Module merton_jump_lognormal 

Merton’s log-utility portfolio with log-normal jumps in the risky asset.

The jump-size multiplier Y is log-normal, ln Y ~ N(m, delta^2), so the jump distribution is a continuum rather than a single deterministic size. As in MertonJump the jump genuinely changes the optimal control, but here the policy has no closed form: it is the root of a transcendental first-order condition.

Dynamics

dS/S  = mu dt + sigma dW + (Y - 1) dN,   ln Y ~ N(m, delta^2)
dx    = x [ r + u (mu - r) ] dt + x u sigma dW + x u (Y - 1) dN

where N is a Poisson process with intensity lambda. Log utility requires wealth to stay positive, so the admissible control is 0 <= u <= 1.

Exact solution

The value function is V(t, x) = ln x + B (T - t) with

B = r + u* (mu - r) - 0.5 (u* sigma)^2 + lambda E[ ln(1 + u* (Y - 1)) ],

and u* solves the transcendental first-order condition

0 = (mu - r) - sigma^2 u* + lambda E[ (Y - 1) / (1 + u* (Y - 1)) ].

The expectations are computed with 32-point Gauss-Hermite quadrature on the standard normal z = (ln Y - m) / delta, and u* is bracketed on [0, 1] and refined by bisection. With delta = 0 the jump degenerates to the deterministic multiplier y = e^m, reducing this model to MertonJump.

Structs

MertonJumpLogNormal

Merton log-utility portfolio with log-normal jumps.