The two-dimensional Itô formula, using

d [W^{X}, W^{Z}]_{t} = ρ d t

, gives the drift of

U (t, Z_{t}, X_{t})

U_{t} + B U_{z} + \frac{1}{2} C^{2} U_{zz} + z Cρ U_{z x} + \frac{1}{2} z^{2} (U_{xx} - U_{x}) .

The PDE makes this zero, leaving only stochastic-integral terms. Thus

M_{t} = U (t, Z_{t}, X_{t})

b

Solution

Substitute

U = e^{θ x} V

. After dividing by

e^{θ x}

, the PDE becomes

V_{t} + (B + θρ z C) V_{z} + \frac{1}{2} C^{2} V_{zz} + \frac{1}{2} θ (θ - 1) z^{2} V = 0,

with terminal condition

V (T, z) = 1

Let

τ = T - t

and set

V (t, z) = exp {P (τ) + Q (τ) z + R (τ) z^{2}} .

Then

\frac{V _{z}}{V} = Q + 2 R z, \frac{V _{zz}}{V} = 2 R + (Q + 2 R z)^{2}, \frac{V _{t}}{V} = - (\dot{P} + \dot{Q} z + \dot{R} z^{2}) .

For

B (z) = a - b z

and

C (z) = c

, matching constant, linear, and quadratic coefficients gives

\dot{R} = 2 c^{2} R^{2} + 2 (θρ c - b) R + \frac{1}{2} θ (θ - 1),

\dot{Q} = 2 a R + (θρ c - b) Q + 2 c^{2} QR,

and

\dot{P} = a Q + c^{2} R + \frac{1}{2} c^{2} Q^{2} .

The terminal condition becomes

P (0) = Q (0) = R (0) = 0

. The first equation is a Riccati equation; once it is solved, the second is linear in

Q

, followed by direct integration for

P

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