker–Planck (forward Kolmogorov) equation. Theorem 2. (Kolmogorov) Assume that conditions (1), (2), (3) are satisfied and that p(·, ·, t, y), a(t, y), b(t, y) ∈ C2,1(R × R+). Then the transition probability density satisfies the equation ∂p ∂ 1 ∂2 = − (a(t, y)p) + (b(t, y)p) , lim p(s, x, t, y) = δ(x − y). ∂t ∂y 2 ∂y2 t→s (5) The Forward Kolmogorov Equation

Remark 1. Assume that initial distribution of Xt is ρ0(x) and set s = 0 (the initial time) in (5). Define Z p(t, y) := p(0, x, t, y) dx.

Integrating the forward Kolmogorov equation (5) with respect to x we obtain the Fokker-Plank equation for p(t, y) ∂p(t, y) ∂ 1 ∂2 = − (a(t, y)p(t, y)) + (b(t, y)p(t, y)) . ∂t ∂y 2 ∂y2 The Fokker-Planck Equation in Arbitrary Dimensions

• the drift and diffusion coefficients of a diffusion process on R2 are defined as: Z 1 lim (y − x)P (s, x, t, dy) = a(s, x) t→s t − s |y−x|<ε • and Z 1 lim (y − x) ⊗ (y − x)P (s, x, t, dy) = b(s, x). t→s t − s |y−x|<ε • The drift a(s, x) is a d-dimensional vector field and the diffusion is a d × d symmetric matrix. • The generator of a d dimensional diffusion process is 1 L = a(s, x) · ∇ + b(s, x): ∇∇ 2 Xd ∂ 1 Xd ∂2 = a + b . j ∂x 2 ij ∂x2 j=1 j i,j=1 j Definition of a Diffusion Process

In Definition 1 we had to truncate the domain of integration since we didn’t know whether the first and second moments exist. If we assume that there exists a δ > 0 such that Z 1 lim |y − x|2+δP (s, x, t, dy) = 0, (6) t→s t − s Rd then we can extend the integration over the whole Rd and use expectations in the definition of the drift and the diffusion coefficient. Indeed, ,let k = 0, 1, 2 and notice that Z Z |y − x|kP (s, x, t, dy) = |y − x|2+δ|y − x|k−(2+δ)P (s, x, t, dy) |y−x|>ε |y−x|>ε Z 1 2+δ 6 2+δ−k |y − x| P (s, x, t, dy) ε |y−x|>ε Z 1 2+δ 6 2+δ−k |y − x| P (s, x, t, dy). ε Rd Definition of a Diffusion Process

Using this estimate together with (6) we conclude that: Z 1 lim |y − x|kP (s, x, t, dy) = 0, k = 0, 1, 2. t→s t − s |y−x|>ε This implies that assumption (6) is sufficient for the sample paths to be continuous (k = 0) and for the replacement of the truncated integrals in (2) and (3) by integrals over Rd (k = 1 and k = 2, respectively). The definitions of the drift and diffusion coefficients become: µ ¯ ¶ Xt − Xs ¯ lim E ¯Xs = x = a(x, s) (7) t→s t − s and µ ¯ ¶ (Xt − Xs) ⊗ (Xt − Xs)¯ lim E ¯Xs = x = b(x, s) (8) t→s t − s Notice also that the continuity condition can be written in the form

P (|Xt − Xs| > ε|Xs = x) = o(t − s). Definition of a Diffusion Process

Now it becomes clear that this condition implies that the probability of large changes in Xt over short time intervals is small. Notice, on the other hand, that the above condition implies that the sample paths of a diffusion process are not differentiable: if they where, then the right hand side of the above equation would have to be 0 when t − s ¿ 1. The sample paths of a diffusion process have the regularity of Brownian paths. A Markovian process cannot be differentiable: we can define the derivative of a sample paths only with processes for which the past and future are not statistically independent when conditioned on the present. Definition of a Diffusion Process

s,x Let us denote the expectation conditioned on Xs = x by E . Notice that the definitions of the drift and diffusion coefficients (7) and (8) can be written in the form

s,x E (Xt − Xs) = a(x, s)(t − s) + o(t − s). and ³ ´ s,x E (Xt − Xs) ⊗ (Xt − Xs) = b(x, s)(t − s) + o(t − s). Consequently, the drift coefficient defines the mean velocity vector for the stochastic process Xt, whereas the diffusion coefficient (tensor) is a measure of the local magnitude of

fluctuations of Xt − Xs about the mean value. hence, we can write locally:

Xt − Xs ≈ a(s, Xs)(t − s) + σ(s, Xs) ξt, T where b = σσ and ξt is a mean zero Gaussian process with

s,x E (ξt ⊗ ξs) = (t − s)I. Definition of a Diffusion Process

Since we have that

Wt − Ws ∼ N (0, (t − s)I), we conclude that we can write locally:

∆Xt ≈ a(s, Xs)∆t + σ(s, Xs)∆Wt.

Or, replacing the differences by differentials:

dXt = a(t, Xt)dt + σ(t, Xt)dWt.

Hence, the sample paths of a diffusion process are governed by a stochastic differential equation (SDE). Proof of Theorems 1 and 2

• the proof of Theorem 1 is based on the Chapman-Kolmogorov equation and the use of a Taylor series expansion for R u(x, s) := f(y)P (dy, t|x, s):

∂u(s, x) 1 ∂2u(s, x) u(z, s)−u(x, s) = (z−x)+ (z−x)2(1+r ), |z−x| 6 ε. ∂x 2 ∂x2 ε • Using this and the CK equation we can obtain a formula for the finite difference u(s , x) − u(s , x) 2 1 . s2 − s1 • We then obtain the backward Kolmogorov equation by passing

to limit s2 → s1 and using the definition of the drift and diffusion coefficients.