Geometric Brownian motion with delay: mean square characterisation

Geometric Brownian Motion with Delay: Mean Square Characterisation

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condit...

Exact solutions for Fokker-Plank equation of geometric Brownian motion with Lie point symmetries

‎In this paper Lie symmetry analysis is applied to find new‎ solution for Fokker Plank equation of geometric Brownian motion‎. This analysis classifies the solution format of the Fokker Plank‎ ‎equation‎.

1 Geometric Brownian motion

where X(t) = σB(t) + μt is BM with drift and S(0) = S0 > 0 is the intial value. We view S(t) as the price per share at time t of a risky asset such as stock. Taking logarithms yields back the BM; X(t) = ln(S(t)/S0) = ln(S(t))− ln(S0). ln(S(t)) = ln(S0) +X(t) is normal with mean μt + ln(S0), and variance σ2t; thus, for each t, S(t) has a lognormal distribution. As we will see in Section 1.4: let...

Simulating Brownian motion ( BM ) and geometric Brownian

2) and 3) together can be summarized by: If t0 = 0 < t1 < t2 < · · · < tk, then the increment rvs B(ti) − B(ti−1), i ∈ {1, . . . k}, are independent with B(ti) − B(ti−1) ∼ N(0, ti − ti−1) (normal with mean 0 and variance ti − ti−1). In particular, B(ti) − B(ti−1) is independent of B(ti−1) = B(ti−1)−B(0). If we only wish to simulate B(t) at one fixed value t, then we need only generate a unit no...

Sigman 1 Geometric Brownian motion