Subexponential solutions of scalar linear integro-differential equations with delay

This paper considers the asymptotic behaviour of solutions of the scalar linear convolution integro-differential equation with delay x0(t) = − n Xi=1 aix(t − i) + Z t 0 k(t − s)x(s) ds, t > 0, x(t) = (t), − t 0, where = max1in i. In this problem, k is a non-negative function in L1(0,1)\C[0,1), i 0, ai > 0 and is a continuous function on [−, 0]. The kernel k is subexponential in the sense that limt!1 k(t)(t)−1 > 0 where is a positive subexponential function. A consequence of this is that k(t)et ! 1 as t ! 1 for every > 0

Asymptotic parabolicity for strongly damped wave equations

For $S$ a positive selfadjoint operator on a Hilbert space, $$\frac{d^2u}{dt}(t) + 2 F(S)\frac{du}{dt}(t) + S^2u(t)=0$$ describes a class of wave equations with strong friction or damping if $F$ is a positive Borel function. Under suitable hypotheses, it is shown that $$u(t)=v(t)+ w(t)$$ where $v$ satisfies $$2F(S)\frac{dv}{dt}(t)+ S^2v(t)=0$$ and $$\frac{w(t)}{\|v(t)\|} \rightarrow 0, \; \text{as} \; t \rightarrow +\infty.$$ The required initial condition $v(0)$ is given in a canonical way in terms of $u(0)$, $u'(0)$

Conditioning an additive functional of a markov chain to stay nonnegative. II, Hitting a high level

Let (X-t)(t >= 0) be a continuous-time irreducible Markov chain on a finite state space E, let v: E -> R \ {0}, and let (phi(t))(t >= 0) be defined by phi(t) = integral(0)(t) v(X-s) ds. We consider the case in which the process (phi(t))(t >= 0) is oscillating and that in which (phi(t))(t >= 0) has a negative drift. In each of these cases, we condition the process (X-t, phi(t))(t >= 0) on the event that (phi(t))(t >= 0) hits level y before hitting 0 and prove weak convergence of the conditioned process as y -> infinity. In addition, we show the relationship between the conditioning of the process (phi(t))(t >= 0) with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (phi(t))(t >= 0) with a negative drift to drift to infinity and the conditioning of it to hit large levels before hitting 0

Option-pricing in incomplete markets: the hedging portfolio plus a risk premium-based recursive approach

Consider a non-spanned security $C_{T}$ in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price $\hat{C_{0}}$ and then hedged. We consider recursive "one-period optimal" self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let $C_{0}(0)$ be its price. Self-financing implies that the residual risk is equal to the sum of the one-period orthogonal hedging errors, $\sum_{t\leq T} Y_{t}(0) e^{r(T -t)}$. To compensate the residual risk, a risk premium $y_{t}\Delta t$ is associated with every $Y_{t}$. Now let $C_{0}(y)$ be the price of the hedging portfolio, and $\sum_{t\leq T}(Y_{t}(y)+y_{t}\Delta t)e^{r(T-t)}$ is the total residual risk. Although not the same, the one-period hedging errors $Y_{t}(0) and Y_{t}(y)$ are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let $\hat{C_{0}}-C_{0}(y)$. A main result follows. Any arbitrage-free price, $\hat{C_{0}}$, is just the price of a hedging portfolio (such as in a complete market), $C_{0}(0)$, plus a premium, $\hat{C_{0}}-C_{0}(0)$. That is, $C_{0}(0)$ is the price of the option's payoff which can be spanned, and $\hat{C_{0}}-C_{0}(0)$ is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of sum $y_{t}\Delta$t$e^{r(T-t)}$ at maturity). We study other applications of option-pricing theory as well

The existence and singularity structure of low regularity solutions of higher-order degenerate hyperbolic equations

This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the initial data (u,\p_t u)(0,x) belongs to suitable conormal classes. In the present paper, we will study low regularity solutions of higher-order degenerate hyperbolic equations in the category of discontinuous and even unbounded functions. More specifically, we are concerned with the local existence and singularity structure of low regularity solutions of the higher-order degenerate hyperbolic equations \p_t(\p_t^2-t^m\Delta_x)u=f(t,x,u) and (\p_t^2-t^{m_1}\Delta_x)(\p_t^2-t^{m_2}\Delta_x)v=f(t,x,v) in \R_+\times\R^n with discontinuous initial data \p_t^iu(0,x)=\phi_i(x) (0\le i\le 2) and \p_t^jv(0,x)=\psi_j(x) (0\le j\le 3), respectively; here m, m_1, m_2\in\N, m_1\neq m_2, x\in\R^n, n\ge 2, and f is C^\infty smooth in its arguments. When the \phi_i and \psi_j are piecewise smooth with respect to the hyperplane \{x_1=0\} at t=0, we show that local solutions u(t,x), v(t,x)\in L^{\infty}((0,T)\times\R^n) exist which are C^\infty away from \G_0\cup \G_m^\pm and \G_{m_1}^\pm\cup\G_{m_2}^\pm in [0,T]\times\R^n, respectively; here \G_0=\{(t,x): t\ge 0, x_1=0\} and the \Gamma_k^\pm = \{(t,x): t\ge 0, x_1=\pm \f{2t^{(k+2)/2}}{k+2}\} are two characteristic surfaces forming a cusp. When the \phi_i and \psi_j belong to C_0^\infty(\R^n\setminus\{0\}) and are homogeneous of degree zero close to x=0, then there exist local solutions u(t,x), v(t,x)\in L_{loc}^\infty((0,T]\times\R^n) which are C^\infty away from \G_m\cup l_0 and \G_{m_1}\cup\G_{m_2} in [0,T]\times\R^n, respectively; here \Gamma_k=\{(t,x): t\ge 0, |x|^2=\f{4t^{k+2}}{(k+2)^2}\} (k=m, m_1, m_2) is a cuspidal conic surface and l_0=\{(t,x): t\ge 0, |x|=0\} is a ray.Comment: 37 pages, 6 figure