A Problem About The Stock Price With Jump-diffusions Considering The Exchange Under Partical Information

This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility,We also consider that the investor's portfolio is affected by the exchange rate risk. More specially, we assume that the asset price S_t has jumps at random times 0 < Ï„₁< Ï„₂ < ….The random time moments Ï„_k can be interpreted as " instances at which a large trade occurs or at which a market maker updates his quotes in reaction to new information " .At random times 0 < Ï„₁ < Ï„₂ < … ,The asset price satisfies:During these times at which asset price S_t has jumps,we have(we assume that Ï„₀ = 0):where B = (B_t)_t≥0 is a standard Brownian motion,θ is a positive function,and x = (x-(t))_t≥0 is a cadlag strong Markov process.The random process x(t) is partial observed by process S_t.And the random process x(t) has the form:μ^x=μ^x(ds, dρ) is a Poisson measure on (R₊ × R,B(R₊) (?)B(R)) with the compensator Ï…^x(ds,dρ) = K(dρ)dt,where K(dρ) is a σ-finite non-negative measure on (R, B(R)).We assume that Ex₀² <∞.The exchange rate e_t satisfies : when the above model satisfies some conditions ,we derive the main result of the paper: Theorem 1: For any bounded function f from the domain of the operator (?) we have:Theorem 2: For any bounded function f from the domain of the operator (?),we have:Corollary 1:The filter Π_t(f) satisfies:Corollary 2:...