Abstracts

In Janssen's ALM risk model, the asset $A$ and the liability $B$ of a company are modeled by the stochastic differential equations $dA(t)= A(t)\mu_A dt + A(t)\sigma_A dZ_A(t)$ and $dB(t) = B(t)\mu_B dt + B(t) \sigma_B dZ_B(t)$, where $Z_A(t)$ and $Z_B(t)$ are standard Brownian motions. Recently, it has been shown by Latouche and Nguyen that Brownian motion can be approximated by a Markovian random walk $\{(X(t),\varphi(t)):t\in\mathbb{R}^{+}\}$, i.e. a two dimensional stochastic process where the phase $\varphi(t)$ is the state at time $t$ of a Markov process which controls the level $X(t)$. The level takes continuous values and varies linearly with a rate depending on the phase. Markovian random walks are widely used in applied probability (telecommunication modeling, risk process,...) because of their tractability and computational efficiency.
In the literature, time-dependent distributions of the level have been studied using Laplace transforms: Ahn and Ramaswami derive time-dependent distributions of a Markovian fluid queue (i.e. random walks with a reflecting boundary at $0$) for the levels $x>0$ in terms of the transform matrix of the busy period duration, i.e. the matrix for which the $(i,j)$-th component is $\mathbb{E}[\exp(-s\tau)1_{\{\varphi(t)=j\}}|X(0)=0,\varphi(0)=i]$, with $\tau=\inf\{t>0:X(t)=0\}$, for $i\in \mathcal{S}_{+}$ and $j\in \mathcal{S}_{-}$.
Here, we use arguments based on the Erlangization method, suggested by Asmussen et al. in a fluid queue context related to ruin theory problems, and so avoid Laplace transform calculations. Furthermore, this method is practical for the computation of joint probabilities thanks to the memoryless property of the exponential distribution.
In this talk, we present the Markovian random walk approximation for the process $\alpha(t)=\ln(A(t)/B(t))$, we perform numerical approximations for $R(u,\theta)=\mathbb{P}[\tau_0>\theta|\alpha(0)=u]$, the non-ruin probability before a given time $\theta$, with $\tau_0=\inf\{t>0:\alpha(t)<0\}$; we provide numerical illustrations for ruin indicators such as $I(u,x,[0,\theta])=\mathbb{P}[\max_{0<t<\theta} \alpha(t)>x|\alpha(0)=u]$; and we show how phase-type models can lead to more elaborate analysis of the process $\alpha(t)$.